❋❮❙■❈❆ ❊❳P❊❘■▼❊◆❚❆▲

❊❞✐♦ ❈✉♥❤❛ ❞❛ ❈♦st❛

✷

❙✉♠ár✐♦

✶ ■◆❚❘❖❉❯➬➹❖ ➚ ❋❮❙■❈❆ ❊❳P❊❘■▼❊◆❚❆▲ ✺✶✳✶ ●r❛♥❞❡③❛s ❢ís✐❝❛s ❡ ♦❜s❡r✈á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶✳✷ ▼❡t♦❞♦❧♦❣✐❛ ❝✐❡♥tí✜❝❛✱ ✉♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

✶✳✷✳✶ ❆ ♥❛t✉r❡③❛ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❝✐❡♥tí✜❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶✳✷✳✷ ❚r✐♥ô♠✐♦✿ ✈❡r❞❛❞❡✲❡✈✐❞ê♥❝✐❛✲❝❡rt❡③❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶✳✷✳✸ ❋♦r♠❛çã♦ ❞♦ ❡s♣ír✐t♦ ❝✐❡♥tí✜❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

✶✳✸ ▼ét♦❞♦ ❝✐❡♥tí✜❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶✳✹ ❈♦♥❝❡✐t♦ ❞❡ ♣❡sq✉✐s❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✷ ▼❊❉■❉❆❙ ❊ ❆▲●❆❘■❙▼❖❙ ❙■●◆■❋■❈❆❚■❱❖❙ ✾✷✳✶ ●r❛♥❞❡③❛s ❋ís✐❝❛s ❡ ❯♥✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷✳✷ ❙✐st❡♠❛ ■♥t❡r♥❛❝✐♦♥❛❧ ❞❡ ❯♥✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷✳✸ ▼❡❞✐❞❛s ❉✐r❡t❛s ❡ ■♥❞✐r❡t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✷✳✹ ❆❧❣❛r✐s♠♦s ❙✐❣♥✐✜❝❛t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✷✳✹✳✶ ▼✉❞❛♥ç❛ ❞❡ ✉♥✐❞❛❞❡ ❡ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✷✳✹✳✷ ❈r✐tér✐♦s ❞❡ ❆♣r♦①✐♠❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✸ ❊❘❘❖❙ ❊ ■◆❈❊❘❚❊❩❆❙ ✶✾✸✳✶ ❱❛❧♦r ♠é❞✐♦ ❡ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✸✳✷ ❉❡s✈✐♦ ❡ ❞❡s✈✐♦ ♠é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸✸✳✸ ❉❡s✈✐♦ ♣❛❞rã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺✸✳✹ ❊rr♦ s✐st❡♠át✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻✸✳✺ Pr♦♣❛❣❛çã♦ ❞❡ ❡rr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻✸✳✻ ❈♦♥✜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ♠❡❞✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✹ ●❘➪❋■❈❖❙ ✷✾✹✳✶ ❱❛r✐á✈❡✐s ❞❡♣❡♥❞❡♥t❡s ❡ ✐♥❞❡♣❡♥❞❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾✹✳✷ ❈♦♥st❛♥t❡s ❡ P❛râ♠❡tr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶✹✳✸ ❈♦♥str✉çã♦ ❞❡ ●rá✜❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✹✳✸✳✶ ❊✐①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷✹✳✸✳✷ ❊s❝❛❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷✹✳✸✳✸ ▼❛r❝❛çã♦ ❞♦s ♣♦♥t♦s ❡ tr❛ç❛❞♦ ❞❛ ❝✉r✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸✹✳✸✳✹ ▲✐♥❡❛r✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺✹✳✸✳✺ ❉❡t❡r♠✐♥❛çã♦ ❞❡ ❝♦♥st❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼✹✳✸✳✻ P❛♣é✐s ❡s♣❡❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✺ ■◆❙❚❘❯▼❊◆❚❖❙ ❉❊ ▼❊❉■❉❆❙ ✹✸✺✳✶ ■♥str✉♠❡♥t♦s ❆♥❛❧ó❣✐❝♦s ❡ ❉✐❣✐t❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸✺✳✷ ■♥str✉♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✸

✹ ❙❯▼➪❘■❖

✻ ❊❳❊❘❈❮❈■❖❙ P❘❖P❖❙❚❖❙ ✹✾

❈❛♣ít✉❧♦ ✶

■◆❚❘❖❉❯➬➹❖ ➚ ❋❮❙■❈❆❊❳P❊❘■▼❊◆❚❆▲

❋❡r♥❛♥❞♦ ❋r❛♥ç❛ ❡ ❊❞✐♦ ❈✉♥❤❛ ❞❛ ❈♦st❛

✧■♥ ♣❤②s✐❝❛❧ s❝✐❡♥❝❡ t❤❡ ✜rst ❡ss❡♥t✐❛❧ st❡♣ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ❧❡❛r♥✐♥❣ ❛♥② s✉❜❥❡❝t ✐s t♦ ✜♥❞♣r✐♥❝✐♣❧❡s ♦❢ ♥✉♠❡r✐❝❛❧ r❡❝❦♦♥✐♥❣ ❛♥❞ ♣r❛❝t✐❝❛❜❧❡ ♠❡t❤♦❞s ❢♦r ♠❡❛s✉r✐♥❣ s♦♠❡ q✉❛❧✐t② ❝♦♥♥❡❝t❡❞✇✐t❤ ✐t✳ ■ ♦❢t❡♥ s❛② t❤❛t ✇❤❡♥ ②♦✉ ❝❛♥ ♠❡❛s✉r❡ ✇❤❛t ②♦✉ ❛r❡ s♣❡❛❦✐♥❣ ❛❜♦✉t✱ ❛♥❞ ❡①♣r❡ss ✐t ✐♥♥✉♠❜❡rs✱ ②♦✉ ❦♥♦✇ s♦♠❡t❤✐♥❣ ❛❜♦✉t ✐t❀ ❜✉t ✇❤❡♥ ②♦✉ ❝❛♥♥♦t ♠❡❛s✉r❡ ✐t✱ ✇❤❡♥ ②♦✉ ❝❛♥♥♦t❡①♣r❡ss ✐t ✐♥ ♥✉♠❜❡rs✱ ②♦✉r ❦♥♦✇❧❡❞❣❡ ✐s ♦❢ ❛ ♠❡❛❣❡r ❛♥❞ ✉♥s❛t✐s❢❛❝t♦r② ❦✐♥❞❀ ✐t ♠❛② ❜❡ t❤❡❜❡❣✐♥♥✐♥❣ ♦❢ ❦♥♦✇❧❡❞❣❡✱ ❜✉t ②♦✉ ❤❛✈❡ s❝❛r❝❡❧② ✐♥ ②♦✉r t❤♦✉❣❤ts ❛❞✈❛♥❝❡❞ t♦ t❤❡ st❛t❡ ♦❢ ❙❝✐❡♥❝❡✱✇❤❛t❡✈❡r t❤❡ ♠❛tt❡r ♠❛② ❜❡✳✧ ✭P♦♣✉❧❛r ▲❡❝t✉r❡s ❛♥❞ ❆❞❞r❡ss❡s✱ ✈♦❧✳ ✶✱ ❙✐r ❲✐❧❧✐❛♠ ❚❤♦♠s♦♥✕ ▲♦r❞ ❑❡❧✈✐♥ ✕ ✶✽✽✸✮

✶✳✶ ●r❛♥❞❡③❛s ❢ís✐❝❛s ❡ ♦❜s❡r✈á✈❡✐s

➱ ✉♠ tr✉ís♠♦ ❛✜r♠❛r q✉❡ ❛ ❋ís✐❝❛ é ✉♠❛ ❝✐ê♥❝✐❛✱ ♣❡❧❛ q✉❛❧ ♦ ❤♦♠❡♠ ❜✉s❝❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡t♦❞♦ ♦ ❯♥✐✈❡rs♦✱ ♣♦✐s ✐♥ú♠❡r♦s sã♦ ♦s r❛♠♦s ❞❛ ❋ís✐❝❛✳ ❊st✉❞❛✲s❡ ❞❡s❞❡ ♦ ♠✐❝r♦ ❛♦ ♠❛❝r♦✱ s✐s✲t❡♠❛s ✐♥❛♥✐♠❛❞♦s ❡✱ ❛té ♠❡s♠♦✱ s❡r❡s ✈✐✈♦s✱ ❝♦♠ ✈❛r✐á✈❡✐s ❡ ❝♦♠♣❧❡①✐❞❛❞❡s ❞✐❢❡r❡♥t❡s✱ ♠✉✐t❛s✈❡③❡s ♥❛❞❛ ♣❡r❝❡♣tí✈❡✐s ❛♦s s❡♥t✐❞♦s ❤✉♠❛♥♦s✳ ❆s t❡♦r✐❛s ❞❛ ❋ís✐❝❛✱ ♥ã♦ r❛r♦✱ r❡q✉❡r❡♠ ❣r❛♥❞❡s❛❜str❛çõ❡s q✉❡✱ ❝♦♥t✉❞♦✱ s❡r✐❛♠ ♠❡r♦s s♦♥❤♦s s❡ ♥ã♦ ❡st✐✈❡ss❡♠ ❛♠♣❛r❛❞♦s ❡♠ s✐st❡♠❛s ❝♦♥✲❝r❡t♦s q✉❡ ❝♦♥st✐t✉❡♠ ♦s ❢❛t♦s ❢ís✐❝♦s ♦❜s❡r✈á✈❡✐s✳ ❉❛í q✉❡ ❛ ❝♦♥st❛t❛çã♦ ❞❡ ✉♠ ❢❛t♦ r❡q✉❡rs❡♠♣r❡ ♠❡❞✐❞❛s ❞❡ ❣r❛♥❞❡③❛s ❢ís✐❝❛s ❛ss♦❝✐❛❞❛s ❛ t❛✐s ❢❛t♦s✳

■♥ú♠❡r❛s ♣♦❞❡♠ s❡r ❛s ❣r❛♥❞❡③❛s ❢ís✐❝❛s ❞❡✜♥✐❞❛s ♣❛r❛ ❛s ♠❡❞✐❞❛s✱ ♣♦✐s ❣r❛♥❞❡ é ♥♦ss❛❝❛♣❛❝✐❞❛❞❡ ❞❡ ❝♦♠♣❛r❛çã♦ ❝♦♠ ❡s❝❛❧❛s ❞❡ r❡❢❡rê♥❝✐❛s q✉❡✱ ♣♦r s✉❛ ✈❡③✱ ♣♦❞❡♠ s❡r ❞❡ ✉s♦s❤❛❜✐t✉❛✐s ♣❡❧❛ s✉❛ s✐♠♣❧✐❝✐❞❛❞❡ ❡ ♣r❛t✐❝✐❞❛❞❡✳ ❆ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ❝♦♠♣r✐♠❡♥t♦✱ ♣♦r ❡①❡♠♣❧♦✱♣♦❞❡ ❝❤❛♠❛r✲s❡ ❥❛r❞❛s✱ ♣és✱ ❛♥♦s✲❧✉③✱ ❝❡♥tí♠❡tr♦✱ ♣♦❧❡❣❛❞❛✱ ♠✐❧❤❛s✱ ❡t❝✳✱ ❝❛❞❛ q✉❛❧ ❝♦♠ ✉♠ ✈❛❧♦r❞❡ ❡s❝❛❧❛ ❝❛r❛❝t❡ríst✐❝♦✳ ◆❛ ♣rát✐❝❛✱ ❛s ❣r❛♥❞❡③❛s ❢ís✐❝❛s sã♦ ♦r❞❡♥❛❞❛s ❡♠ s✐st❡♠❛s ❞❡ ✉♥✐❞❛❞❡s✱t❛✐s ❝♦♠♦ ❙✐st❡♠❛ ●❛✉ss✐❛♥♦ ❞❡ ❯♥✐❞❛❞❡s✱ ❙✐st❡♠❛ ▼étr✐❝♦ ❉❡❝✐♠❛❧ ❡ ❙✐st❡♠❛ ■♥t❡r♥❛❝✐♦♥❛❧❞❡ ❯♥✐❞❛❞❡s ✭❙■✮✱ ❡♥tr❡ ♦✉tr♦s✳

✺

✻ ❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖ ➚ ❋❮❙■❈❆ ❊❳P❊❘■▼❊◆❚❆▲

✶✳✷ ▼❡t♦❞♦❧♦❣✐❛ ❝✐❡♥tí✜❝❛✱ ✉♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦

✶✳✷✳✶ ❆ ♥❛t✉r❡③❛ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❝✐❡♥tí✜❝♦

❆ ❜✉s❝❛ ❞❡ ♥♦ss♦s ❡st✉❞♦s é ♦ s❛❜❡r✱ ♦ q✉❡ ❞❡s♣♦♥t❛ ♣❡❧♦ ♠♦t❡ ❞❛ ❝✉r✐♦s✐❞❛❞❡✳ ❈♦♥t✉❞♦✱♣❛r❛ ❜✉s❝❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♣r❡❝✐s❛♠♦s ❡s❝❧❛r❡❝❡r✿ ♦ q✉❡ é ♦ ❝♦♥❤❡❝✐♠❡♥t♦❄ ❊♠ ♣♦✉❝❛s ❧✐♥❤❛s♣♦❞❡♠♦s ❞✐③❡r q✉❡ ❝♦♥❤❡❝✐♠❡♥t♦ é ❛ r❡❧❛çã♦ q✉❡ s❡ ❡st❛❜❡❧❡❝❡ ❡♥tr❡ ♦ s✉❥❡✐t♦ q✉❡ ❝♦♥❤❡❝❡ ❡ ♦♦❜❥❡t♦ ❝♦♥❤❡❝✐❞♦✳ ❖ ❝♦♥❤❡❝✐♠❡♥t♦✱ s❡❣✉♥❞♦ ❈❡r✈♦ ❡ ❇❡r✈✐❛♥❬✶❪✱ ♣♦❞❡ s❡ ❛♣r❡s❡♥t❛r ❡♠ ♥í✈❡✐s❞❡ ♣r♦❢✉♥❞✐❞❛❞❡ q✉❡✱ ❞❡ ♠♦❞♦ ❣r♦ss❡✐r♦✱ ♣♦❞❡♠♦s ❝❧❛ss✐✜❝❛r ❞❡s❞❡ ❛ s✉♣❡r✜❝✐❛❧✐❞❛❞❡✱ ❝♦♠♦❡♠♣ír✐❝♦✱ ❝✐❡♥tí✜❝♦✱ ✜❧♦só✜❝♦✱ ❛té ❛ ♣r♦❢✉♥❞✐❞❛❞❡ ❞❛ ❝r❡♥ç❛✱ ♦ t❡♦❧ó❣✐❝♦✳

❈♦♥❤❡❝✐♠❡♥t♦ ❡♠♣ír✐❝♦✿ t❛♠❜é♠ ❝❤❛♠❛❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ✈✉❧❣❛r✱ é ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦♣♦✈♦✱ ♦❜t✐❞♦ ❛♦ ❛❝❛s♦✱ ❛♣ós ✐♥ú♠❡r❛s t❡♥t❛t✐✈❛s✳ ❈❛r❛❝t❡r✐③❛✲s❡ ♣♦r s❡r ❛♠❡tó❞✐❝♦ ❡ ❛ss✐s✲t❡♠át✐❝♦✳

❈♦♥❤❡❝✐♠❡♥t♦ ❝✐❡♥tí✜❝♦✿ é ♦ ❝♦♥❤❡❝✐♠❡♥t♦ q✉❡ ✈❛✐ ❛❧é♠ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❡♠♣ír✐❝♦✱♣r♦❝✉r❛♥❞♦ ❝♦♥❤❡❝❡r✱ ❞❡s❝r❡✈❡♥❞♦ ❝♦♥❝❡✐t♦s ❞❡ ❢♦r♠❛ r❛❝✐♦♥❛❧✱ ❛❧é♠ ❞♦ ❢❛t♦ ❢❡♥♦♠❡♥♦❧ó❣✐❝♦✱❜✉s❝❛♥❞♦ ❛s ❝❛✉s❛s ❡ ❞❡s❝r❡✈❡♥❞♦ ❧❡✐s✱ ♣❡❧♦ ❡♠♣r❡❣♦ ❞♦ ♠ét♦❞♦ ❝✐❡♥tí✜❝♦✱ q✉❡ ♣❡r♠✐t❡✲♥♦st❡r ❡st✐♠❛t✐✈❛s ❡ r❡♣r♦❞✉t✐❜✐❧✐❞❛❞❡ ❞❡ r❡s✉❧t❛❞♦s✱ ♣r♦♠♦✈❡♥❞♦ ❛ ✐♥t❡r❛çã♦ ❝♦♥s❝✐❡♥t❡ ❝♦♠ ♦❯♥✐✈❡rs♦✳ ❖ ❝♦♥❤❡❝✐♠❡♥t♦ ❝✐❡♥tí✜❝♦ é ❞❡ ❝r✉❝✐❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣♦✐s ♣❡r♠✐t❡ ❛ ❡✜❝✐ê♥❝✐❛ ❞❛❡♠♣r❡❣❛❜✐❧✐❞❛❞❡ ❞♦s r❡❝✉rs♦s ❛♦ ❛❧❝❛♥❝❡ ❞♦ ❤♦♠❡♠✱ ♦✉ s❡❥❛✱ ❛ ❡✜❝á❝✐❛ ❞❛s ❛çõ❡s ❞❡ ♥♦ss❛s✈✐❞❛s✳

❈♦♥❤❡❝✐♠❡♥t♦ ✜❧♦só✜❝♦✿ ❞✐st✐♥❣✉❡✲s❡ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❝✐❡♥tí✜❝♦ ♣❡❧♦ ♦❜❥❡t♦ ❞❡ ✐♥✈❡s✲t✐❣❛çã♦ ❡ ♣❡❧♦ ♠ét♦❞♦ ❛♣❧✐❝❛❞♦✳ ❖ ♦❜❥❡t♦ ❞❛ ✜❧♦s♦✜❛ é ❝♦♥st✐t✉í❞♦ ❞❡ r❡❛❧✐❞❛❞❡s ♠❡❞✐❛t❛s✱✐♠♣❡r❝❡♣tí✈❡✐s ❛♦s s❡♥t✐❞♦s✱ ❡ q✉❡ ♣♦r s❡r❡♠ ❞❡ ♦r❞❡♠ s✉♣r❛ s❡♥sí✈❡✐s✱ ✉❧tr❛♣❛ss❛♠ ❛ ❡①♣❡r✐ê♥✲❝✐❛ t❛♥❣í✈❡❧ ✭❛♣❧✐❝❛ ♦ ♠ét♦❞♦ r❛❝✐♦♥❛❧✮✳

❈♦♥❤❡❝✐♠❡♥t♦ t❡♦❧ó❣✐❝♦✿ ♣♦r ✈❡③✱ é ♦ ❝♦♥❤❡❝✐♠❡♥t♦ r❡✈❡❧❛❞♦ ✲ r❡❧❛t✐✈♦ ❛♦ ❞✐✈✐♥♦ ✲ ❛❝❡✐t♦♣❡❧❛ ❢é t❡♦❧ó❣✐❝❛✳ ➱ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❡r❞❛❞❡s ❛ q✉❡ ♦s ❤♦♠❡♥s ❝❤❡❣❛r❛♠✱ ♥ã♦ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❛✐♥t❡❧✐❣ê♥❝✐❛✱ ♠❛s ♠❡❞✐❛♥t❡ ❛ ❛❝❡✐t❛çã♦ ❞♦s ❞❛❞♦s ❞❡ ✉♠❛ r❡✈❡❧❛çã♦ ❞✐✈✐♥❛ ✭❡s❝r✐t♦s s❛❣r❛❞♦s✱♣♦r ❡①❡♠♣❧♦✮✳ ❚❛✐s ✈❡r❞❛❞❡s ♣❛ss❛♠ ❛ s❡r ❝♦♥s✐❞❡r❛❞❛s ❝♦♠♦ ✜❞❡❞✐❣♥❛s ❡✱ ♣♦r ✐ss♦✱ sã♦ ❛❝❡✐t❛s♣❡❧♦s q✉❡ ♣❛❝t✉❛♠ ❛ ♠❡s♠❛ ❢é✳

✶✳✷✳✷ ❚r✐♥ô♠✐♦✿ ✈❡r❞❛❞❡✲❡✈✐❞ê♥❝✐❛✲❝❡rt❡③❛

❱❡r❞❛❞❡✿ ♦ q✉❡ é ✈❡r❞❛❞❡❄ ❙❡❣✉♥❞♦ ❇❡r✈✐❛♥✱ ✈❡r❞❛❞❡ é ♦ ❡♥❝♦♥tr♦ ❞♦ ❤♦♠❡♠ ❝♦♠ ♦ ❞❡s✈❡❧❛✲♠❡♥t♦✱ ❝♦♠ ♦ ❞❡s♦❝✉❧t❛♠❡♥t♦ ❡ ❝♦♠ ❛ ♠❛♥✐❢❡st❛çã♦ ❞♦ s❡r✳ ❖ s❡r ❞❛s ❝♦✐s❛s q✉❡ s❡ ♠❛♥✐❢❡st❛✱t♦r♥❛✲s❡ tr❛♥s❧ú❝✐❞♦✱ ✈✐sí✈❡❧ ❛♦ ♦❧❤❛r✱ à ✐♥t❡❧✐❣ê♥❝✐❛ ❡ à ❝♦♠♣r❡❡♥sã♦ ❞♦ ❤♦♠❡♠✳ P♦❞❡✲s❡ ❞✐③❡rq✉❡ ❤á ✈❡r❞❛❞❡ q✉❛♥❞♦ ♦ ❤♦♠❡♠ ✭✐♥t❡❧✐❣ê♥❝✐❛✮ ♣❡r❝❡❜❡ ❡ ❞✐③ ♦ s❡r q✉❡ s❡ ❞❡s✈❡❧❛✱ q✉❡ s❡♠❛♥✐❢❡st❛✳

❊✈✐❞ê♥❝✐❛✿ ❛ ✈❡r❞❛❞❡ só r❡s✉❧t❛ q✉❛♥❞♦ ❤♦✉✈❡r ❡✈✐❞ê♥❝✐❛✳ ❊✈✐❞ê♥❝✐❛ é ♠❛♥✐❢❡st❛çã♦ ❞♦s❡r✱ é ♣♦✐s✱ ♦ ❝r✐tér✐♦ ❞❛ ✈❡r❞❛❞❡✳

❈❡rt❡③❛✿ ❈❡rt❡③❛ é ♦ ❡st❛❞♦ ❞❡ ❡s♣ír✐t♦ q✉❡ ❝♦♥s✐st❡ ♥❛ ❛❞❡sã♦ ✜r♠❡ ❛ ✉♠❛ ✈❡r❞❛❞❡✱ s❡♠t❡♠♦r ❞❡ ❡♥❣❛♥♦✳ ❊ss❡ ❡st❛❞♦ ❞❡ ❡s♣ír✐t♦ s❡ ❢✉♥❞❛♠❡♥t❛ ♥❛ ❡✈✐❞ê♥❝✐❛✱ ♥♦ ❞❡s✈❡❧❛♠❡♥t♦ ❞♦ s❡r✳

✶✳✷✳✸ ❋♦r♠❛çã♦ ❞♦ ❡s♣ír✐t♦ ❝✐❡♥tí✜❝♦

P♦✉❝♦ ❛❞✐❛♥t❛r✐❛ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❡ ♦ ❡♠♣r❡❣♦ ❞♦ ✐♥str✉♠❡♥t♦ ♠❡t♦❞♦❧ó❣✐❝♦✱ s❡♠ ❛q✉❡❧❡ r✐❣♦r❡ s❡r✐❡❞❛❞❡ q✉❡ ♦ tr❛❜❛❧❤♦ ❝✐❡♥tí✜❝♦ ❞❡✈❡rá s❡r r❡✈❡st✐❞♦✳ ❖ ❡s♣ír✐t♦ ❝✐❡♥tí✜❝♦ ❛ss✉♠❡ ❛ ❛t✐t✉❞❡❞❡ ❤✉♠✐❧❞❛❞❡ ❡ ❞❡ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ s✉❛s ❧✐♠✐t❛çõ❡s✱ ❞❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❛❝❡rt♦s ❡ ❡♥❣❛♥♦s✳ ➱✐♠♣❛r❝✐❛❧✱ ♣♦✐s ♥ã♦ ❞✐st♦r❝❡ ♦s ❢❛t♦s✳ ❘❡s♣❡✐t❛ ❡s❝r✉♣✉❧♦s❛♠❡♥t❡ ❛ ✈❡r❞❛❞❡✳ ❆ ❤♦♥❡st✐❞❛❞❡ ❞♦❝✐❡♥t✐st❛ ❡stá r❡❧❛❝✐♦♥❛❞❛✱ ✉♥✐❝❛♠❡♥t❡✱ ❝♦♠ ❛ ✈❡r❞❛❞❡ ❞♦s ❢❛t♦s q✉❡ ✐♥✈❡st✐❣❛✳

✶✳✸✳ ▼➱❚❖❉❖ ❈■❊◆❚❮❋■❈❖ ✼

✶✳✸ ▼ét♦❞♦ ❝✐❡♥tí✜❝♦

P❛r❛ ❢✉♥❞❛♠❡♥t❛r✱ ♣♦❞❡♠♦s ❝♦♠❡ç❛r ❛r❣✉✐♥❞♦✿ ♦ q✉❡ é ♠ét♦❞♦❄ P♦r r❡s♣♦st❛✱ ♣♦❞❡♠♦s ❞✐③❡r✿é ❛ ♦r❞❡♠ q✉❡ s❡ ❞❡✈❡ ✐♠♣♦r ❛♦s ❞✐❢❡r❡♥t❡s ♣r♦❝❡ss♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❛t✐♥❣✐r ✉♠ ✜♠ ❞❛❞♦ ♦✉✉♠ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✳ ◆❛s ❝✐ê♥❝✐❛s✱ ❡♥t❡♥❞❡✲s❡ ♣♦r ♠ét♦❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣r♦❝❡ss♦s q✉❡ ♦❡s♣ír✐t♦ ❤✉♠❛♥♦ ❞❡✈❡ ❡♠♣r❡❣❛r ♥❛ ✐♥✈❡st✐❣❛çã♦ ❡ ❞❡♠♦♥str❛çã♦ ❞❛ ✈❡r❞❛❞❡✳

✏▼ét♦❞♦ ❝✐❡♥tí✜❝♦ é ❛ ❧ó❣✐❝❛ ❣❡r❛❧✱ tá❝✐t❛ ♦✉ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❡♠♣r❡❣❛❞❛♣❛r❛ ❛♣r❡❝✐❛r ♦s ♠ér✐t♦s ❞❡ ✉♠❛ ♣❡sq✉✐s❛✑ ❬✶❪✳

❆s té❝♥✐❝❛s q✉❡ ❝♦♠♣õ❡♠ ♦ ♠ét♦❞♦ ❝✐❡♥tí✜❝♦ ❞❡✈❡♠ s❡ ❛❞❛♣t❛r ❛♦ ♦❜❥❡t✐✈♦ ❞❡ ✐♥✈❡st✐❣❛çã♦✳❊♠ ❣❡r❛❧✱ s❡❞✐♠❡♥t❛♠✲s❡ ❡♠

✶✳ ♦❜s❡r✈❛çã♦ ❡ ❞❡✜♥✐çã♦ ❞♦ ♣r♦❜❧❡♠❛❀

✷✳ ❢♦r♠✉❧❛çã♦ ❞❡ ✉♠❛ ❤✐♣ót❡s❡❀

✸✳ ❡①♣❡r✐♠❡♥t❛çã♦ ✭t❡st❡ ❞❛ ❤✐♣ót❡s❡✮❀

✹✳ ❛❜♦r❞❛❣❡♥s ✐♥❞✉t✐✈❛ ❡ ❞❡❞✉t✐✈❛❀

✺✳ ❛♥á❧✐s❡ ❡ sí♥t❡s❡❀

✻✳ ❡♠♣r❡❣♦ ❞❛ t❡♦r✐❛❀

✼✳ ❞♦✉tr✐♥❛ ✭❛①✐♦♠❛s✱ ♣♦st✉❧❛❞♦s ❡✴♦✉ ❧❡✐s ❝✐❡♥tí✜❝❛s✮✳

❆s té❝♥✐❝❛s ♦✉ ♣r♦❝❡ss♦s ❛ s❡❣✉✐r s❡rã♦ ❞❡s❡♥✈♦❧✈✐❞♦s ❝♦♠ r❡s♣❡✐t♦ ❛♦ ♠ét♦❞♦ ❡①♣❡r✐♠❡♥t❛❧✳❖❜s❡r✈❛çã♦✿ ♦❜s❡r✈❛r é ❛♣❧✐❝❛r ❛t❡♥t❛♠❡♥t❡ ♦s s❡♥t✐❞♦s ✭❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡✮ ❛ ✉♠

♦❜❥❡t♦✱ ♣❛r❛ ❞❡❧❡ ❛❞q✉✐r✐r ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❝❧❛r♦ ❡ ♣r❡❝✐s♦✳ ❆ ♦❜s❡r✈❛çã♦ ❞❡✈❡ s❡r ❛t❡♥t❛✱ ❡①❛t❛❡ ❝♦♠♣❧❡t❛✱ ❛❧é♠ ❞❡ s❡r ♣r❡❝✐s❛✱ s✉❝❡ss✐✈❛ ❡ ♠❡tó❞✐❝❛✳

❍✐♣ót❡s❡✿ é ❛ s✉♣♦s✐çã♦ ❞❡ ✉♠❛ ❝❛✉s❛ ♦✉ ❞❡ ✉♠❛ ❧❡✐ ❞❡st✐♥❛❞❛ ❛ ❡①♣❧✐❝❛r ♣r♦✈✐s♦r✐❛♠❡♥t❡✉♠ ❢❡♥ô♠❡♥♦ ❛té q✉❡ ♦s ❢❛t♦s ❛ ✈❡♥❤❛♠ ❝♦♥tr❛❞✐③❡r ♦✉ ❛✜r♠❛r✳ ❆s ❤✐♣ót❡s❡s ♣♦ss✉❡♠ ❝♦♠♦❢✉♥çã♦ ♣rát✐❝❛ ♦r✐❡♥t❛r ♦ ♣❡sq✉✐s❛❞♦r✱ ❝♦❧♦❝❛♥❞♦✲♦ ♥❛ ❞✐r❡çã♦ ❞❛ ❝❛✉s❛ ♦✉ ❧❡✐ q✉❡ s❡ ♣r♦❝✉r❛✳❆s ❤✐♣ót❡s❡s ♣♦ss✉❡♠ ❛✐♥❞❛ ❢✉♥çõ❡s t❡ór✐❝❛s✱ ❝♦♠♦ ❝♦♦r❞❡♥❛r ❡ ❝♦♠♣❧❡t❛r ♦s r❡s✉❧t❛❞♦s ❥á♦❜t✐❞♦s✱ ❛❣r✉♣❛♥❞♦✲♦s ♥✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❧❡t♦ ❞❡ ❢❛t♦s✱ ❛✜♠ ❞❡ ❢❛❝✐❧✐t❛r ❛ s✉❛ ✐♥t❡❧✐❣✐❜✐❧✐❞❛❞❡❡ ❡st✉❞♦✳ ◗✉❛♥t♦ ❛ ♥❛t✉r❡③❛ ❞❛ ❤✐♣ót❡s❡✱ ❡st❛ ♥ã♦ ❞❡✈❡ ❝♦♥tr❛❞✐③❡r ♥❡♥❤✉♠❛ ✈❡r❞❛❞❡ ❥á ❛❝❡✐t❛♦✉ ❡①♣❧✐❝❛❞❛✱ ❞❡✈❡♥❞♦ s❡r s✐♠♣❧❡s ❡ ✈❡r✐✜❝á✈❡❧ ♣❡❧♦s ❢❛t♦s✳

❊①♣❡r✐♠❡♥t❛çã♦✿ ❝♦♥s✐st❡ ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ♣r♦❝❡ss♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ ✈❡r✐✜❝❛r ❛s ❤✐♣ót❡s❡s✳❉✐❢❡r❡ ❞❛ ♦❜s❡r✈❛çã♦ ♣♦rq✉❡ ♦❜❡❞❡❝❡ ❛ ✉♠❛ ✐❞❡✐❛ ❞✐r❡tr✐③ ❡ ♥ã♦ s✐♠♣❧❡s♠❡♥t❡ ♣♦rq✉❡ ✐♠♣❧✐❝❛❛ ✐♥t❡r✈❡♥çã♦ ❞♦ sá❜✐♦ ❡♠ ✈✐st❛ ❞❡ ♠♦❞✐✜❝❛r ♦s ❢❡♥ô♠❡♥♦s✳ ❖ ♣r✐♥❝í♣✐♦ ❣❡r❛❧ ♥❛ q✉❛❧ s❡❢✉♥❞❛♠❡♥t❛♠ ♦s ♣r♦❝❡ss♦s ❞❡ ❡①♣❡r✐♠❡♥t❛çã♦ é ♦ ❞♦ ❞❡t❡r♠✐♥✐s♠♦✱ q✉❡ s❡ ❡♥✉♥❝✐❛ ❛ss✐♠✿ ♥❛s♠❡s♠❛s ❝✐r❝✉♥stâ♥❝✐❛s✱ ❛s ♠❡s♠❛s ❝❛✉s❛s ♣r♦❞✉③❡♠ ♦s ♠❡s♠♦s ❡❢❡✐t♦s ✲ ♦✉ ❛✐♥❞❛ ✲ ❛s ❧❡✐s ❞❛♥❛t✉r❡③❛ sã♦ ✜①❛s ❡ ✐♠✉tá✈❡✐s✳

■♥❞✉çã♦ ❡ ❞❡❞✉çã♦✿ ❛ ✐♥❞✉çã♦ ❡ ❛ ❞❡❞✉çã♦ sã♦ ❢♦r♠❛s ❞❡ r❛❝✐♦❝í♥✐♦ ♦✉ ❞❡ ❛r❣✉♠❡♥t❛çã♦❡✱ ❝♦♠♦ t❛✐s✱ sã♦ ❢♦r♠❛s ❞❡ r❡✢❡①ã♦ ❡ ♥ã♦ s✐♠♣❧❡s ♣❡♥s❛♠❡♥t♦✳

❆♥á❧✐s❡ ❡ sí♥t❡s❡✿ ❛ ❛♥á❧✐s❡ é ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ✉♠ t♦❞♦ ❡♠ s✉❛s ♣❛rt❡s✳ ❆ sí♥t❡s❡ é❛ r❡❝♦♥str✉çã♦ ❞♦ t♦❞♦ ❞❡❝♦♠♣♦st♦ ♣❡❧❛ ❛♥á❧✐s❡✳ ❆ ❛♥á❧✐s❡ é ♦ ♣r♦❝❡ss♦ q✉❡ ♣❛rt❡ ❞♦ ♠❛✐s❝♦♠♣❧❡①♦ ♣❛r❛ ♦ ♠❡♥♦s ❝♦♠♣❧❡①♦ ❡ ❛ sí♥t❡s❡ ♣❛rt❡ ❞♦ ♠❛✐s s✐♠♣❧❡s ♣❛r❛ ♦ ♠❡♥♦s s✐♠♣❧❡s✳❚❛♥t♦ ❛ ❛♥á❧✐s❡ q✉❛♥t♦ ❛ sí♥t❡s❡ sã♦ ♥❡❝❡ssár✐❛s✱ ♣♦✐s s❡♠ ❛ ❛♥á❧✐s❡✱ t♦❞♦ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ é❝♦♥❢✉s♦ ❡ s✉♣❡r✜❝✐❛❧✱ ❡♥q✉❛♥t♦ s❡♠ ❛ sí♥t❡s❡✱ ❡❧❡ é ✐♥❝♦♠♣❧❡t♦✳

❚❡♦r✐❛✿ é ♦ t❡r♠♦ ❡♠♣r❡❣❛❞♦ ♣❛r❛ s✐❣♥✐✜❝❛r ✉♠ r❡s✉❧t❛❞♦ ❛ q✉❡ t❡♥❞❡♠ ❛s ❝✐ê♥❝✐❛s✳ ❊st❛s♥ã♦ s❡ ❝♦♥t❡♥t❛♠ ❛♣❡♥❛s ❝♦♠ ❛ ❢♦r♠✉❧❛çã♦ ❞❛s ❧❡✐s✳ ❆♦ ❝♦♥trár✐♦✱ ✉♠❛ ✈❡③ ❞❡t❡r♠✐♥❛❞❛s ❛s

✽ ❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖ ➚ ❋❮❙■❈❆ ❊❳P❊❘■▼❊◆❚❆▲

❧❡✐s✱ ♣r♦❝✉r❛♠ ✐♥t❡r♣r❡tá✲❧❛s ♦✉ ❡①♣❧✐❝á✲❧❛s✳ ❆ t❡♦r✐❛ ❞✐st✐♥❣✉❡✲s❡ ❞❛ ❤✐♣ót❡s❡✱ ✉♠❛ ✈❡③ q✉❡ ❛❤✐♣ót❡s❡ é ✈❡r✐✜❝á✈❡❧ ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ❡ ❛ t❡♦r✐❛ ♥ã♦✳ ❆s t❡♦r✐❛s ❝♦♦r❞❡♥❛♠ ❡ ✉♥✐✜❝❛♠ ♦s❛❜❡r ❝✐❡♥tí✜❝♦✳

❉♦✉tr✐♥❛✿ ❛ ❝✐ê♥❝✐❛ ✈✐s❛ ❡①♣❧✐❝❛r ♦s ❢❡♥ô♠❡♥♦s✳ P❛r❛ ✐ss♦ ♦❜s❡r✈❛✱ ❛♥❛❧✐s❛✱ ❧❡✈❛♥t❛ ❤✐♣ót❡✲s❡s ❡ ❛s ✈❡r✐✜❝❛✱ ❡♠ ❝♦♥❢r♦♥t♦ ❝♦♠ ♦s ❢❛t♦s✱ ♣❡❧❛ ❡①♣❡r✐♠❡♥t❛çã♦✱ ❡ ✐♥❞✉③ ❛ ❧❡✐✱ ❝♦❧♦❝❛♥❞♦✲❛♥✉♠ ❝♦♥t❡①t♦ ♠❛✐s ❛♠♣❧♦ ❛tr❛✈és ❞❡ t❡♦r✐❛s✳ ❆ ❞♦✉tr✐♥❛ ♣r♦♣õ❡ ❞✐r❡tr✐③❡s ♣❛r❛ ❛çã♦✳ ◆✉♠❛❞♦✉tr✐♥❛ ❤á ✐❞❡✐❛s ♠♦r❛✐s✱ ♣♦s✐çõ❡s ✜❧♦só✜❝❛s ❡ ♣♦❧ít✐❝❛s✱ ❡ ❛t✐t✉❞❡s ♣s✐❝♦❧ó❣✐❝❛s✳ ❍á t❛♠❜é♠✱s✉❜❥❛❝❡♥t❡s✱ ✐♥t❡r❡ss❡s ✐♥❞✐✈✐❞✉❛✐s✱ ✐♥t❡r❡ss❡s ❞❡ ❝❧❛ss❡s ♦✉ ♥❛çõ❡s✳ ❆ ❞♦✉tr✐♥❛ é✱ ❛ss✐♠✱ ✉♠❡♥❝❛❞❡❛♠❡♥t♦ ❞❡ ❝♦rr❡♥t❡s✱ ❞❡ ♣❡♥s❛♠❡♥t♦ q✉❡ ♥ã♦ s❡ ❧✐♠✐t❛♠ ❛ ❝♦♥st❛t❛r ❡ ❛ ❡①♣❧✐❝❛r ♦s ❢❡♥ô✲♠❡♥♦s✱ ♠❛s ❛♣r❡❝✐❛♠✲♥♦s ❡♠ ❢✉♥çã♦ ❞❡ ❞❡t❡r♠✐♥❛❞❛s ❝♦♥❝❡♣çõ❡s ét✐❝❛s ❡✱ á ❧✉③ ❞❡st❡s ❥✉í③♦s✱♣r❡❝♦♥✐③❛♠ ❝❡rt❛s ♠❡❞✐❞❛s ❡ ♣r♦í❜❡♠ ♦✉tr❛s✳

✶✳✹ ❈♦♥❝❡✐t♦ ❞❡ ♣❡sq✉✐s❛

❆ ♣❡sq✉✐s❛ é ✉♠❛ ❛t✐✈✐❞❛❞❡ ✈♦❧t❛❞❛ ♣❛r❛ ❛ s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✱ ❛tr❛✈és ❞♦ ❡♠♣r❡❣♦ ❞❡♣r♦❝❡ss♦s ❝✐❡♥tí✜❝♦s✳ ❆ ♣❡sq✉✐s❛ ♣❛rt❡ ❞❡ ✉♠❛ ❞ú✈✐❞❛ ♦✉ ♣r♦❜❧❡♠❛ ❡✱ ❝♦♠ ♦ ✉s♦ ❞♦ ♠ét♦❞♦❝✐❡♥tí✜❝♦✱ ❜✉s❝❛ ♣♦r r❡s♣♦st❛s ❡ ♦✉ s♦❧✉çõ❡s✳

❈❛♣ít✉❧♦ ✷

▼❊❉■❉❆❙ ❊ ❆▲●❆❘■❙▼❖❙❙■●◆■❋■❈❆❚■❱❖❙

❱✐t♦r ❍✉❣♦ ●❛r❝✐❛ ❡ ❊❞✐♦ ❈✉♥❤❛ ❞❛ ❈♦st❛

❆ ❋ís✐❝❛ é ✉♠❛ ❝✐ê♥❝✐❛ ❡♠♣ír✐❝❛ ♦✉ ❡①♣❡r✐♠❡♥t❛❧✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡ t♦❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❢ís✐❝♦é ♦r✐✉♥❞♦ ❞❡ ♠❡❞✐❞❛s✳ ❆s ❧❡✐s✱ ♦s ♣r✐♥❝í♣✐♦s✱ ♦s ♠♦❞❡❧♦s✱ ❞❡s❞❡ ♦ ♠✉♥❞♦ ♠✐❝r♦s❝♦♣✐❝❛♠❡♥t❡♣❡q✉❡♥♦ ❞❛s ♣❛rtí❝✉❧❛s ❡❧❡♠❡♥t❛r❡s ❛té ♦ ♠✉♥❞♦ ♠❛❝r♦s❝♦♣✐❝❛♠❡♥t❡ ❣r❛♥❞❡ ❞❛s ❣❛❧á①✐❛s✱ t✉❞♦✐ss♦ é r❡s✉❧t❛❞♦ ❞❛ ♠❡❞✐❞❛ ❞❡ ❞❡t❡r♠✐♥❛❞❛s q✉❛♥t✐❞❛❞❡s✱ ❝❤❛♠❛❞❛s ❞❡ ❣r❛♥❞❡③❛s ❢ís✐❝❛s✳ ❖♣r♦❝❡ss♦ ❞❡ ♠❡❞✐çã♦ ❞❡✈❡ s❡❣✉✐r ❛s ♥♦r♠❛s ✐♥t❡r♥❛❝✐♦♥❛❧♠❡♥t❡ ❝♦♥✈❡♥❝✐♦♥❛❞❛s✱ q✉❡ ❢❛③❡♠ ♣❛rt❡❞❛ ❣r❛♥❞❡ ár❡❛ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ▼❡tr♦❧♦❣✐❛✳ ❊✈✐❞❡♥t❡♠❡♥t❡✱ ❛s ♠❡❞✐❞❛s ❢ís✐❝❛ssã♦ ❛♣❡♥❛s ✉♠❛ ❞❛s ♣❛rt❡s ❞❡ss❛ ♠❛tér✐❛✱ q✉❡ ✐♥❝❧✉✐ ❛✐♥❞❛ ♠❡❞✐❞❛s q✉í♠✐❝❛s ❡ ❜✐♦❧ó❣✐❝❛s✱ ❞❡♥tr❡♦✉tr❛s✳

✷✳✶ ●r❛♥❞❡③❛s ❋ís✐❝❛s ❡ ❯♥✐❞❛❞❡s

●❡♥❡r✐❝❛♠❡♥t❡✱ ❣r❛♥❞❡③❛ é ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ♦✉ ❞❡ ✉♠ s✐st❡♠❛✱ ❞❡ ✉♠ ❝♦r♣♦♦✉ ❞❡ ✉♠❛ s✉❜stâ♥❝✐❛✱ ♦✉ ✉♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ✉♠ ❢❡♥ô♠❡♥♦✱ q✉❡ ♣♦❞❡ s❡r ♠❡❞✐❞❛✱ ✐st♦ é✱❡①♣r❡ss❛ ♣♦r ✉♠❛ q✉❛♥t✐❞❛❞❡ r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠ ♥ú♠❡r♦ ❡ ✉♠❛ ✉♥✐❞❛❞❡ ❡✱ t❛♠❜é♠✱ ♣♦r ✉♠♣r♦❝❡❞✐♠❡♥t♦ ❞❡ ♠❡❞✐çã♦ ❡ ❝♦♥❞✐çõ❡s ❛♠❜✐❡♥t❡s✳ P♦r ❡①❡♠♣❧♦✱ ✉♠ ❞❛❞♦ ❝♦r♣♦ ❛♣r❡s❡♥t❛ ❝♦♠♦♣r♦♣r✐❡❞❛❞❡ t❛♠❛♥❤♦ ✭r❡❧❛❝✐♦♥❛❞♦ ❛ s✉❛ ❛❧t✉r❛✱ ❝♦♠♣r✐♠❡♥t♦ ❡ ❧❛r❣✉r❛✮ ❡ q✉❛♥t✐❞❛❞❡ ❡ t✐♣♦ ❞❡♠❛tér✐❛✱ r❡❧❛❝✐♦♥❛❞❛ ❛ s✉❛ ♠❛ss❛ ❡ ❛♦ t✐♣♦ ❞❡ ♠❛t❡r✐❛❧ ❞♦ q✉❛❧ ♦ ❝♦r♣♦ é ❝♦♥st✐t✉í❞♦✳ ❙❡ é❞❡ ♣❧ást✐❝♦ ♦✉ ❞❡ ❢❡rr♦✱ s❡ é ✏❧❡✈❡✑ ♦✉ ✏♣❡s❛❞♦✑✳ ❆ss✐♠✱ t❛✐s ♣r♦♣r✐❡❞❛❞❡s ♣♦❞❡♠ s❡r ❞❡s❝r✐t❛sq✉❛♥t✐t❛t✐✈❛♠❡♥t❡✱ ✐st♦ é✱ ♣♦r ♥ú♠❡r♦s✳ ◆♦ ❡♥t❛♥t♦✱ ♥ã♦ ❜❛st❛ ✉♠ ♥ú♠❡r♦ ♣❛r❛ r❡♣r❡s❡♥t❛rt❛❧ ❣r❛♥❞❡③❛✳ ❙❛❜❡♠♦s q✉❡ ♦ t❛♠❛♥❤♦ ❞❡ ✉♠ ❝♦r♣♦ ❡ s❡✉ ♣❡s♦ sã♦ q✉❛♥t✐❞❛❞❡s ♦❜t✐❞❛s ❞❡♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s✿ ♦ t❛♠❛♥❤♦ é ♦❜t✐❞♦ ♣❡❧❛ ❝♦♠♣❛r❛çã♦ ❞✐r❡t❛ ❝♦♠ ♦✉tr♦ ❝♦r♣♦✱ t♦♠❛❞♦❝♦♠♦ r❡❢❡rê♥❝✐❛✱ ♣❛r❛ q✉❡ ♣♦ss❛♠♦s ❞✐③❡r s❡ ❡st❡ ❝♦r♣♦ é ♣❡q✉❡♥♦ ♦✉ ❣r❛♥❞❡✱ ❡♥q✉❛♥t♦ q✉❡ s❡✉♣❡s♦ ❡stá r❡❧❛❝✐♦♥❛❞♦ ❛♦ ♥♦ss♦ ❡s❢♦rç♦ ❢ís✐❝♦ ✭s❡♥t✐❞♦ ♥♦s ♠ús❝✉❧♦s✮ ♣❛r❛ ❡r❣✉❡r ❡ ♠♦✈✐♠❡♥t❛rt❛❧ ❝♦r♣♦✳

❊①✐st❡♠ ❣r❛♥❞❡③❛s ❢ís✐❝❛s q✉❡ sã♦ ❞✐❢❡r❡♥t❡s✱ ♣♦ré♠✱ ❞❡ ♠❡s♠❛ ♥❛t✉r❡③❛✱ ♥♦ s❡♥t✐❞♦ ❞❡q✉❡ ♣♦❞❡♠ s❡r ♠❡❞✐❞❛s ❞✐r❡t❛♠❡♥t❡ ♣❡❧♦ ♠❡s♠♦ ♣r♦❝❡❞✐♠❡♥t♦✳ ➱ ♦ ❝❛s♦ ❞♦ ❝♦♠♣r✐♠❡♥t♦✱ ❞❛❧❛r❣✉r❛ ❡ ❞❛ ❛❧t✉r❛ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❝♦r♣♦✱ q✉❡ sã♦ ❝❛r❛❝t❡ríst✐❝❛s ❞✐❢❡r❡♥t❡s ❞❡st❡ ❝♦r♣♦✳ ◆♦❡♥t❛♥t♦✱ t♦❞❛s ♣♦❞❡♠ s❡r ♠❡❞✐❞❛s ♣❡❧♦ ♠❡s♠♦ ♣r♦❝❡❞✐♠❡♥t♦✱ ❝♦♠♣❛r❛♥❞♦✲❛s ❞✐r❡t❛♠❡♥t❡ ❝♦♠✉♠ ♣❛❞rã♦✳ ❉✐③❡♠♦s q✉❡ ❡st❛s três ❣r❛♥❞❡③❛s ❢ís✐❝❛s ✭❝♦♠♣r✐♠❡♥t♦✱ ❧❛r❣✉r❛ ❡ ❛❧t✉r❛✮ ♣♦ss✉❡♠❛ ♠❡s♠❛ ❞✐♠❡♥sã♦✳ ❏á ❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ♠❛ss❛ ❞❡✈❡ s❡r ❞❡t❡r♠✐♥❛❞❛ ❞❡ ♠❛♥❡✐r❛ ❞✐❢❡r❡♥t❡✱❡♥tã♦ s✉❛ ❞✐♠❡♥sã♦ é ❞✐❢❡r❡♥t❡ ❞❛ ❞✐♠❡♥sã♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦✳ ❖ ♠❡s♠♦ ♦❝♦rr❡ ❝♦♠ ❛ ❣r❛♥❞❡③❛

✾

✶✵ ❈❆P❮❚❯▲❖ ✷✳ ▼❊❉■❉❆❙ ❊ ❆▲●❆❘■❙▼❖❙ ❙■●◆■❋■❈❆❚■❱❖❙

❢ís✐❝❛ t❡♠♣♦✳ ❆ ✐❞❛❞❡ ❞❡ ✉♠❛ ♣❡ss♦❛ ❡ ♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ♣❛r❛ q✉❡ ❛ ❧✉❛ ❞❡✐①❡ ❞❡ ❡st❛r ♥❛❢❛s❡ ❞❡ ❧✉❛ ❝❤❡✐❛ ❡ ✈♦❧t❡ à ❡st❛ ♠❡s♠❛ ❢❛s❡✱ sã♦ ❣r❛♥❞❡③❛s ❢ís✐❝❛s ❞✐❢❡r❡♥t❡s ❡ ❝❛r❛❝t❡ríst✐❝❛s ❞❡s✐st❡♠❛s ❞✐❢❡r❡♥t❡s❀ ♣♦ré♠✱ ❞❡ ♠❡s♠❛ ♥❛t✉r❡③❛ ❡ t❡♥❞♦ ❛ ♠❡s♠❛ ❞✐♠❡♥sã♦✳

❙✉r❣❡ ❞✐st♦ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✏❛❧❣♦ ❛ ♠❛✐s✑ ♥❛ ❡s♣❡❝✐✜❝❛çã♦ ❞❡ ✉♠❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛✿ ♣r❡✲❝✐s❛♠♦s t❛♠❜é♠ ❞❡ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ q✉❡ é ✉♠❛ ❣r❛♥❞❡③❛ ❡s❝❛❧❛r r❡❛❧✱ ❞❡✜♥✐❞❛ ❡ ❛❝❡✐t❛♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ❝♦♥✈❡♥çã♦✳ ◗✉❛✐sq✉❡r ❞✉❛s ❣r❛♥❞❡③❛s ❞♦ ♠❡s♠♦ t✐♣♦ ♦✉ ❡s♣é❝✐❡ ♣♦❞❡♠s❡r ❝♦♠♣❛r❛❞❛s ❝♦♠ ❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ♣❛r❛ ❡①♣r❡ss❛r ♥✉♠❡r✐❝❛♠❡♥t❡ ❛ r❛③ã♦ ❡♥tr❡ ❛♠✲❜❛s✳ ❋r❡q✉❡♥t❡♠❡♥t❡ ✉s❛✲s❡ ❛ ♣❛❧❛✈r❛ ✏✉♥✐❞❛❞❡✑ ❝♦♠ ♦ ♥♦♠❡ ❞❛ ❣r❛♥❞❡③❛ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱✉♥✐❞❛❞❡ ❞❡ ár❡❛ ❡ ✉♥✐❞❛❞❡ ❞❡ t❡♠♣❡r❛t✉r❛✳ ▼❡❞✐r ✉♠❛ ❣r❛♥❞❡③❛✱ ❝♦♠♦ ❥á ♠❡♥❝✐♦♥❛❞♦✱ s✐❣♥✐✜❝❛❝♦♠♣❛rá✲❧❛ ❝♦♠ ♦✉tr❛ ❣r❛♥❞❡③❛✱ ❞❛ ♠❡s♠❛ ❡s♣é❝✐❡✱ ❝♦♥✈❡♥❝✐♦♥❛❞❛ ❝♦♠♦ s❡♥❞♦ ♦ ♣❛❞rã♦ ❞❡ss❛❡s♣é❝✐❡✱ s❡♥❞♦ ❡st❡ ♣❛❞rã♦ ❛ ♣ró♣r✐❛ ✉♥✐❞❛❞❡✳ P♦rt❛♥t♦✱ ♠❡❞✐r é ✈❡r✐✜❝❛r ❛ r❡❧❛çã♦ ♥✉♠ér✐❝❛❡①✐st❡♥t❡ ❡♥tr❡ ❛ ❣r❛♥❞❡③❛ ❡ ♦ ♣❛❞rã♦✳

❙❡❥❛ G ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✉♠❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ q✉❛❧q✉❡r✱ ❡ U ❛ ✉♥✐❞❛❞❡ ❝♦♥✈❡♥❝✐♦♥❛❞❛ ♣❛r❛❡ss❛ ❣r❛♥❞❡③❛✱ ❡♥tã♦✱ ♦ ✈❛❧♦r ♥✉♠ér✐❝♦ M(G) ❞❡ss❛ q✉❛♥t✐❞❛❞❡ é ❛ r❛③ã♦✿

M(G) =G

U, ✭✷✳✶✳✶✮

❞❡ ♠♦❞♦ q✉❡ ♦ r❡s✉❧t❛❞♦ ❞❡ss❛ ♠❡❞✐çã♦ t♦♠❛ ❛ ❢♦r♠❛

G = M(G)U . ✭✷✳✶✳✷✮

❖❜s❡r✈❡♠♦s q✉❡ ❣r❛♥❞❡③❛s ❢ís✐❝❛s ❞❡ ♠❡s♠❛ ♥❛t✉r❡③❛ ♣♦❞❡♠ ❛♣r❡s❡♥t❛r ❞✐❢❡r❡♥t❡s ✉♥✐❞❛❞❡s✱❝♦♠♦ é ♦ ❝❛s♦ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ✉♠❛ ♣❡ss♦❛ ❝❛♠✐♥❤❛♥❞♦✱ q✉❡ ♣♦❞❡ s❡ ❡①♣r❡ss❛ ❡♠ ♠❡tr♦s ♣♦rs❡❣✉♥❞♦✱ ♣❛ss♦s ♣♦r ♠✐♥✉t♦ ♦✉ ♠✐❧❤❛s ♣♦r ❤♦r❛✱ ❞❡♥tr❡ ♦✉tr❛s✳ ➱ ✐♠♣♦rt❛♥t❡ s❛❧✐❡♥t❛r q✉❡ t♦❞❛♠❡❞✐❞❛ ❞❡ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ❞❡✈❡ s❡r ❛❝♦♠♣❛♥❤❛ ❞❡ ✉♠❛ ✉♥✐❞❛❞❡✳

❯♥✐❞❛❞❡s ❞❡ ♠❡❞✐❞❛ ❞❡ ❣r❛♥❞❡③❛s ❞❡ ♠❡s♠❛ ❞✐♠❡♥sã♦ ♣♦❞❡♠ t❡r ♦ ♠❡s♠♦ ♥♦♠❡ ❡ sí♠❜♦❧♦✱♠❡s♠♦ q✉❡ ❛s ❣r❛♥❞❡③❛s ♥ã♦ s❡❥❛♠ ❞♦ ♠❡s♠♦ t✐♣♦✳ P♦r ❡①❡♠♣❧♦✱ ♠❡tr♦ ♣♦r s❡❣✉♥❞♦ ❡ m/s sã♦✱r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦ ♥♦♠❡ ❡ ♦ sí♠❜♦❧♦ ❞❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ✉♠ ❛✉t♦♠ó✈❡❧✳❙❡r✈❡ t❛♠❜é♠ ♣❛r❛ ✐♥❢♦r♠❛r ❛ t❛①❛ ❞❡ ♣r♦❞✉çã♦ ❞❡ ✜♦✱ r❡❛❧✐③❛❞❛ ♣♦r ✉♠❛ ♠áq✉✐♥❛ ❡①tr✉s♦r❛❞❡ ♣❧ást✐❝♦ ✭❛ ❝❛❞❛ s❡❣✉♥❞♦ ❛ ♠áq✉✐♥❛ ♣r♦❞✉③ m ♠❡tr♦s ❞❡ ✜♦✮✳ ▼❛s✱ ❡♠ ❛❧❣✉♥s ❝❛s♦s✱ ♥♦♠❡s❡s♣❡❝✐❛✐s ❞❡ ✉♥✐❞❛❞❡s ❞❡ ♠❡❞✐❞❛ sã♦ ✉t✐❧✐③❛❞♦s ❡①❝❧✉s✐✈❛♠❡♥t❡ ♣❛r❛ ❣r❛♥❞❡③❛s ❞❡ ✉♠ t✐♣♦ ❡s✲♣❡❝í✜❝♦✳ P♦r ❡①❡♠♣❧♦✱ ❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ♣❛r❛ ❢r❡q✉ê♥❝✐❛s é s❡❣✉♥❞♦ ❡❧❡✈❛❞♦ ❛ ♠❡♥♦s ✉♠✱♦✉ s❡❥❛✱ 1/s ♦✉ ❛✐♥❞❛ s−1✱ ❡ é ❝❤❛♠❛❞❛ ❞❡ ❤❡rt③✱ r❡♣r❡s❡♥t❛❞❛ ♣♦r Hz✳ ▼✉✐t❛s ✉♥✐❞❛❞❡s ❝r✐❛❞❛s❛♣r❡s❡♥t❛♠ ♦ ♥♦♠❡ ❞❡ ❛❧❣✉♠ ❝✐❡♥t✐st❛ ♣❛r❛ ❤♦♠❡♥❛❣❡á✲❧♦✳ ❙ã♦ ♦s ❝❛s♦s ❞♦ ❥♦✉❧❡ ✭✉♥✐❞❛❞❡ ❞❡❡♥❡r❣✐❛✮✱ ❞♦ ♥❡✇t♦♥ ✭✉♥✐❞❛❞❡ ❞❡ ❢♦rç❛✮ ❡ ♦ ♣ró♣r✐♦ ❤❡rt③ ✭✉♥✐❞❛❞❡ ❞❡ ❢r❡q✉ê♥❝✐❛✮✳ ❊♠ ❛❧❣✉♥s❝❛s♦s sã♦ ❞❛❞♦s ♥♦♠❡s ❡s♣❡❝✐❛✐s às ✉♥✐❞❛❞❡s ❞❡ ♠❡❞✐❞❛ ❞❡ ❣r❛♥❞❡③❛s ❛❞✐♠❡♥s✐♦♥❛✐s q✉❡ sã♦♥ú♠❡r♦s✱ ❝♦♠♦ ♦ r❛❞✐❛♥♦ ❡ ♦ ❞❡❝✐❜❡❧✱ ♣❛r❛ ❛ ❡s♣❡❝✐✜❝❛çã♦ ❞❡ â♥❣✉❧♦ ❡ ✐♥t❡♥s✐❞❛❞❡ ❞❡ ♥í✈❡❧s♦♥♦r♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♠ ✉♠ s✐st❡♠❛ ❞❡ ✉♥✐❞❛❞❡s ❝♦❡r❡♥t❡ ❛❞♦t❛✲s❡ ♣♦r ❝♦♥✈❡♥çã♦ ✉♠❛ú♥✐❝❛ ✉♥✐❞❛❞❡ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❝❛❞❛ ❣r❛♥❞❡③❛ ❢✉♥❞❛♠❡♥t❛❧✳ P♦r ❡①❡♠♣❧♦ ♥♦ ❙■✱ ❙✐st❡♠❛ ■♥✲t❡r♥❛❝✐♦♥❛❧ ❞❡ ❯♥✐❞❛❞❡s✱ ♦ q✉✐❧♦❣r❛♠❛ é ❛ ✉♥✐❞❛❞❡ ❞❡ ❜❛s❡ ♣❛r❛ ♠❛ss❛✱ ❡♥q✉❛♥t♦ q✉❡ ♥♦ s✐st❡♠❛❈●❙ ✭❝❡♥tí♠❡tr♦✲❣r❛♠❛✲s❡❣✉♥❞♦✮✱ ♦ ❣r❛♠❛ é ❛ ✉♥✐❞❛❞❡ ❞❡ ❜❛s❡ ♣❛r❛ ♠❛ss❛✳ ❊✈✐❞❡♥t❡♠❡♥t❡✱♣♦❞❡ ❛❝♦♥t❡❝❡r ❞❡ ✉♠❛ ❣r❛♥❞❡③❛ ❞❡r✐✈❛❞❛ t❡r ❛ ♠❡s♠❛ ❞✐♠❡♥sã♦ ❞❡ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ❜❛s❡ ❡✱❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ♠❡s♠❛ ✉♥✐❞❛❞❡✳ P♦r ❡①❡♠♣❧♦✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝❤✉✈❛ ♣♦❞❡ s❡r ♠❡♥s✉r❛❞❛❝♦♠♦ ✈♦❧✉♠❡ ♣♦r ár❡❛✱ q✉❡ t❡♠ ♦ ♠❡tr♦ ❝♦♠♦ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡r✐✈❛❞❛ ❝♦❡r❡♥t❡ ♥♦ ❙■✳ ◗✉❛♥❞♦s❡ ❝♦♥s✐❞❡r❛ ♦ ♥ú♠❡r♦ ❞❡ ❡♥t✐❞❛❞❡s✱ ❡♠ q✉❛❧q✉❡r s✐st❡♠❛ ❞❡ ✉♥✐❞❛❞❡s✱ ♦ ♥ú♠❡r♦ ✉♠ ✭1✮✱ ♣♦❞❡s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ❜❛s❡✱ ❛❞✐♠❡♥s✐♦♥❛❧✱ ❝❧❛r❛♠❡♥t❡✳

❱✐♠♦s✱ ❡♥tã♦✱ q✉❡ t♦❞❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ♥❡❝❡ss✐t❛ ❞❡ ✉♠ ♥ú♠❡r♦ ❡ ✉♠❛ ✉♥✐❞❛❞❡ ♣❛r❛ s❡r❡s♣❡❝✐✜❝❛❞❛✳ ◆♦ ❡♥t❛♥t♦✱ ❡①✐st❡♠ ✈ár✐❛s ♦✉tr❛s ❣r❛♥❞❡③❛s ❢ís✐❝❛s q✉❡ ♥❡❝❡ss✐t❛♠ t❛♠❜é♠ ❞❡❞✐r❡çã♦ ❡ s❡♥t✐❞♦ ♣❛r❛ q✉❡ s✉❛ ❞❡s❝r✐çã♦ ❡st❡❥❛ ❝♦♠♣❧❡t❛✳ ➱ ♦ ❝❛s♦ ❞❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ✈❡❧♦❝✐❞❛❞❡✳❯♠ ❝❛rr♦ ♣♦❞❡ ❡st❛r s❡ ♠♦✈✐♠❡♥t❛♥❞♦ ❛ 45 km/h ♥❛ ❞✐r❡çã♦ ♥♦rt❡✳ ◆♦ ❡♥t❛♥t♦✱ ♦ ♠❡s♠♦ ❝❛rr♦

✷✳✷✳ ❙■❙❚❊▼❆ ■◆❚❊❘◆❆❈■❖◆❆▲ ❉❊ ❯◆■❉❆❉❊❙ ✶✶

♣♦❞❡ ❛✐♥❞❛ s❡ ♠♦✈✐♠❡♥t❛r ❛ 45 km/h ♥❛ ❞✐r❡çã♦ ❧❡st❡✱ sã♦ s✐t✉❛çõ❡s ❢ís✐❝❛s ❞✐❢❡r❡♥t❡s✳ ❉❡ ♠♦❞♦s❡♠❡❧❤❛♥t❡✱ s❡ ❛❧❣✉é♠ ❧❤❡ ❞✐ss❡r q✉❡ ❛♣❧✐❝♦✉ ✉♠❛ ❢♦rç❛ ❞❡ 2, 5N s♦❜r❡ ✉♠ ❝♦r♣♦ q✉❛❧q✉❡r ✈♦❝êq✉❡st✐♦♥❛rá✿ ✏♠❛s ❡st❛ ❢♦rç❛ ❢♦✐ ❛♣❧✐❝❛❞❛ ❞❡ ❜❛✐①♦ ♣❛r❛ ❝✐♠❛✱ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛ ♦✉❡♠ ❛❧❣✉♠❛ ♦✉tr❛ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦❄✑✳ ❆❧é♠ ❞✐ss♦ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r✱ ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡✱ q✉❡ ❛❛♣❧✐❝❛çã♦ ❞❡ ❞✐❢❡r❡♥t❡s ❢♦rç❛s ❛ ✉♠ ♠❡s♠♦ ❝♦r♣♦ ♣♦❞❡ r❡s✉❧t❛r ❡♠ ❡❢❡✐t♦s ❞✐❢❡r❡♥t❡s✱ ❞❡ ❛❝♦r❞♦❝♦♠ ♦ ♣♦♥t♦ ❞❡ ❛♣❧✐❝❛çã♦ ❞❡st❛s ❢♦rç❛s ❡ ❞❛ ❞✐r❡çã♦ ❡♠ q✉❡ sã♦ ❛♣❧✐❝❛❞❛s✳ P♦r ❡①❡♠♣❧♦✱ ♦ ❡❢❡✐t♦❞❛ ❛çã♦ ❞❡ ❞✉❛s ❢♦rç❛s ❞❡ ♠❡s♠❛ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦ é ❞✐❢❡r❡♥t❡ ❞❛q✉❡❧❡ ❡♠ q✉❡ ❛s ♠❡s♠❛s ❢♦rç❛ssã♦ ❛♣❧✐❝❛❞❛s ❡♠ ❞✐r❡çõ❡s ♣❡r♣❡♥❞✐❝✉❧❛r❡s✳ ➱ ✉♠ ❢❛t♦ ❡①♣❡r✐♠❡♥t❛❧ q✉❡ ❛s ❢♦rç❛s s❛t✐s❢❛③❡♠ ❛r❡❣r❛ ❞❛ s♦♠❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦✳

❆ss✐♠✱ t♦❞❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ q✉❡ ♥❡❝❡ss✐t❛ ❞❡ ❞✐r❡çã♦ ✭❡ s❡♥t✐❞♦✮✱ ❛❧é♠ ❞❡ ✐♥t❡♥s✐❞❛❞❡ ❡✉♥✐❞❛❞❡✱ ♣❛r❛ s❡r ❝♦♠♣❧❡t❛♠❡♥t❡ ❡s♣❡❝✐✜❝❛❞❛✱ ❡ q✉❡ s❛t✐s✜③❡r ❛ r❡❣r❛ ❞❛ s♦♠❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦✱s❡rá ❝❤❛♠❛❞❛ ❞❡ ❣r❛♥❞❡③❛ ✈❡t♦r✐❛❧✳ ❈❛s♦ ❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ✜q✉❡ ❝♦♠♣❧❡t❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛❛♣❡♥❛s ♣♦r ✉♠ ♥ú♠❡r♦ ❡ ✉♠❛ ✉♥✐❞❛❞❡✱ ❡❧❛ s❡rá ❝❤❛♠❛❞❛ ❞❡ ❣r❛♥❞❡③❛ ❡s❝❛❧❛r✳ ●r❛♥❞❡③❛s ❝♦♠♦❝♦♠♣r✐♠❡♥t♦✱ ♠❛ss❛✱ t❡♠♣♦✱ t❡♠♣❡r❛t✉r❛ ❡ ✐♥t❡♥s✐❞❛❞❡ ❞❡ ❝♦rr❡♥t❡ ❡❧étr✐❝❛ sã♦ ❡①❡♠♣❧♦s ❞❡❣r❛♥❞❡③❛s ❡s❝❛❧❛r❡s ❡ ❣r❛♥❞❡③❛s ❝♦♠♦ ❞❡s❧♦❝❛♠❡♥t♦✱ ❢♦rç❛✱ ✈❡❧♦❝✐❞❛❞❡ ❡ ❝❛♠♣♦ ❡❧étr✐❝♦ sã♦❡①❡♠♣❧♦s ❞❡ ❣r❛♥❞❡③❛s ✈❡t♦r✐❛✐s✳

❉á✲s❡ ♦ ♥♦♠❡ ❞❡ s✐st❡♠❛ ❞❡ ❣r❛♥❞❡③❛s ❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❣r❛♥❞❡③❛s q✉❡ ♠❛♥té♠ ❡♥tr❡ s✐r❡❧❛çõ❡s ❜❡♠ ❝♦♥❤❡❝✐❞❛s✱ ❡♠ q✉❡ ❤á ❝❡rt♦ ♥ú♠❡r♦ ❞❡ ❣r❛♥❞❡③❛s ❞❡ ❜❛s❡ ♦✉ ❢✉♥❞❛♠❡♥t❛✐s q✉❡sã♦ ♠✉t✉❛♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ♦✉ s❡❥❛✱ ✉♠❛ ❞❡ss❛s ❣r❛♥❞❡③❛s ♥ã♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣♦r✉♠ ♣r♦❞✉t♦ ❞❡ ♣♦tê♥❝✐❛s ❞❛s ❞❡♠❛✐s ❣r❛♥❞❡③❛s ❞❡ ❜❛s❡✱ ❡ ❤á ❣r❛♥❞❡③❛s ❞❡r✐✈❛❞❛s q✉❡ ♣♦❞❡♠s❡r ❡①♣r❡ss❛s ♣♦r ✉♠ ♣r♦❞✉t♦ ❞❡ ♣♦tê♥❝✐❛s ❞❛s ❣r❛♥❞❡③❛s ❞❡ ❜❛s❡✳ P♦r ❡①❡♠♣❧♦✱ ❛s ❣r❛♥❞❡③❛s❞❡ ❜❛s❡ ❞♦ ❙✐st❡♠❛ ■♥t❡r♥❛❝✐♦♥❛❧ ❞❡ ●r❛♥❞❡③❛s ✭■❙◗ ✕ ■♥t❡r♥❛t✐♦♥❛❧ ❙②st❡♠ ♦❢ ◗✉❛♥t✐t✐❡s✮sã♦ ❝♦♠♣r✐♠❡♥t♦✱ ♠❛ss❛✱ t❡♠♣♦✱ ❝♦rr❡♥t❡ ❡❧étr✐❝❛✱ t❡♠♣❡r❛t✉r❛ t❡r♠♦❞✐♥â♠✐❝❛✱ q✉❛♥t✐❞❛❞❡ ❞❡s✉❜stâ♥❝✐❛ ❡ ✐♥t❡♥s✐❞❛❞❡ ❧✉♠✐♥♦s❛❀ ❡♥tã♦✱ ❛ ✈❡❧♦❝✐❞❛❞❡✱ ❞❡✜♥✐❞❛ ♣❡❧♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ❝♦♠✲♣r✐♠❡♥t♦ ♣♦r ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦✱ é ✉♠❛ ❣r❛♥❞❡③❛ ❞❡r✐✈❛❞❛✶✳ ❖ ❝♦♥❥✉♥t♦ ❞❛s ❣r❛♥❞❡③❛s❢✉♥❞❛♠❡♥t❛✐s é ❜❛s❡ ❞❡ ✉♠ s✐st❡♠❛ q✉❡ ♣♦❞❡ ❝♦♥t❡r t♦❞❛s ❛s ❣r❛♥❞❡③❛s ❢ís✐❝❛s ❞❡r✐✈❛❞❛s✱ ❝♦♠♦♦ ❙✐st❡♠❛ ■♥t❡r♥❛❝✐♦♥❛❧ ❞❡ ❯♥✐❞❛❞❡s ✭❙■✮✳

✷✳✷ ❙✐st❡♠❛ ■♥t❡r♥❛❝✐♦♥❛❧ ❞❡ ❯♥✐❞❛❞❡s

❖ ♣r♦♣ós✐t♦ ❞♦ ❇✉r❡❛✉ ■♥t❡r♥❛❝✐♦♥❛❧ ❞❡ P❡s♦s ❡ ▼❡❞✐❞❛s é ❞❡✜♥✐r ❡ ♣r♦♠♦✈❡r ♦ ❙✐st❡♠❛ ■♥t❡r✲♥❛❝✐♦♥❛❧ ❞❡ ❯♥✐❞❛❞❡s ✭❙■✮✱ ♦ q✉❛❧ t❡♠ s✐❞♦ ✉s❛❞♦ ❡♠ t♦❞♦ ♦ ♠✉♥❞♦ ❝♦♠♦ ❛ ❧✐♥❣✉❛❣❡♠ ♣r❡❢❡r✐❞❛❞❛ ❈✐ê♥❝✐❛ ❡ ❚❡❝♥♦❧♦❣✐❛ ❞❡s❞❡ ✶✾✹✽✱ ❛tr❛✈és ❞❡ ✉♠❛ r❡s♦❧✉çã♦ ❞❛ 9a ❈♦♥❢❡rê♥❝✐❛ ●❡r❛❧ ❞❡ P❡s♦s❡ ▼❡❞✐❞❛s✳ ❆t✉❛❧♠❡♥t❡ ❡❧❡ ❝♦♥t❛ ❝♦♠ s❡t❡ ✉♥✐❞❛❞❡s✲♣❛❞rã♦ ♣❛r❛ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❡ ❣r❛♥❞❡③❛s❢ís✐❝❛s✳ ❙ã♦ ❡❧❛s✿

✉♥✐❞❛❞❡ ❞❡ t❡♠♣♦✿ ❖ s❡❣✉♥❞♦ ✭s✮ é ❛ ❞✉r❛çã♦ ❞❡ 9.192.631.770 ♣❡rí♦❞♦s ❞❛ r❛❞✐❛çã♦ ❝♦r✲r❡s♣♦♥❞❡♥t❡ ❛ tr❛♥s✐çã♦ ❡♥tr❡ ❞♦✐s ♥í✈❡✐s ❤✐♣❡r✜♥♦s ❞♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❞♦ át♦♠♦ ❞❡❝és✐♦ 133❀

✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦✿ ♦ ♠❡tr♦ ✭m✮ é ♦ t❛♠❛♥❤♦ ❞♦ ❝❛♠✐♥❤♦ ♣❡r❝♦rr✐❞♦ ♣❡❧❛ ❧✉③ ♥♦

✈á❝✉♦ ❞✉r❛♥t❡ ♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ❞❡1

299.792.458❞❡ ✉♠ s❡❣✉♥❞♦✷❀

✶◆ã♦ ❝♦♥❢✉♥❞❛ ❡st❛ ❞❡✜♥✐çã♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛✱ q✉❡ é ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❛ ♣♦s✐çã♦✱♦✉ ❞❡r✐✈❛❞❛ ❞❛ ♣♦s✐çã♦ ❡♠ r❡❧❛çã♦ ❛♦ t❡♠♣♦✳ ❆q✉✐ q✉❡r❡♠♦s ♠❡♥❝✐♦♥❛r q✉❡✱ ✉♠❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ s❡♥❞♦ ❞❡r✐✈❛❞❛❞❡ ♦✉tr❛✱ s✐❣♥✐✜❝❛ q✉❡ ❡st❛ é ❡s❝r✐t❛ ❡♠ t❡r♠♦s ❞❛q✉❡❧❛s❀ é ❞❡❞✉③✐❞❛✳ ➱ ♦ ❝❛s♦ ❞❛ ❞❡♥s✐❞❛❞❡ ❞❡ ✉♠ ❝♦r♣♦✱ ❞❡✜♥✐❞❛❝♦♠♦ ❛ r❛③ã♦ ❡♥tr❡ ♠❛ss❛ ❡ ❝♦♠♣r✐♠❡♥t♦ ❛♦ ❝✉❜♦ ✭✈♦❧✉♠❡✮✳

✷❙❡❣✉❡ q✉❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③✱ ♦✉ s❡❥❛✱ ❞❛ r❛❞✐❛çã♦ ❡❧❡tr♦♠❛❣♥ét✐❝❛ ♥♦ ✈á❝✉♦ é ❡①❛t❛♠❡♥t❡ 299.792.458

♠❡tr♦s ♣♦r s❡❣✉♥❞♦✱ r❡♣r❡s❡♥t❛❞❛ ♣♦r c✳

✶✷ ❈❆P❮❚❯▲❖ ✷✳ ▼❊❉■❉❆❙ ❊ ❆▲●❆❘■❙▼❖❙ ❙■●◆■❋■❈❆❚■❱❖❙

✉♥✐❞❛❞❡ ❞❡ ♠❛ss❛✿ ♦ q✉✐❧♦❣r❛♠❛ ✭kg✮ é ❛ ✉♥✐❞❛❞❡ ❞❡ ♠❛ss❛❀ é ✐❣✉❛❧ ❛ ♠❛ss❛ ❞♦ ♣r♦tót✐♣♦✐♥t❡r♥❛❝✐♦♥❛❧ ❞♦ q✉✐❧♦❣r❛♠❛❀

✉♥✐❞❛❞❡ ❞❡ ❝♦rr❡♥t❡ ❡❧étr✐❝❛✿ ♦ ❛♠♣èr❡ ✭A✮ é ❛ ❝♦rr❡♥t❡ ❝♦♥st❛♥t❡ q✉❡✱ s❡ ♠❛♥t✐❞❛ ❡♠ ❞♦✐s✜♦s ❝♦♥❞✉t♦r❡s r❡t✐❧í♥❡♦s ❡ ♣❛r❛❧❡❧♦s ❡♥tr❡ s✐✱ ❞❡ t❛♠❛♥❤♦ ✐♥✜♥✐t♦ ❡ ❝♦♠ s❡çã♦ tr❛♥s✈❡rs❛❧❝✐r❝✉❧❛r ❞❡s♣r❡③í✈❡❧✱ s❡♣❛r❛❞♦s ♣❡❧❛ ❞✐stâ♥❝✐❛ ❞❡ ✉♠ ♠❡tr♦✱ ♥♦ ✈á❝✉♦✱ ♣r♦❞✉③ ❡♥tr❡ ♦s ✜♦s✉♠❛ ❢♦rç❛ ❞❡ 2× 10−7 ♥❡✇t♦♥ ♣♦r ♠❡tr♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦❀

✉♥✐❞❛❞❡ ❞❡ t❡♠♣❡r❛t✉r❛ t❡r♠♦❞✐♥â♠✐❝❛✿ ♦ ❦❡❧✈✐♥ ✭K✮✱ ✉♥✐❞❛❞❡ ❞❡ t❡♠♣❡r❛t✉r❛ t❡r♠♦❞✲

✐♥â♠✐❝❛✱ é ❛ ❢r❛çã♦1

273, 16❞❛ t❡♠♣❡r❛t✉r❛ t❡r♠♦❞✐♥â♠✐❝❛ ❞♦ ♣♦♥t♦ tr✐♣❧♦ ❞❛ á❣✉❛❀

✉♥✐❞❛❞❡ ❞❡ ✐♥t❡♥s✐❞❛❞❡ ❧✉♠✐♥♦s❛✿ ❛ ❝❛♥❞❡❧❛ ✭cd✮ é ❛ ✐♥t❡♥s✐❞❛❞❡ ❧✉♠✐♥♦s❛✱ ♥✉♠❛ ❞❛❞❛❞✐r❡çã♦✱ ❞❡ ✉♠❛ ❢♦♥t❡ q✉❡ ❡♠✐t❡ r❛❞✐❛çã♦ ♠♦♥♦❝r♦♠át✐❝❛ ❞❡ ❢r❡q✉ê♥❝✐❛ 540× 1012 ❤❡rt③

❡ q✉❡ ♣♦ss✉✐ ❛ r❛❞✐❛çã♦ ❝♦♠ ✐♥t❡♥s✐❞❛❞❡ ❞❡1

683✇❛tt ♣♦r ❡s❢❡r♦✲r❛❞✐❛♥♦ ♥❛q✉❡❧❛ ❞✐r❡çã♦❀

✉♥✐❞❛❞❡ ❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ s✉❜stâ♥❝✐❛✿ ♦ ♠♦❧❡ ♦✉ ♠♦❧ ✭mol✮ é ❛ q✉❛♥t✐❞❛❞❡ ❞❡ s✉❜stâ♥❝✐❛❞❡ ✉♠ s✐st❡♠❛ q✉❡ ❝♦♥té♠ t❛♥t❛s ❡♥t✐❞❛❞❡s ❡❧❡♠❡♥t❛r❡s q✉❛♥t♦ ❛s q✉❡ ❡①✐st❡♠ ❡♠ 0, 012q✉✐❧♦❣r❛♠❛s ❞❡ ❝❛r❜♦♥♦ 12✳

❉❡✈❡ s❡r ♦❜s❡r✈❛❞♦ q✉❡ ❡st❛s ❞❡✜♥✐çõ❡s r❡q✉❡r❡♠ ❝♦♥❞✐çõ❡s ❡s♣❡❝✐❛✐s ❞❡ ❧❛❜♦r❛tór✐♦ ♣❛r❛ ❛ s✉❛❡s♣❡❝✐✜❝❛çã♦✳ P♦r ❡①❡♠♣❧♦✱ ♥❛ ❞❡✜♥✐çã♦ ❞♦ s❡❣✉♥❞♦✱ ❡①✐❣❡✲s❡ q✉❡ ♦s át♦♠♦s ❞❡ ❝és✐♦ ❡st❡❥❛♠❡♠ r❡♣♦✉s♦ ❡ ♥♦ ③❡r♦ ❛❜s♦❧✉t♦✸✳ ❆ ❞❡✜♥✐çã♦ ❞❡ t❡♠♣❡r❛t✉r❛ t❡r♠♦❞✐♥â♠✐❝❛ s❡ r❡❢❡r❡ ❛ á❣✉❛t❡♥❞♦ ❛ ❝♦♠♣♦s✐çã♦ ✐s♦tó♣✐❝❛ ❞❡✜♥✐❞❛ ❡①❛t❛♠❡♥t❡ ♣❡❧❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s ❞❡ q✉❛♥t✐❞❛❞❡ ❞❡s✉❜stâ♥❝✐❛✿ 0, 000 155 76 ♠♦❧❡s ❞❡ 2H ♣♦r ♠♦❧ ❞❡ 1H❀ 0, 000 379 9 ♠♦❧❡s ❞❡ 17O ♣♦r ♠♦❧ ❞❡ 16O✱❡ 0, 002 005 2 ♠♦❧❡s ❞❡ 18O ♣♦r ♠♦❧ ❞❡ 16O✳ ▼❛✐♦r❡s ❞❡t❛❧❤❡s ❝♦♥s✉❧t❛r ❬✷❪✳

✷✳✸ ▼❡❞✐❞❛s ❉✐r❡t❛s ❡ ■♥❞✐r❡t❛s

P♦❞❡♠♦s ❝❧❛ss✐✜❝❛r ♦s ♣r♦❝❡ss♦s ❞❡ ♠❡❞✐❞❛✱ ❞❡♣❡♥❞❡♥❞♦ ❞❛ ♥❛t✉r❡③❛ ❞❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ❡ ❞♦♣❛❞rã♦ ❞❡ ♠❡❞✐❞❛ ❛❞♦t❛❞♦✱ ❡♠ ❞♦✐s t✐♣♦s✳

▼❡❞✐❞❛ ❞✐r❡t❛✿ r❡❛❧✐③❛♠♦s ✉♠❛ ♠❡❞✐❞❛ ❞✐r❡t❛ q✉❛♥❞♦ ❝♦♠♣❛r❛♠♦s ❞✐r❡t❛♠❡♥t❡ ❛ ❣r❛♥❞❡③❛❢ís✐❝❛ ❞❡ ✐♥t❡r❡ss❡ ❝♦♠ ♦ ♣❛❞rã♦ ♣ré✲❡st❛❜❡❧❡❝✐❞♦✳

▼❡❞✐❞❛ ✐♥❞✐r❡t❛✿ r❡❛❧✐③❛♠♦s ✉♠❛ ♠❡❞✐❞❛ ✐♥❞✐r❡t❛♠❡♥t❡ q✉❛♥❞♦ ❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ❞❡ ✐♥t❡r❡ss❡❢♦r ❞❡t❡r♠✐♥❛❞❛ ❛tr❛✈és ❞❡ ♦♣❡r❛çõ❡s ♠❛t❡♠át✐❝❛s q✉❡ ❡♥✈♦❧✈❛♠ ♦✉tr❛s ❣r❛♥❞❡③❛s ❢ís✐❝❛s✱♠❡❞✐❞❛s ❞✐r❡t❛♠❡♥t❡✳

❖s ❡①❡♠♣❧♦s ❛ s❡❣✉✐r ✐❧✉str❛♠ ❡st❡s ❞♦✐s ♣r♦❝❡❞✐♠❡♥t♦s✳

❊①❡♠♣❧♦ ✶✳ ❉❡s❡❥❛♠♦s ♠❡❞✐r ♦ ❝♦♠♣r✐♠❡♥t♦ ❡ ❛ ❧❛r❣✉r❛ ❞❡ ✉♠❛ ❢♦❧❤❛ ❡ ❛❞♦t❛r❡♠♦s ❝♦♠♦♣❛❞rã♦ ❞❡ ♠❡❞✐❞❛ ✉♠❛ ❝❛♥❡t❛✱ ❝♦♠♦ ♠♦str❛❞❛s ♥❛ ✜❣✉r❛ ✷✳✸✳✶✳ ❆ ♠❡❞✐❞❛ ❞✐r❡t❛ t❛♥t♦ ❞♦❝♦♠♣r✐♠❡♥t♦ q✉❛♥t♦ ❞❛ ❧❛r❣✉r❛ ❞❛ ❢♦❧❤❛ sã♦ ♦❜t✐❞❛s ❝♦♠♣❛r❛♥❞♦✲❛s ❞✐r❡t❛♠❡♥t❡ ❝♦♠ ♦ ♣❛❞rã♦♣ré✲❡st❛❜❡❧❡❝✐❞♦ ❡ ✈❡r✐✜❝❛♥❞♦✲s❡ q✉❛♥t❛s ✉♥✐❞❛❞❡s ❞❛ ♠❡❞✐❞❛ ♣❛❞rã♦ ❝❛❜❡♠ ♥♦ ❝♦♠♣r✐♠❡♥t♦ ❡♥❛ ❧❛r❣✉r❛ ❞❛ ❢♦❧❤❛✳

✸◆❛ r❡❛❧✐❞❛❞❡ ♦s át♦♠♦s ❝❤❡❣❛♠ ❛ t❡♠♣❡r❛t✉r❛s ❞❛ ♦r❞❡♠ ❞❡ 10−9

K✱ ♠❛s ♥ã♦ ♦ ③❡r♦ ❛❜s♦❧✉t♦✳

✷✳✸✳ ▼❊❉■❉❆❙ ❉■❘❊❚❆❙ ❊ ■◆❉■❘❊❚❆❙ ✶✸

❋✐❣✉r❛ ✷✳✸✳✶✿ ▼❡❞✐❞❛ ❞✐r❡t❛✳ ❖ ❝♦♠♣r✐♠❡♥t♦ ❡ ❛ ❧❛r❣✉r❛ ❞❛ ❢♦❧❤❛ sã♦ ♦❜t✐❞♦s ❝♦♠♣❛r❛♥❞♦✲♦s❞✐r❡t❛♠❡♥t❡ ❝♦♠ ♦ ♣❛❞rã♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❡s❝♦❧❤✐❞♦✱ ♥♦ ❝❛s♦✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❝❛♥❡t❛✳ ❈♦♠♦é ❢á❝✐❧ ❞❡ ✈❡r✐✜❝❛r✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❢♦❧❤❛ ❡stá ❡♥tr❡ ✉♠❛ ❡ ❞✉❛s ✉♥✐❞❛❞❡s✱ ❡♥q✉❛♥t♦ q✉❡ ❛❧❛r❣✉r❛ t❡♠ ✉♠ ❝♦♠♣r✐♠❡♥t♦ ♠❡♥♦r ❞♦ q✉❡ ✉♠❛ ✉♥✐❞❛❞❡✳

❊①❡♠♣❧♦ ✷✳ ❉❡s❡❥❛♠♦s ♠❡❞✐r ❛ ár❡❛ ❞❛ ♠❡s♠❛ ❢♦❧❤❛ ✉s❛❞❛ ♥♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✳ ❚❛❧ ♠❡❞✐❞❛♣♦❞❡ s❡r ♦❜t✐❞❛ ✐♥❞✐r❡t❛♠❡♥t❡ ♦✉ ❞✐r❡t❛♠❡♥t❡✳ ❚❡♥❞♦ ❡♠ ✈✐st❛ q✉❡ ♠❡❞✐♠♦s ♦ ❝♦♠♣r✐♠❡♥t♦ ❡❛ ❧❛r❣✉r❛ ❞❛ ❢♦❧❤❛✱ ♣❡❧♦ ♣r♦❞✉t♦ comprimento× largura ♦❜t❡♠♦s ❛ ♠❡❞✐❞❛ ✐♥❞✐r❡t❛ ❞❛ ár❡❛ ❞❛❢♦❧❤❛✱ ♣♦✐s ❡st❛ ❢♦✐ ♦❜t✐❞❛ ❛tr❛✈és ❞❛ ♦♣❡r❛çã♦ ♠❛t❡♠át✐❝❛ ♣r♦❞✉t♦ ❞❡ ❞✉❛s ♠❡❞✐❞❛s ❞✐r❡t❛s✳▼❛s ♣♦❞❡♠♦s ♠❡❞✐r ❛ ár❡❛ ❞✐r❡t❛♠❡♥t❡✱ ❜❛st❛ q✉❡ t❡♥❤❛♠♦s ✉♠ ♣❛❞rã♦ ❞❡ ár❡❛ ♣❛r❛ ❝♦♠♣❛rá✲❧♦ ❞✐r❡t❛♠❡♥t❡ ❝♦♠ ❛ ár❡❛ ❛ s❡r ♠❡❞✐❞❛ ❡ ✈❡r✐✜❝❛r q✉❛♥t❛s ✉♥✐❞❛❞❡s ❞❛ ár❡❛ ♣❛❞rã♦ ❝❛❜❡♠ ♥❛ár❡❛ ❛ s❡r ♠❡❞✐❞❛✳ ❆ ✜❣✉r❛ ✷✳✸✳✷ ♠♦str❛ ❛ ♠❡❞✐❞❛ ❞✐r❡t❛ ❞❛ ár❡❛ ❞❛ ❢♦❧❤❛ ❡ ❢♦✐ ✉s❛❞❛✱ ❝♦♠♦♣❛❞rã♦ ❞❡ ár❡❛✱ ❛ ár❡❛ ❞❡ ✉♠ ❞✐sq✉❡t❡ ❞❡ ❝♦♠♣✉t❛❞♦r✳

❋✐❣✉r❛ ✷✳✸✳✷✿ ▼❡❞✐❞❛ ❞✐r❡t❛ ❞❛ ár❡❛✳ ❚❡♥❞♦ ❡s❝♦❧❤✐❞♦ ♦ ♣❛❞rã♦ ❞❡ ár❡❛ ✭❞✐sq✉❡t❡ ❞❡ ❝♦♠♣✉t❛❞♦r✮✱♣♦❞❡♠♦s ❝♦♠♣❛rá✲❧♦ ❞✐r❡t❛♠❡♥t❡ ❝♦♠ ❛ ár❡❛ ❞❛ ❢♦❧❤❛✱ ❝♦♥t❛♥❞♦ q✉❛♥t❛s ✉♥✐❞❛❞❡s ❞♦ ♣❛❞rã♦❝♦rr❡s♣♦♥❞❡ ❛ ár❡❛ ❞❛ ❢♦❧❤❛✳ ❱❡r✐✜q✉❡ q✉❡ ❛ ár❡❛ ❞❛ ❢♦❧❤❛ ❡stá ♥✉♠❡r✐❝❛♠❡♥t❡ ♣ró①✐♠❛ ❞❛ ár❡❛❞❡ ❞❡③♦✐t♦ ❞✐sq✉❡t❡s✳

❖❜s❡r✈❡ q✉❡ ♥♦s ❞♦✐s ❡①❡♠♣❧♦s ❛❝✐♠❛✱ ♦s ✈❛❧♦r❡s ♠❡❞✐❞♦s ♥ã♦ ❝♦rr❡s♣♦♥❞❡r❛♠ ❛ ✈❛❧♦r❡s✐♥t❡✐r♦s ❞❛ ✉♥✐❞❛❞❡ ❡ ✏s♦❜r♦✉✑ ♣❛rt❡ ❞❛ ✉♥✐❞❛❞❡ ♣❛❞rã♦ ✉s❛❞❛ ♥❛s ♠❡❞✐❞❛s✱ t❛♥t♦ ❞❡ ❝♦♠♣r✐✲♠❡♥t♦ ❡ ❧❛r❣✉r❛ q✉❛♥t♦ ❞❡ ár❡❛✳ P❛r❛ ❡s♣❡❝✐✜❝❛r♠♦s ❝♦♠♣❧❡t❛♠❡♥t❡ ♦ ✈❛❧♦r ❞❛ ♠❡❞✐❞❛✱ ♦✉❡s♣❡❝✐✜❝❛r♠♦s ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❞❡✈❡♠♦s ❞✐✈✐❞✐r ♦ ♣❛❞rã♦ ❡♠ s✉❜♠ú❧t✐♣❧♦s ❡ ❡st✐♠❛r ❛ ❢r❛çã♦

✶✹ ❈❆P❮❚❯▲❖ ✷✳ ▼❊❉■❉❆❙ ❊ ❆▲●❆❘■❙▼❖❙ ❙■●◆■❋■❈❆❚■❱❖❙

❞❛ ✉♥✐❞❛❞❡ ✉s❛❞❛ ♣❛r❛ ❝♦♠♣❧❡t❛r ♦ ✈❛❧♦r ❞❛ ♠❡❞✐❞❛✳ ❊♠ ❣❡r❛❧✱ ❢❛③❡♠♦s ✉♠❛ ❞✐✈✐sã♦ ❡♠ ❞❡③♣❛rt❡s ✐❣✉❛✐s ♣❛r❛ ❝♦♥t❛❜✐❧✐③❛r♠♦s ♦s ❞é❝✐♠♦s ❞❛ ✉♥✐❞❛❞❡ q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ♥✉♠❛ ♠❡❞✐❞❛✳

◆♦ ❝❛s♦ ❞❛ ♠❡❞✐❞❛ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❢♦❧❤❛✱ ✈❡r✐✜❝❛✲s❡ ❢❛❝✐❧♠❡♥t❡ q✉❡ ❡st❡ é ♠❛✐♦r ❞♦ q✉❡✉♠❛ ✉♥✐❞❛❞❡ ❡ ♠❡♥♦r ❞♦ q✉❡ ❞✉❛s ✉♥✐❞❛❞❡s✳ ❙❡ ✐♠❛❣✐♥❛r♠♦s ✉♠❛ s✉❜❞✐✈✐sã♦ ❞❛ ❝❛♥❡t❛ ❡♠ ❞❡③♣❛rt❡s ✐❣✉❛✐s ❛ ♠❡❞✐❞❛ ♣♦❞❡rá s❡r ❡s❝r✐t❛ ❝♦♠♦ 1, 7 ♦✉ 1, 8 ✉♥✐❞❛❞❡s✳ ❉❡ ♠♦❞♦ s❡♠❡❧❤❛♥t❡✱ ❛♠❡❞✐❞❛ ❞❛ ❧❛r❣✉r❛ ♣♦❞❡ s❡r 0, 8 ♦✉ 0, 9✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ♠❡❞✐❞❛ ✐♥❞✐r❡t❛ ❞❛ ár❡❛ ❞❛ ❢♦❧❤❛ é♦ ♣r♦❞✉t♦✹ 1, 7×0, 8 = 1, 36 ♦✉ 1, 7×0, 9 = 1, 53 ♦✉ 1, 8×0, 8 = 1, 44 ♦✉ ❛✐♥❞❛ 1, 8×0, 9 = 1, 62✳P♦r ♦✉tr♦ ❧❛❞♦✱ ❛ ♠❡❞✐❞❛ ❞✐r❡t❛ ❞❛ ár❡❛ ❞❛ ❢♦❧❤❛ ✉t✐❧✐③❛♥❞♦ ❛ ár❡❛ ❞♦ ❞✐sq✉❡t❡ ❝♦♠♦ ✉♥✐❞❛❞❡ ❞❡ár❡❛ ❡stá ❡♥tr❡ 17 ❡ 18 ❞✐sq✉❡t❡s ❡ ♣♦❞❡rí❛♠♦s ❡s❝r❡✈❡r 17, 2 ❞✐sq✉❡t❡s✳ P♦rt❛♥t♦✱ ❛ ♠❡s♠❛ ár❡❛❞❛ ❢♦❧❤❛ ♣♦❞❡ s❡r ❡①♣r❡ss❛ t❛♥t♦ ❡♠ ✏✉♥✐❞❛❞❡s ❞❡ ❝❛♥❡t❛ ❛♦ q✉❛❞r❛❞♦✑ q✉❛♥t♦ ❡♠ ✏✉♥✐❞❛❞❡s❞❡ ❞✐sq✉❡t❡✑✳ ❊♠ q✉❛❧q✉❡r ❞♦s ❝❛s♦s ♣♦❞❡✲s❡✱ ❡✈✐❞❡♥t❡♠❡♥t❡✱ ❝♦♥✈❡rt❡r ❛♠❜❛s ❡♠ ✉♥✐❞❛❞❡s ❞♦❙✐st❡♠❛ ■♥t❡r♥❛❝✐♦♥❛❧✱ ♦✉ s❡❥❛✱ ❡♠ ♠❡tr♦s ❡ ❡♠ ♠❡tr♦s q✉❛❞r❛❞♦s✳

❊st❡ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡ ❡st✐♠❛r ❛ ♣❛rt❡ ❞❡❝✐♠❛❧ ❞❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ♥♦s ❧❡✈❛ ❛♦s ❝♦♥❝❡✐t♦s❞❡ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s ❞❡ ✉♠❛ ♠❡❞✐❞❛ ❡ ❛❧❣❛r✐s♠♦ ❞✉✈✐❞♦s♦✳

✷✳✹ ❆❧❣❛r✐s♠♦s ❙✐❣♥✐✜❝❛t✐✈♦s

❱✐♠♦s ❛♥t❡r✐♦r♠❡♥t❡ q✉❡ ♦ ❛t♦ ❞❡ ♠❡❞✐r ✉♠❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ♥❛❞❛ ♠❛✐s é ❞♦ q✉❡ ❝♦♠♣❛rá✲❧❛ ❝♦♠ ❛❧❣✉♠ ♣❛❞rã♦✳ ◆♦s ❡①❡♠♣❧♦s ✶ ❡ ✷ ❛❝✐♠❛ ❡♠ q✉❡ ❛s ❞✐♠❡♥sõ❡s ❞❡ ✉♠❛ ❢♦❧❤❛ ❢♦r❛♠♠❡❞✐❞❛s ❝♦♠ ✉♠❛ ❝❛♥❡t❛✱ ✈❡r✐✜❝♦✉✲s❡ q✉❡ t❡♠♦s ❛ ❝❡rt❡③❛ ❞❡ q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ é ♠❛✐♦r ❞♦q✉❡ ♦ ❞❡ ✉♠❛ ❝❛♥❡t❛ ❡ ♠❡♥♦r ❞♦ q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❞✉❛s ❝❛♥❡t❛s✱ ♦ q✉❡ ♥♦s ❧❡✈♦✉ ❛ ✐♠❛❣✐♥❛r✉♠❛ s✉❜❞✐✈✐sã♦ ❞❛ ❝❛♥❡t❛ ❡♠ ❞❡③ ♣❛rt❡s ✐❣✉❛✐s✱ ❧❡✈❛♥❞♦ ❛♦ ✈❛❧♦r 1, 7 ♦✉ 1, 8✳ ❖❜s❡r✈❡ ❡♥tã♦q✉❡ ❡st❛ ♣❛rt❡ ❞❡❝✐♠❛❧ ♥❡❝❡ss✐t❛ s❡r ❡st✐♠❛❞❛✱ ♦✉ s❡❥❛✱ ❛✈❛❧✐❛❞❛✱ ❡ ♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛❧❣❛r✐s♠♦é ❝❤❛♠❛❞♦ ❞❡ ❛❧❣❛r✐s♠♦ ❞✉✈✐❞♦s♦✳ P❛r❛ ♦ ♣r❡s❡♥t❡ ❝❛s♦✱ ❛ ♠❡❞✐❞❛ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❢♦❧❤❛❛♣r❡s❡♥t❛ ❞♦✐s ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s✿ ✉♠ q✉❡ t❡♠♦s ❝❡rt❡③❛ ❡ ♦✉tr♦ q✉❡ é ❞✉✈✐❞♦s♦✳ ❙❡t❡♠♦s ❞ú✈✐❞❛ ♥♦ ❛❧❣❛r✐s♠♦ ❞❛ ♣r✐♠❡✐r❛ ❝❛s❛ ❞❡❝✐♠❛❧✱ t❡r❡♠♦s ♠❛✐s ❞ú✈✐❞❛ ❛✐♥❞❛ ♥❛ s❡❣✉♥❞❛❝❛s❛ ❞❡❝✐♠❛❧✳ P♦r ✐ss♦✱ ❛ ♠❡❞✐❞❛ 1, 75 ♦✉ 1, 78 ♥ã♦ ❢❛③ s❡♥t✐❞♦❀ ♥ã♦ ♣♦❞❡♠♦s✱ ♥❡st❡ ❝❛s♦✱ ✐♥❞✐❝❛r♦ ✈❛❧♦r ❞❛ ❣r❛♥❞❡③❛ ❝♦♠ ♣r❡❝✐sã♦ ❞❡ ❝❡♥tés✐♠♦s ❞❡ ✉♥✐❞❛❞❡✳

❆❧❣❛r✐s♠♦ ❞✉✈✐❞♦s♦✿ é ❛q✉❡❧❡ ❛❧❣❛r✐s♠♦ ❞❛ ♠❡❞✐❞❛ q✉❡ ♣r❡❝✐s❛ s❡r ❛✈❛❧✐❛❞♦ ❡ t♦❞❛ ♠❡❞✐❞❛♣♦❞❡ t❡r✱ ♥♦ ♠á①✐♠♦✱ ✉♠ ❛❧❣❛r✐s♠♦ ❞✉✈✐❞♦s♦✳

❆❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s✿ ♦s ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s ❞❡ ✉♠❛ ♠❡❞✐❞❛ sã♦ t♦❞♦s ♦s ❛❧❣❛r✐s✲♠♦s q✉❡ t❡♠♦s ❝❡rt❡③❛ ♠❛✐s ♦ ❛❧❣❛r✐s♠♦ ❞✉✈✐❞♦s♦✳ ❙❡❣✉❡✱ ❡♥tã♦✱ q✉❡ ♦ ❛❧❣❛r✐s♠♦ ❞✉✈✐❞♦s♦❞❡ ✉♠❛ ♠❡❞✐❞❛ t❛♠❜é♠ é ✉♠ ❛❧❣❛r✐s♠♦ s✐❣♥✐✜❝❛t✐✈♦✳

❙❡ ♠❡❞✐r♠♦s ❛ ♠❡s♠❛ ❢♦❧❤❛ ❝♦♠ ❞✐❢❡r❡♥t❡s ré❣✉❛s✱ ♣♦❞❡r❡♠♦s ♦❜t❡r ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ♣❛r❛ ❛♠❡❞✐❞❛✱ ♦✉ s❡❥❛✱ ❞✐❢❡r❡♥t❡s q✉❛♥t✐❞❛❞❡s ❞❡ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s✱ ♣♦rq✉❡ ❛s ❞✐❢❡r❡♥t❡s ré❣✉❛s♣♦❞❡rã♦ ❛♣r❡s❡♥t❛r ❡s❝❛❧❛s ❞✐❢❡r❡♥t❡s✳ ❯♠ ré❣✉❛ ♣♦❞❡rá ❛♣r❡s❡♥t❛r ♦ ❝❡♥tí♠❡tr♦ ❝♦♠♦ ❛ ♠❡♥♦r❞✐✈✐sã♦ ❞❡ ❡s❝❛❧❛✱ ❡♥q✉❛♥t♦ ♦✉tr❛ ♣♦❞❡rá ❛♣r❡s❡♥t❛r ♦ ♠✐❧í♠❡tr♦ ❝♦♠♦ ❛ ♠❡♥♦r ❞✐✈✐sã♦ ❞❡❡s❝❛❧❛✳

❆ ♠❡❞✐❞❛ r❡❛❧✐③❛❞❛ ❝♦♠ ❛ ré❣✉❛ ❝❡♥t✐♠❡tr❛❞❛ ♣❡r♠✐t❡ q✉❡ ❡st✐♠❡♠♦s ♦ ❞é❝✐♠♦ ❞❡ ❝❡♥✲tí♠❡tr♦✱ ❡♥q✉❛♥t♦ q✉❡ ❛ ré❣✉❛ ♠✐❧✐♠❡tr❛❞❛ ♣❡r♠✐t❡ ❛ ❡st✐♠❛t✐✈❛ ❞♦ ❞é❝✐♠♦ ❞❡ ♠✐❧í♠❡tr♦✳ ➱❡✈✐❞❡♥t❡ q✉❡ ♦ ❛✉♠❡♥t♦ ❞❛ ♣r❡❝✐sã♦ ❞♦ ❛♣❛r❡❧❤♦ ❞❡ ♠❡❞✐❞❛ ❛✉♠❡♥t❛ ♦ ♥ú♠❡r♦ ❞❡ ❛❧❣❛r✐s♠♦ss✐❣♥✐✜❝❛t✐✈♦s ❞❡ ✉♠❛ ♠❡❞✐❞❛ ♠❛s ♥ã♦ ❛✉♠❡♥t❛ ♦ ♥ú♠❡r♦ ❞❡ ❛❧❣❛r✐s♠♦s ❞✉✈✐❞♦s♦s✳ ❊❧❡ ❛✉♠❡♥t❛❛ ♣r❡❝✐sã♦✳

■♠❛❣✐♥❡ ❛❣♦r❛ q✉❡ ✉♥❛♠♦s ❞✉❛s ❢♦❧❤❛s ❡ q✉❡r❡♠♦s s❛❜❡r ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛s ❞✉❛s ❥✉♥t❛s✳➱ ó❜✈✐♦ q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ t♦t❛❧ s❡rá ❞❛❞♦ ♣❡❧❛ s♦♠❛ ❞♦s ❞♦✐s ❝♦♠♣r✐♠❡♥t♦s✳ ◆♦ ❡♥t❛♥t♦✱

✹P♦st❡r✐♦r♠❡♥t❡ ✈❡r❡♠♦s ♦s ❝r✐tér✐♦s ❞❡ ❛rr❡❞♦♥❞❛♠❡♥t♦ ❡ ♦♣❡r❛çã♦ ❝♦♠ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s✳

✷✳✹✳ ❆▲●❆❘■❙▼❖❙ ❙■●◆■❋■❈❆❚■❱❖❙ ✶✺

✜s✐❝❛♠❡♥t❡ ♥ã♦ s♦♠❛♠♦s ♥ú♠❡r♦s✱ s♦♠❛♠♦s ❣r❛♥❞❡③❛s ❢ís✐❝❛s✱ ❞❡ ♠❡s♠❛ ❞✐♠❡♥sã♦✱ ❞❡s❝r✐t❛s♣♦r ♥ú♠❡r♦s q✉❡ sã♦ ♦❜t✐❞♦s ❛tr❛✈és ❞❡ ✉♠ ♣r♦❝❡ss♦ ❞❡ ♠❡❞✐❞❛✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡s♦♠❛r ✏♥ú♠❡r♦s ♠❛t❡♠át✐❝♦s✑ é ❞✐❢❡r❡♥t❡ ❞❡ s♦♠❛r ✏♥ú♠❡r♦s ❝♦♠ s✐❣♥✐✜❝❛❞♦ ❢ís✐❝♦✑✳ P♦r ❡①✲❡♠♣❧♦✱ s✉♣♦♥❤❛ q✉❡ ✉♠ ❞♦s ❝♦♠♣r✐♠❡♥t♦s t❡♥❤❛ s✐❞♦ ♠❡❞✐❞♦ ❝♦♠ ❛ ré❣✉❛ ❝❡♥t✐♠❡tr❛❞❛ ❡ q✉❡♦ ♦✉tr♦ ❝♦♠♣r✐♠❡♥t♦ t❡♥❤❛ s✐❞♦ ♠❡❞✐❞♦ ❝♦♠ ❛ ré❣✉❛ ♠✐❧✐♠❡tr❛❞❛✳ ❉✐❣❛♠♦s q✉❡ ❛s ♠❡❞✐❞❛ss❡❥❛♠ 5, 3 cm ❡ 53, 7mm✱ ♦✉ s❡❥❛✱ 5, 37 cm✳ ◗✉❛❧ ♦ ❝♦♠♣r✐♠❡♥t♦ t♦t❛❧❄ Pr✐♠❡✐r❛♠❡♥t❡ ❞❡✈❡♠♦s❝♦❧♦❝❛r t♦❞❛s ❛s ♠❡❞✐❞❛s ♥❛ ♠❡s♠❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ♣❛r❛ ❡❢❡t✉❛r♠♦s ❛ s♦♠❛✳ ❖ r❡s✉❧t❛❞♦✭♠❛t❡♠át✐❝♦✮ é 10, 67 cm✳ ▼❛s ❧❡♠❜r❡♠♦✲♥♦s ❞❡ q✉❡ ❡st❡s ♥ú♠❡r♦s ♣♦ss✉❡♠ ✉♠ s✐❣♥✐✜❝❛❞♦❢ís✐❝♦ ❡ ❢♦r❛♠ ♦❜t✐❞♦s ❝♦♠ ♦ ✉s♦ ❞❡ ❞✐❢❡r❡♥t❡s ✐♥str✉♠❡♥t♦s ❞❡ ♠❡❞✐❞❛s✳ P❡❧♦ ❢❛t♦ ❞❛s ❞✉❛s♠❡❞✐❞❛s ❛♣r❡s❡♥t❛r❡♠✱ ❝❛❞❛ ✉♠❛✱ ✉♠ ❛❧❣❛r✐s♠♦ ❞✉✈✐❞♦s♦✱ é ❝♦rr❡t♦ ❛✜r♠❛r q✉❡ ❛ s♦♠❛ t❛♠✲❜é♠ ❞❡✈❛ ❛♣r❡s❡♥t❛r ✉♠ ❛❧❣❛r✐s♠♦ ❞✉✈✐❞♦s♦ ♣♦✐s✱ s❡ ✉s❛r♠♦s q✉❛❧q✉❡r ✉♠❛ ❞❛s ❞✉❛s ré❣✉❛s♣❛r❛ ♠❡❞✐r ♦ ❝♦♠♣r✐♠❡♥t♦ t♦t❛❧✱ t❛❧ ♠❡❞✐❞❛ ❞❡✈❡rá ♣♦ss✉✐r ❛♣❡♥❛s ✉♠ ❛❧❣❛r✐s♠♦ ❞✉✈✐❞♦s♦✳❆ ♣r✐♠❡✐r❛ ❞❛s ♠❡❞✐❞❛s ❛❝✐♠❛ ♣♦❞❡r✐❛ t❡r s✐❞♦ 5, 4 cm✱ ♦ q✉❡ ❧❡✈❛r✐❛ ❛♦ r❡s✉❧t❛❞♦ 10, 77 cm✳❆ s❡❣✉♥❞❛ ♠❡❞✐❞❛ ♣♦❞❡r✐❛ t❡r s✐❞♦ 5, 36 cm✱ ❧❡✈❛♥❞♦ ❛♦ r❡s✉❧t❛❞♦ 10, 66 cm✳ ➱ ❢á❝✐❧ ✈❡r✐✜❝❛rq✉❡✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞♦ ✈❛❧♦r ❡st✐♠❛❞♦ ♣❛r❛ ♦ ❞é❝✐♠♦ ❞❡ ♠✐❧í♠❡tr♦ ❛♦ s❡r ✉s❛❞❛ ❛ ré❣✉❛♠✐❧✐♠❡tr❛❞❛✱ ♦ r❡s✉❧t❛❞♦ ❞❛ s♦♠❛ ❛♣r❡s❡♥t❛ ✉♠ ❛❧❣❛r✐s♠♦ ❞✉✈✐❞♦s♦ ♥❛ ❝❛s❛ ❞♦s ❞é❝✐♠♦s ❞❡ ❝❡♥✲tí♠❡tr♦ ❛❧é♠ ❞♦ ❞✉✈✐❞♦s♦ ♥❛ ❝❛s❛ ❞♦s ❞é❝✐♠♦s ❞❡ ♠✐❧í♠❡tr♦✳ P❡❧♦ ❢❛t♦ ❞❡ ✉♠❛ ♠❡❞✐❞❛ ♣♦❞❡r❛♣r❡s❡♥t❛r ❛♣❡♥❛s ✉♠ ❛❧❣❛r✐s♠♦ ❞✉✈✐❞♦s♦ ❡✱ s❡♥❞♦ ❛ ré❣✉❛ ❝❡♥t✐♠❡tr❛❞❛ ♠❡♥♦s ♣r❡❝✐s❛ ❞♦ q✉❡❛ ré❣✉❛ ♠✐❧✐♠❡tr❛❞❛✱ ❞❡✈❡♠♦s ❛♣r♦①✐♠❛r ♦ r❡s✉❧t❛❞♦ ♥♦ ❛❧❣❛r✐s♠♦ s✐❣♥✐✜❝❛t✐✈♦ ❝♦rr❡s♣♦♥❞❡♥t❡❛♦ ❞é❝✐♠♦ ❞❡ ❝❡♥tí♠❡tr♦✳ ❊♥tã♦ ♥❡❝❡ss✐t❛♠♦s ❞❡ ✉♠ ♦✉ ♠❛✐s ❝r✐tér✐♦s ❞❡ ❛♣r♦①✐♠❛çã♦✱ t❛♠❜é♠❝❤❛♠❛❞♦s ❞❡ ❝r✐tér✐♦s ❞❡ ❛rr❡❞♦♥❞❛♠❡♥t♦✱ ♣❛r❛ ♦s ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s ❞❡ ✉♠❛ ♠❡❞✐❞❛✳

❯♠ ♦✉tr♦ ❡①❡♠♣❧♦ é q✉❛♥❞♦ ❛ ❣r❛♥❞❡③❛ é ❞❡t❡r♠✐♥❛❞❛ ❛tr❛✈és ❞❛s ♦♣❡r❛çõ❡s ❞❡ ♣r♦✲❞✉t♦ ♦✉ ❞✐✈✐sã♦✳ ❱❛♠♦s s✉♣♦r q✉❡ ❛s ♠❡❞✐❞❛s ❞❡ ✉♠❛ ❢♦❧❤❛ s❡❥❛♠ 53, 3 cm ❡ 234, 5mm✱ ♦✉s❡❥❛✱ 23, 45 cm✳ ❈♦♥s✐❞❡r❡♠♦s ❛ ár❡❛ ❞❛ ❢♦❧❤❛ ♥♦✈❛♠❡♥t❡✳ ▼❛t❡♠❛t✐❝❛♠❡♥t❡✱ ❛ ár❡❛ s❡r✐❛1.249, 885 cm2✳ ▼❛s✱ ♣❡❧❛ ✐♥❝❡rt❡③❛ q✉❡ t❡♠♦s ♥♦ ú❧t✐♠♦ ❛❧❣❛r✐s♠♦ ❞❡ ❝❛❞❛ ♠❡❞✐❞❛✱ ♦s ✈❛❧✲♦r❡s ♣♦❞❡r✐❛♠ s❡r ❞✐❢❡r❡♥t❡s✳ ❈❛s♦ ❛s ♠❡❞✐❞❛s ❢♦ss❡♠ 53, 2 cm ❡ 234, 5mm✱ ♦ r❡s✉❧t❛❞♦ s❡r✐❛1.247, 54 cm2✳ ❖❜s❡r✈❡ ♦s r❡s✉❧t❛❞♦s ❛❜❛✐①♦ ♥♦s q✉❛✐s ♦s ❛❧❣❛r✐s♠♦s s✉❜❧✐♥❤❛❞♦s r❡♣r❡s❡♥t❛♠♦s ❛❧❣❛r✐s♠♦s ❞✉✈✐❞♦s♦s✳

2 3, 4 5× 5 3, 3

7 0 3 5+ 7 0 3 51 1 7 2 51 2 4 9, 8 8 5

2 3, 4 5× 5 3, 2

4 6 9 0+ 7 0 3 51 1 7 2 51 2 4 7, 5 4 0

2 3, 4 3× 5 3, 3

7 0 2 9+ 7 0 2 91 1 7 1 51 2 4 8, 8 1 9

2 3, 4 3× 5 3, 2

4 6 8 6+ 7 0 2 91 1 7 1 51 2 4 6, 4 7 6

❖❜s❡r✈❡♠♦s q✉❡ ♥❡st❡s ❝❛s♦s✱ ❛ ❛❧t❡r❛çã♦ ♥♦ ❛❧❣❛r✐s♠♦ ❞✉✈✐❞♦ ❧❡✈❛ ❛ ❛❧t❡r❛çõ❡s ♥♦s ❛❧❣❛r✐s✲♠♦s q✉❡ ❡stã♦ ✏❞❡♥tr♦ ❞❛ ❝♦♥t❛✑ ♥ã♦ ❛❢❡t❛♥❞♦ ♦s ❞♦✐s ♣r✐♠❡✐r♦s ❛❧❣❛r✐s♠♦s ❞❛ r❡s♣♦st❛✳ ❆ss✐♠✱♦ r❡s✉❧t❛❞♦ ❞❡✈❡ ❛♣r❡s❡♥t❛r ❛♣❡♥❛s três ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s ✭❞♦✐s q✉❡ t❡♠♦s ❝❡rt❡③❛ ♠❛✐s♦ ❞✉✈✐❞♦s♦✮✳ ❊st❡ ❝❛s♦ t❛♠❜é♠ r❡q✉❡r ❝r✐tér✐♦s ❞❡ ❛rr❡❞♦♥❞❛♠❡♥t♦ ♣♦rq✉❡ t♦❞♦s ♦s r❡s✉❧t❛✲❞♦s sã♦ ❞❛ ❢♦r♠❛ 124 · · · s❡♥❞♦ q✉❡ ❛s r❡t✐❝ê♥❝✐❛s ❝♦rr❡s♣♦♥❞❡♠ ❛♦s ❛❧❣❛r✐s♠♦s q✉❡ ❞❡✈❡♠ s❡r❞❡s❝❛rt❛❞♦s ❡ ❡♠ t♦❞♦s ♦s ❝❛s♦s ❡❧❡s sã♦ ❞✐❢❡r❡♥t❡s✳ ➱ ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡ ♦s r❡s✉❧t❛❞♦s ❛❝✐♠❛

✶✻ ❈❆P❮❚❯▲❖ ✷✳ ▼❊❉■❉❆❙ ❊ ❆▲●❆❘■❙▼❖❙ ❙■●◆■❋■❈❆❚■❱❖❙

❛♣r❡s❡♥t❛rã♦ ❛♣❡♥❛s três ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s✱ ❝♦✐♥❝✐❞✐♥❞♦ ❝♦♠ ♦ ♥ú♠❡r♦ ❞❡ ❛❧❣❛r✐s♠♦ss✐❣♥✐✜❝❛t✐✈♦s ❞❛ ♠❡❞✐❞❛ ❝♦♠ ♠❡♥♦r q✉❛♥t✐❞❛❞❡ ❞❡ s✐❣♥✐✜❝❛t✐✈♦s✳

❊♠ ♠✉✐t❛s s✐t✉❛çõ❡s ❢❛③✲s❡ ♥❡❝❡ssár✐❛ ❛ ❝♦♥✈❡rsã♦ ❞❡ ✉♥✐❞❛❞❡s✳ ◆♦s ❝❛s♦s ❛❝✐♠❛✱ ❛ ár❡❛❡stá ♠❡❞✐❞❛ ❡♠ cm2✱ s❡♥❞♦ q✉❡ ❛ ♠❡s♠❛ ár❡❛ ♣♦❞❡r✐❛ s❡r ❡①♣r❡ss❛ ❡♠ ♣♦❧❡❣❛❞❛ q✉❛❞r❛❞❛ ♦✉❡♠ ♠✐❧í♠❡tr♦s q✉❛❞r❛❞♦s✱ ♣♦r ❡①❡♠♣❧♦✳ P❛r❛ q✉❡ ❛ ♠❡❞✐❞❛ s❡❥❛ ❡s❝r✐t❛ ❡♠ ♠❡tr♦s q✉❛❞r❛❞♦s✱♣♦r ❡①❡♠♣❧♦✱ ❧❡♠❜r❛♠♦s q✉❡ 1m = 1000mm✱ ❡♥tã♦ 1m2 = 1m× 1m = 1000mm× 1000mm✱ ♦✉s❡❥❛✱ 1m2 = 1× 106mm2 ♦✉ 1mm2 = 1× 10−6m2✳ ❆ss✐♠✱ ♥❡❝❡ss✐t❛♠♦s t❛♠❜é♠ ❞❡ r❡❣r❛s ♣❛r❛❛ ♠✉❞❛♥ç❛ ❞❡ ✉♥✐❞❛❞❡ ❡ ♣❛r❛ ❡s❝r❡✈❡r ✉♠❛ ♠❡❞✐❞❛ ✉s❛♥❞♦ ❛ ♣♦tê♥❝✐❛ ❞❡ 10✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦♥♦t❛çã♦ ❝✐❡♥tí✜❝❛✳

✷✳✹✳✶ ▼✉❞❛♥ç❛ ❞❡ ✉♥✐❞❛❞❡ ❡ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛

❚❡♠♦s ❞✉❛s r❡❣r❛s ❜ás✐❝❛s ♣❛r❛ ♠✉❞❛♥ç❛ ❞❡ ✉♥✐❞❛❞❡ ❡ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛✳

✶➟ r❡❣r❛✿ ❛ ♠✉❞❛♥ç❛ ❞❡ ✉♥✐❞❛❞❡ ♣r❡s❡r✈❛ ♦ ♥ú♠❡r♦ ❞❡ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s ❞❡ ✉♠❛ ♠❡✲❞✐❞❛✳

✷➟ r❡❣r❛✿ ✉♠❛ ♠❡❞✐❞❛ ❞❡✈❡ ❛♣r❡s❡♥t❛r ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s✱ q✉❛♥❞♦❡s❝r✐t❛ ♥❛ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛✳

❆ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛ é ✉t✐❧✐③❛❞❛ q✉❛♥❞♦ ♣r❡❝✐s❛♠♦s ❡s♣❡❝✐✜❝❛r ♠❡❞✐❞❛s ✏♠✉✐t♦ ❣r❛♥❞❡s✑ ♦✉✏♠✉✐t♦ ♣❡q✉❡♥❛s✑✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♦r❞❡♠ ❞❡ ❣r❛♥❞❡③❛ ❞❛ ♠❡❞✐❞❛✱ q✉❛♥❞♦ ❝♦♠♣❛r❛❞❛ ❝♦♠ ❛✉♥✐❞❛❞❡✳ ❆ ♠❛ss❛ ❞❛ ❚❡rr❛ ✭∼ 1024 kg✮✱ ❛ t❡♠♣❡r❛t✉r❛ ♠é❞✐❛ ♥♦ ✐♥t❡r✐♦r ❞♦ ❙♦❧ ✭∼ 106 K✮ ❡ ♦t❛♠❛♥❤♦ ❞❡ ✉♠❛ ❜❛❝tér✐❛ ✭∼ 10−6 m✮ sã♦ ❡①❡♠♣❧♦s✳ ❯♠❛ ♠❡❞✐❞❛ ❡s❝r✐t❛ ♥❛ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛❞❡✈❡ ❛♣r❡s❡♥t❛r ❛♣❡♥❛s ✉♠ ❛❧❣❛r✐s♠♦ ❛♥t❡s ❞❛ ✈ír❣✉❧❛✱ s❡❣✉✐❞❛ ❞♦s ❞❡♠❛✐s ❛❧❣❛r✐s♠♦s ❡ ❛♣♦tê♥❝✐❛ ❞❡ ❞❡③ ❝♦rr❡s♣♦♥❞❡♥t❡✱ s❡♥❞♦ q✉❡ ♦ ♣r✐♠❡✐r♦ ❛❧❣❛r✐s♠♦ ❞❡✈❡ s❡r ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ ✉♠❡ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ ♥♦✈❡✳ ❆ ❞✐stâ♥❝✐❛ ❚❡rr❛✲❙♦❧ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦ 1, 50× 108 km ❡ ❛ ♠❛ss❛❞♦ ❡❧étr♦♥ ❝♦♠♦ 9, 1× 10−31 kg✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡st❛s ❞✉❛s r❡❣r❛s ♦❜t❡♠♦s ❛

✸➟ r❡❣r❛✿ ③❡r♦s à ❡sq✉❡r❞❛ ♥ã♦ sã♦ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s ✭❡❧❡s ❛♣❡♥❛s ♣♦s✐❝✐♦♥❛♠ ❛ ✈ír❣✉❧❛✮❡ ③❡r♦s à ❞✐r❡✐t❛ sã♦ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s✳

❉❡ ❢❛t♦✱ ♦s ③❡r♦s à ❞✐r❡✐t❛ sã♦ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s ♣♦rq✉❡ ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ❛ ♣r❡❝✐sã♦ ❞♦✐♥str✉♠❡♥t♦ ✉t✐❧✐③❛❞♦❀ ✜s✐❝❛♠❡♥t❡ 5, 0m é ❞✐❢❡r❡♥t❡ ❞❡ 5, 000m✳ ❈♦♠♦ ❡①❡♠♣❧♦ 0, 036 kg ♣♦ss✉✐❞♦✐s ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s ❡ 0, 03600 kg ♣♦ss✉✐ q✉❛tr♦ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s✳ ❊s❝r✐t♦s ♥❛♥♦t❛çã♦ ❝✐❡♥tí✜❝❛ ❡❧❡s ✜❝❛♠ 3, 6× 10−2kg ❡ 3, 600× 10−2kg✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❊♠ ✈ár✐❛s s✐t✉❛çõ❡s ❛ ♣♦tê♥❝✐❛ ♣♦❞❡rá s❡r ♣♦s✐t✐✈❛ ♦✉ ♥❡❣❛t✐✈❛✱ t❛♠❜é♠ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛♦r❞❡♠ ❞❡ ❣r❛♥❞❡③❛ ❞❛ ♠❡❞✐❞❛✱ ❞❡ ♠♦❞♦ q✉❡ ♣r❡✜①♦s sã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ❢❛❝✐❧✐t❛r ❛ ♥♦t❛çã♦✳❙❛❜❡♠♦s q✉❡ ♦ ♠✐❧í♠❡tr♦ é ❛ ♠✐❧és✐♠❛ ♣❛rt❡ ❞♦ ♠❡tr♦✱ r❡♣r❡s❡♥t❛❞♦ ♣♦rmm✱ ❡ q✉❡ ♦ q✉✐❧ô♠❡tr♦❝♦rr❡s♣♦♥❞❡ ❛ ♠✐❧ ♠❡tr♦s✱ s❡♥❞♦ r❡♣r❡s❡♥t❛❞♦ ♣♦r km✱ ❛ss✐♠ ❝♦♠♦ ♦ q✉✐❧♦❣r❛♠❛ ❝♦rr❡s♣♦♥❞❡❛ ♠✐❧ ❣r❛♠❛s ❡ é r❡♣r❡s❡♥t❛❞♦ ♣♦r kg✳ ❊♠ ❛♠❜♦s ♦s ❝❛s♦s✱ ♦ ♣r❡✜①♦ ❝♦rr❡s♣♦♥❞❡♥t❡ é ✏q✉✐❧♦✑❡ ✈❛❧❡ ♠✐❧ ✈❡③❡s ❛ ✉♥✐❞❛❞❡✳ ❆ t❛❜❡❧❛ ✷✳✶ ❛♣r❡s❡♥t❛ ♦s ♣r❡✜①♦s r❡❝♦♠❡♥❞❛❞♦s ♣❡❧♦ ❇✉r❡❛✉■♥t❡r♥❛❝✐♦♥❛❧ ❞❡ P❡s♦s ❡ ▼❡❞✐❞❛s✳

✷✳✹✳ ❆▲●❆❘■❙▼❖❙ ❙■●◆■❋■❈❆❚■❱❖❙ ✶✼

❙✉❜♠ú❧t✐♣❧♦s❖r❞❡♠ ❞❡ ❣r❛♥❞❡③❛ Pr❡✜①♦ ❆❜r❡✈✐❛t✉r❛

10−18 ❛tt♦ ❛10−15 ❢❡♠t♦ ❢10−12 ♣✐❝♦ ♣10−9 ♥❛♥♦ ♥10−6 ♠✐❝r♦ µ10−3 ♠✐❧✐ ♠10−2 ❝❡♥t✐ ❝10−1 ❞❡❝✐ ❞

▼ú❧t✐♣❧♦s❖r❞❡♠ ❞❡ ❣r❛♥❞❡③❛ Pr❡✜①♦ ❆❜r❡✈✐❛t✉r❛

101 ❞❡❝❛ ❞❛102 ❤❡❝t♦ ❤103 q✉✐❧♦ ❦106 ♠❡❣❛ ▼109 ❣✐❣❛ ●1012 t❡r❛ ❚1015 ♣❡t❛ P1018 ❡①❛ ❊

❚❛❜❡❧❛ ✷✳✶✿ Pr❡✜①♦s r❡❝♦♠❡♥❞❛❞♦s ♣❡❧♦ ❇✉r❡❛✉ ■♥t❡r♥❛❝✐♦♥❛❧ ❞❡ P❡s♦s ❡ ▼❡❞✐❞❛s✳

✷✳✹✳✷ ❈r✐tér✐♦s ❞❡ ❆♣r♦①✐♠❛çõ❡s

❆❞✐çã♦ ❡ ❙✉❜tr❛çã♦

◆❛ ❛❞✐çã♦ ❡ ♥❛ s✉❜tr❛çã♦ ♦♣❡r❛♠♦s ♥♦r♠❛❧♠❡♥t❡ ❛s ❣r❛♥❞❡③❛s ❢ís✐❝❛s✱ ❞❡s❞❡ q✉❡ ♣♦ss✉❛♠ ❛♠❡s♠❛ ❞✐♠❡♥sã♦ ❡ ❡st❡❥❛♠ ♥❛ ♠❡s♠❛ ✉♥✐❞❛❞❡✱ ♣❛r❛ ❞❡♣♦✐s ✉s❛r ♦ ❝r✐tér✐♦ ❞❡ ❛♣r♦①✐♠❛çã♦✳ ❉♦q✉❡ ❢♦✐ ❞✐s❝✉t✐❞♦ ❛❝✐♠❛✱ ❝♦♥❝❧✉í♠♦s q✉❡

♦ r❡s✉❧t❛❞♦ ❞❡ ✉♠❛ s♦♠❛ ♦✉ ❞❡ ✉♠❛ s✉❜tr❛çã♦ ❞❡✈❡ ♣♦ss✉✐r ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❝❛s❛s ❞❡❝✐♠❛✐s❞❛ ♠❡❞✐❞❛ ❝♦♠ ♠❡♥♦r q✉❛♥t✐❞❛❞❡ ❞❡ ❝❛s❛s ❞❡❝✐♠❛✐s✳

▼✉❧t✐♣❧✐❝❛çã♦ ❡ ❉✐✈✐sã♦

◆❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ♥❛ ❞✐✈✐sã♦ ♦♣❡r❛♠♦s ♥♦r♠❛❧♠❡♥t❡ ❝♦♠ ❛s ♠❡❞✐❞❛s ❡ ♦ r❡s✉❧t❛❞♦✱ ❡♠ t❡r♠♦s❞♦ ♥ú♠❡r♦ ❝♦rr❡t♦ ❞❡ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s✱ é ♦❜t✐❞♦ ❝♦♠ ♦ ❝r✐tér✐♦ ❞❡ q✉❡

♦ ♣r♦❞✉t♦ ♦✉ ❛ ❞✐✈✐sã♦ ❞❡ ❞✉❛s ❣r❛♥❞❡③❛s ❢ís✐❝❛s ❞❡✈❡ ❝♦♥t❡r ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❛❧❣❛r✐s♠♦ss✐❣♥✐✜❝❛t✐✈♦s ❞❛ ♠❡❞✐❞❛ ❝♦♠ ♠❡♥♦r q✉❛♥t✐❞❛❞❡ ❞❡ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s✳

❖♣❡r❛çõ❡s ▼✐st❛s

➪s ✈❡③❡s ✉♠❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ é ❞❡t❡r♠✐♥❛❞❛ ❛tr❛✈és ❞❡ ✈ár✐❛s ♦♣❡r❛çõ❡s ♠❛t❡♠át✐❝❛s ❡♥tr❡♦✉tr❛s ❣r❛♥❞❡③❛s✱ ❝❤❛♠❛❞❛s ♦♣❡r❛çõ❡s ♠✐st❛s✳ ❖ ♣❡rí♦❞♦ ❞❡ ✉♠ ✉♠ ♣ê♥❞✉❧♦ s✐♠♣❧❡s ✭T =

2π

√

l

g❡♥✈♦❧✈❡♥❞♦ ✉♠❛ ❞✐✈✐sã♦ ❡ ✉♠❛ r❛❞✐❝✐❛çã♦✮ ❡ ♦ ✈❛❧♦r ♠é❞✐♦ ❞❛ ❡♥❡r❣✐❛ ❝✐♥ét✐❝❛ ❞❡ ✉♠❛

♣❛rtí❝✉❧❛ ✭< K >= 1

2m < v2 > ❡♥✈♦❧✈❡♥❞♦ s♦♠❛ ❞❡ ♣♦tê♥❝✐❛s✮ sã♦ ❞♦✐s ❡①❡♠♣❧♦s s✐♠♣❧❡s ❡

♣❛r❛ ❡st❡s ❝❛s♦s t❛♠❜é♠ ❞❡✈❡♠ s❡r ❛❞♦t❛❞♦s ❝r✐tér✐♦s ❞❡ ♦♣❡r❛çõ❡s ❡ ❞❡ ❛♣r♦①✐♠❛çõ❡s✳ ❆q✉✐❛❞♦t❛r❡♠♦s ❛

✹➟ r❡❣r❛✿ ♦ r❡s✉❧t❛❞♦ ❞❡ ✉♠❛ r❛❞✐❝✐❛çã♦✱ ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ♦✉ ❤✐♣❡r❜ó❧✐❝❛s✱ ❡①♣♦♥❡♥❝✐❛❧♦✉ ❧♦❣❛rít♠✐❝❛ s❡rá ❛♣r❡s❡♥t❛❞♦ ❝♦♠ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s ❞❡ s❡✉❛r❣✉♠❡♥t♦❀

✺➟ r❡❣r❛✿ ♦ r❡s✉❧t❛❞♦ ❞❡ ✉♠❛ ♦♣❡r❛çã♦ ♠✐st❛ ♣♦❞❡ s❡r ♦❜t✐❞♦ ❢❛③❡♥❞♦✲s❡ t♦❞❛s ❛s ♦♣❡r❛çõ❡s❡ ❛rr❡❞♦♥❞❛♥❞♦✲s❡ ❛♦ ✜♥❛❧ ❞❡ ♠❛♥❡✐r❛ q✉❡ ♦ r❡s✉❧t❛❞♦ ❛♣r❡s❡♥t❡ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s ❞❛ ♠❡❞✐❞❛ ❝♦♠ ♠❡♥♦r q✉❛♥t✐❞❛❞❡ ❞❡ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s✳

❈♦♠ ❡st❛s r❡❣r❛s ✐r❡♠♦s ❛❞♦t❛r ♦ s❡❣✉✐♥t❡s ❝r✐tér✐♦s ❞❡ ❛rr❡❞♦♥❞❛♠❡♥t♦ ❡♠ ♦♣❡r❛çõ❡s ❝♦♠❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s✳

✶✽ ❈❆P❮❚❯▲❖ ✷✳ ▼❊❉■❉❆❙ ❊ ❆▲●❆❘■❙▼❖❙ ❙■●◆■❋■❈❆❚■❱❖❙

✶➸ ❝r✐tér✐♦✿ s❡ ❛ ♣❛rt❡ ❛ s❡r ❞❡s♣r❡③❛❞❛ ❢♦r ♠❛✐♦r ❞♦ 5, 50, 500, 5000, · · · ✱ ❛❞✐❝✐♦♥❛✲s❡ ✉♠❛✉♥✐❞❛❞❡ ♥♦ ❛❧❣❛r✐s♠♦ ✐♠❡❞✐❛t❛♠❡♥t❡ ❛♥t❡r✐♦r ❛ ♣❛rt❡ ❞❡s♣r❡③❛❞❛✳

✷➸ ❝r✐tér✐♦✿ s❡ ❛ ♣❛rt❡ ❛ s❡r ❞❡s♣r❡③❛❞❛ ❢♦r ♠❡♥♦r ❞♦ q✉❡ 5, 50, 500, 5000, · · · ✱ ♦ ❛❧❣❛r✐s♠♦✐♠❡❞✐❛t❛♠❡♥t❡ ❛♥t❡r✐♦r ❛ ♣❛rt❡ ❛ s❡r ❞❡s♣r❡③❛❞❛ ♣❡r♠❛♥❡❝❡ ♦ ♠❡s♠♦✳

✸➸ ❝r✐tér✐♦✿ s❡ ❛ ♣❛rt❡ ❛ s❡r ❞❡s♣r❡③❛❞❛ ❢♦r ✐❣✉❛❧ ❛ 5, 50, 500, 5000, · · · ✱ ❡♥tã♦ ❛❞✐❝✐♦♥❛✲s❡ ✉♠❛✉♥✐❞❛❞❡ ❛♦ ❛❧❣❛r✐s♠♦ ✐♠❡❞✐❛t❛♠❡♥t❡ ❛♥t❡r✐♦r ❛ ♣❛rt❡ ❛ s❡r ❞❡s♣r❡③❛❞❛ ❝❛s♦ ❡st❡ s❡❥❛ ✉♠♥ú♠❡r♦ í♠♣❛r❀ s❡ ♥ã♦✱ ❡❧❡ ♣❡r♠❛♥❡❝❡ ❝♦♠ ♦ ♠❡s♠♦ ✈❛❧♦r✳

❈❛♣ít✉❧♦ ✸

❊❘❘❖❙ ❊ ■◆❈❊❘❚❊❩❆❙

❊❞✐♦ ❈✉♥❤❛ ❞❛ ❈♦st❛

❆ ♣r✐♥❝í♣✐♦✱ ❛ ♠❡❞✐❞❛ ❞✐r❡t❛ é ❛❧❣♦ s✐♠♣❧❡s ❞❡ s❡r ❢❡✐t♦✳ ❈♦♠♦ ✈✐st♦ ❛♥t❡r✐♦r♠❡♥t❡✱ é ✉♠ ♣r♦✲❝❡ss♦ q✉❡ ❝♦♥s✐st❡ ❡♠ ❝♦♠♣❛r❛r ❞✐r❡t❛♠❡♥t❡ ❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ❛ s❡r ♠❡❞✐❞❛ ❝♦♠ ♦ ✈❛❧♦r ♣❛❞rã♦✱❛❞♦t❛❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛✳ ◆♦ ❡♥t❛♥t♦✱ ❡st❡ ✏s✐♠♣❧❡s✑ ♣r♦❝❡❞✐♠❡♥t♦ é s❡♠♣r❡ ❛❝♦♠♣❛♥❤❛❞♦ ❞❡❡rr♦s✱ ✐♥❡r❡♥t❡s ❛♦ ♣r♦❝❡ss♦ ❞❡ ♠❡❞✐❞❛✳ ❖ ✐♥str✉♠❡♥t♦ ❞❡ ♠❡❞✐❞❛ ✉t✐❧✐③❛❞♦ ❛♣r❡s❡♥t❛ ❡rr♦s✱❞❡✈✐❞♦ ❛♦ ♣r♦❝❡ss♦ ❞❡ ❢❛❜r✐❝❛çã♦ ❞♦ ♣ró♣r✐♦ ✐♥str✉♠❡♥t♦✱ ❡rr♦s ❞❡s✈✐♦ ❛ ♠❛r❝❛çã♦ ❞❛ ❡s❝❛❧❛✳◆♦ ❝❛s♦ ❞❡ ✉♠❛ ré❣✉❛ ♠❡tá❧✐❝❛ ✉s❛❞❛ ♣❛r❛ ♠❡❞✐r ❝♦♠♣r✐♠❡♥t♦s✱ ❡rr♦s sã♦ ✐♥tr♦❞✉③✐❞♦s ♣❡❧♦s✐♠♣❧❡s ❢❛t♦ ❞❡ ❛ ré❣✉❛ s♦❢r❡r ♠✉❞❛♥ç❛ ❡♠ s❡✉ ❝♦♠♣r✐♠❡♥t♦ ♣♦r ♠✉❞❛♥ç❛s ♥❛ t❡♠♣❡r❛t✉r❛✳❉❡ ❢❛t♦✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ♦❜❥❡t♦ ♣♦❞❡ ❛♣r❡s❡♥t❛r ✈❛❧♦r❡s ❞✐❢❡r❡♥t❡s ❡♠ ❤♦rár✐♦s ❞✐❢❡r❡♥t❡s❞♦ ❞✐❛✳

❆❧é♠ ❞❡st❡s ❡rr♦s✱ ❡①✐st❡♠ ❛q✉❡❧❡s ❝✉❥❛ ♦r✐❣❡♠ ❡stá ♥♦ ♣ró♣r✐♦ ❡①♣❡r✐♠❡♥t❛❞♦r✱ ✐st♦ é✱ ♥❛♣❡ss♦❛ q✉❡ ❡stá ✉t✐❧✐③❛♥❞♦ ✉♠ ✐♥str✉♠❡♥t♦ ❞❡ ♠❡❞✐❞❛ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ ✈❛❧♦r ❞❡ ✉♠❛ ❣r❛♥❞❡③❛❢ís✐❝❛✳ P♦r ❡①❡♠♣❧♦✱ ❛♦ ❧❡r ♦ ✈❛❧♦r ❞❡ ✉♠❛ ♠❡❞✐❞❛ ❞❡ ❡s♣❡ss✉r❛ ✉t✐❧✐③❛♥❞♦ ✉♠ ♣❛q✉í♠❡tr♦✱ ❛♣❡ss♦❛ ♣♦❞❡ ♥ã♦ ❡st❛r s❡❣✉r❛♥❞♦ ♦ ♣❛q✉í♠❡tr♦ ❞❡✈✐❞❛♠❡♥t❡✱ ♦ q✉❡ ♣♦❞❡ ❧❡✈❛r ❛ ✉♠ ❡rr♦ ❞❡♣❛r❛❧❛①❡✳ ❖✉ s❡❥❛✱ ♦ ❡①♣❡r✐♠❡♥t❛❞♦r ♣♦❞❡ ♥ã♦ ❡st❛r ✉t✐❧✐③❛❞♦ ❝♦rr❡t❛♠❡♥t❡ ♦ ✐♥str✉♠❡♥t♦❞❡ ♠❡❞✐❞❛✳ ❈♦♥s✐❞❡r❡✱ ❝♦♠♦ ♦✉tr♦ ❡①❡♠♣❧♦✱ ❛ t♦♠❛❞❛ ❞❡ t❡♠♣♦ ❞❡ q✉❡❞❛ ❞❡ ✉♠❛ ♣❡q✉❡♥❛❡s❢❡r❛✳ ❆♦ ❧❛r❣❛r ❛ ❡s❢❡r❛ ❞❡ ✉♠❛ ❛❧t✉r❛ ♣ré✲❞❡t❡r♠✐♥❛❞❛ ❞✐s♣❛r❛✲s❡ ✉♠ ❝r♦♥ô♠❡tr♦ ❡ tr❛✈❛✲s❡q✉❛♥❞♦ ❛ ❡s❢❡r❛ t♦❝❛r ♦ ❝❤ã♦✳ ❘❡♣❡t✐❞❛s ♠❡❞✐❞❛s ❧❡✈❛rã♦ ❛ ❞✐❢❡r❡♥t❡s r❡s✉❧t❛❞♦s✱ ♣♦✐s ♥✉♥❝❛❝♦♥s❡❣✉✐r❡♠♦s ❞✐s♣❛r❛r ♦ ❝r♦♥ô♠❡tr♦ ❡①❛t❛♠❡♥t❡ ♥♦ ✐♥st❛♥t❡ ❡♠ q✉❡ ❛ ❡s❢❡r❛ ❢♦r s♦❧t❛ ❡ ♥✉♥❝❛❝♦♥s❡❣✉✐r❡♠♦s tr❛✈❛r ♦ ❝r♦♥ô♠❡tr♦ ♥♦ ♠❡s♠♦ ✐♥st❛♥t❡ ♥♦ q✉❛❧ ❛ ❡s❢❡r❛ t♦❝❛ ♦ s♦❧♦✳ ❆ss✐♠✱ ✉♠❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛s ❢♦r♥❡❝❡rá ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛s ❞❡ t❡♠♣♦ ❡ ❛ ♣❡r❣✉♥t❛ q✉❡ t❡♠♦s é✿q✉❛❧ ♦ ✈❛❧♦r ❝♦rr❡t♦ ❞♦ t❡♠♣♦ ❞❡ q✉❡❞❛❄ ➱ ❡✈✐❞❡♥t❡ q✉❡✱ r❡♣❡t✐♥❞♦✲s❡ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ✈ár✐❛s✈❡③❡s✱ ♦❜t❡r❡♠♦s ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❛❧♦r❡s q✉❡ ✜❝❛rã♦ ♣ró①✐♠♦s ✉♥s ❞♦s ♦✉tr♦s✱ ♣♦rq✉❡ ❛ ♣rát✐❝❛❞❡ ❞✐s♣❛r❛r ❡ tr❛✈❛r ♦ ❝r♦♥ô♠❡tr♦ ♥♦s r❡✜♥❛rá ❛ ❝♦♦r❞❡♥❛çã♦ ♠♦t♦r❛ ❞❡ ♠♦❞♦ ❛ ❝♦♥s❡❣✉✐r♠♦s✈❛❧♦r❡s ❝❛❞❛ ✈❡③ ♠❛✐s ♣ró①✐♠♦s ❞❛q✉❡❧❡ q✉❡ ❞❡✈❡ s❡r ♦ ✈❛❧♦r r❡❛❧✳ ❆✐♥❞❛ ❛ss✐♠✱ ❡①✐st❡ ✉♠ ❡rr♦❛ss♦❝✐❛❞♦ ❛♦ ❡①♣❡r✐♠❡♥t❛❞♦r✱ ❡ ❡st❡ ❡rr♦ ❧❡✈❛ ❛ ✉♠❛ ✐♥❝❡rt❡③❛ ♥♦ ✈❛❧♦r ❞❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ q✉❡s❡ ❡stá ❞❡t❡r♠✐♥❛♥❞♦✳

❚❛♠❜é♠ ❡①✐st❡♠ ❛q✉❡❧❡s ❡rr♦s q✉❡ sã♦ ♦❜t✐❞♦s ♣♦r ❝♦♥t❛ ❞❡ ♠❡❞✐❞❛s ✐♥❞✐r❡t❛s ❞❡ ❣r❛♥❞❡③❛s❢ís✐❝❛s✳ ◆♦ ❝❛s♦ ❝♦♥s✐❞❡r❛❞♦ ❛❝✐♠❛✱ ❛s ♠❡❞✐❞❛s ❞♦ t❡♠♣♦ ❞❡ q✉❡❞❛ ❡ ❞❛ ❛❧t✉r❛ ❞❡ q✉❡❞❛♣❡r♠✐t❡♠ ❛ ♦❜t❡♥çã♦ ❞♦ ✈❛❧♦r ❞❛ ❛❝❡❧❡r❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡✳ ❚♦❞❛✈✐❛✱ ❡①✐st✐♥❞♦ ❡rr♦s ♥♦ ♣r♦❝❡ss♦❞❡ ♠❡❞✐❞❛ ❞♦ t❡♠♣♦ ❞❡ q✉❡❞❛✱ ❡ t❛♠❜é♠ ♥♦ ✈❛❧♦r ❞❛ ❛❧t✉r❛ ❞❡ q✉❡❞❛✱ ♦❜t❡r❡♠♦s ✉♠ ❡rr♦ ♥♦✈❛❧♦r ❞❡ g✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ♣r♦♣❛❣❛çã♦ ❞❡ ❡rr♦s✱ ♦✉ s❡❥❛✱ ❡rr♦s ♦r✐✉♥❞♦s ❞♦s ♣r♦❝❡ss♦s ❞❡♠❡❞✐❞❛ ❡ ❞♦s ❝á❧❝✉❧♦s ❛r✐t♠ét✐❝♦s ❧❡✈❛♠ ❛ ❡rr♦s ♥❛s ❣r❛♥❞❡③❛s ♠❡❞✐❞❛s ✐♥❞✐r❡t❛♠❡♥t❡✳

▼❡s♠♦ s❡ ♣✉❞éss❡♠♦s ❡❧✐♠✐♥❛r t♦❞♦s ❡ss❡s ❡rr♦s✱ ❛♦ ♠❡♥♦s ♠✐♥✐♠✐③á✲❧♦s✱ ❛✐♥❞❛ ❛ss✐♠✱ ❡①✲✐st✐r✐❛♠ ❡rr♦s ♥♦s ♣r♦❝❡ss♦s ❞❡ ♠❡❞✐❞❛s✱ ❡rr♦s q✉❡ ♥ã♦ ♣♦❞❡r✐❛♠ s❡r ❞❡t❡❝t❛❞♦s tã♦ ♣♦✉❝♦

✶✾

✷✵ ❈❆P❮❚❯▲❖ ✸✳ ❊❘❘❖❙ ❊ ■◆❈❊❘❚❊❩❆❙

❡❧✐♠✐♥❛❞♦s ♦✉ r❡❞✉③✐❞♦s✳ ❙ã♦ ❡rr♦s ♣r♦✈❡♥✐❡♥t❡s ❞❡ s✐t✉❛çõ❡s q✉❡ ❢♦❣❡♠ ❛♦ ❝♦♥tr♦❧❡ ❞♦ ❡①♣❡r✲✐♠❡♥t❛❞♦r✳ ❙ã♦ ♦s ❝❛s♦s ❞❡ ❡rr♦s ✐♥tr♦❞✉③✐❞♦s ❞❡✈✐❞♦ ❛ ♠✉❞❛♥ç❛s ❞❡ t❡♠♣❡r❛t✉r❛ ❡ ♣r❡ssã♦❛t♠♦s❢ér✐❝❛ ❞♦ ❛♠❜✐❡♥t❡ ♦♥❞❡ s❡ ❡stá r❡❛❧✐③❛♥❞♦ ♦ ❡①♣❡r✐♠❡♥t♦ ❡ ❡rr♦s ♣r♦❞✉③✐❞♦s ♣♦r ✈✐❜r❛çõ❡s❞♦ ❛♣❛r❛t♦ ❡①♣❡r✐♠❡♥t❛❧ ❞❡✈✐❞♦ ❛ ❛❣❡♥t❡s ❡①t❡r♥♦s✱ ❡♥tr❡ ♦✉tr♦s✳

❙❡❣✉❡✱ ❡♥tã♦✱ q✉❡ ♣r❡❝✐s❛♠♦s ✐♥❞✐❝❛r ♦ ❣r❛✉ ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❞❛ ♠❡❞✐❞❛ ❡ ❞❡s❝♦❜r✐r ❛s ❢♦♥t❡s❞❡ ❡rr♦s ♣❛r❛ ❡❧✐♠✐♥á✲❧❛s q✉❛♥❞♦ ♣♦ssí✈❡❧✳ ❯♠❛ ✈❡③ ❝♦♥❤❡❝✐❞❛s ❡st❛s ❢♦♥t❡s ❞❡ ❡rr♦s ♣♦❞❡♠♦s❛♦ ♠❡♥♦s ♠✐♥✐♠✐③❛r ♦s s❡✉s ❡❢❡✐t♦s s♦❜r❡ ❛s ♠❡❞✐❞❛s✳ ❚❡♥❞♦ ❡♠ ✈✐st❛ q✉❡ ♠✉✐t♦s ❞❡st❡s ❡rr♦s♣♦❞❡♠ ♦❝♦rr❡r ❛♦ ❛❝❛s♦✱ ♦✉ q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞♦s ✈❛❧♦r❡s ❞❛ ♠❡❞✐❞❛ ❞❡ ✉♠❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ♥ã♦✐♥✢✉❡♥❝✐❛♠ ♦s ♣ró①✐♠♦s ✈❛❧♦r❡s ❞❛ ♠❡s♠❛ ❣r❛♥❞❡③❛✱ ♣♦❞❡♠♦s ✉t✐❧✐③❛r ♦s ❝♦♥❝❡✐t♦s ❞❛ t❡♦r✐❛ ❞❛s♣r♦❜❛❜✐❧✐❞❛❞❡s ❡ ❞❛ ❡st❛tíst✐❝❛ ♣❛r❛ ♦ tr❛t❛♠❡♥t♦ ❞❡ ❞❛❞♦s✱ ❛✜♠ ❞❡ ♦❜t❡r ♦s ✈❛❧♦r❡s ❞❡s❡❥❛❞♦s❞❛s ❣r❛♥❞❡③❛s ❝♦♠ ✉♠ ❝❡rt♦ ❣r❛✉ ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❡ ♣r❡❝✐sã♦✳

✸✳✶ ❱❛❧♦r ♠é❞✐♦ ❡ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧

❙❡❥❛ K ✉♠❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ❛ s❡r ♠❡❞✐❞❛ ❡ s❡❥❛ k1, k2, · · · , kN♦ ❝♦♥❥✉♥t♦ ❞♦s N ✈❛❧♦r❡s ♠❡❞✐❞♦s♣❛r❛ ❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❣r❛♥❞❡③❛ ❢ís✐❝❛✳ ❱❛♠♦s s✉♣♦r q✉❡ t♦❞❛s ❛s ♠❡❞✐❞❛s s❡❥❛♠ r❡❛❧✐③❛❞❛s ❞❛♠❡s♠❛ ♠❛♥❡✐r❛✱ s❡❣✉✐♥❞♦ ♦s ♠❡s♠♦s ♣r♦❝❡❞✐♠❡♥t♦s ❡ ❝r✐tér✐♦s✱ ❡ s❡♥❞♦ ✉t✐❧✐③❛❞♦s ♦s ♠❡s♠♦s✐♥str✉♠❡♥t♦s ❞❡ ♠❡❞✐❞❛✳ ❉✐③❡♠♦s q✉❡ ❛s ♠❡❞✐❞❛s sã♦ ❞❡s❝♦rr❡❧❛❝✐♦♥❛❞❛s s❡ ✉♠❛ ♠❡❞✐❞❛ ♥ã♦❛❢❡t❛ ❛ ♦✉tr❛✳ ❆ t❛❜❡❧❛ ✸✳✶ ❛❜❛✐①♦ ♠♦str❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛s ❞❡ t❡♠♣♦ ❞❡ q✉❡❞❛ ❞❡ ✉♠❛♣❡q✉❡♥❛ ❜♦❧❛ ❞❡ s✐❧✐❝♦♥❡✳

t✭s✮ ✵✱✹✺ ✵✱✹✹ ✵✱✹✻ ✵✱✹✺ ✵✱✹✸ ✵✱✹✻ ✵✱✹✺ ✵✱✹✺ ✵✱✹✹ ✵✱✹✺ ✵✱✹✷ ✵✱✹✺✵✱✹✻ ✵✱✹✺ ✵✱✹✺ ✵✱✹✹ ✵✱✹✸ ✵✱✹✻ ✵✱✹✺ ✵✱✹✺ ✵✱✹✷ ✵✱✹✹ ✵✱✹✺ ✵✱✹✹✵✱✹✸ ✵✱✹✺ ✵✱✹✺ ✵✱✹✹ ✵✱✹✺ ✵✱✹✻ ✵✱✹✺ ✵✱✹✹ ✵✱✹✹ ✵✱✹✸ ✵✱✹✹ ✵✱✹✺✵✱✹✹ ✵✱✹✷ ✵✱✹✼ ✵✱✹✻ ✵✱✹✺ ✵✱✹✺ ✵✱✹✸ ✵✱✹✶ ✵✱✹✹ ✵✱✹✷ ✵✱✹✺ ✵✱✹✸

❚❛❜❡❧❛ ✸✳✶✿ ❚❛❜❡❧❛ ❝♦♠ ❞❛❞♦s r❡❢❡r❡♥t❡s ❛ q✉❡❞❛ ❞❡ ✉♠❛ ❡s❢❡r❛ ❞❡ s✐❧✐❝♦♥❡✳ ❆ ❡s❢❡r❛ ❢♦✐ s♦❧t❛❞❛ ❛❧t✉r❛ ✭❛♣r♦①✐♠❛❞❛✮ ❞❡ 1, 00m ❡ ♦ t❡♠♣♦ ❞❡ q✉❡❞❛ ❛té ♦ ❝❤ã♦ ❢♦✐ ♠❡❞✐❞♦ ✉t✐❧✐③❛♥❞♦✲s❡ ✉♠❝r♦♥ô♠❡tr♦✳

❖ ♣r♦❝❡ss♦ ❞❡ ♠❡❞✐❞❛ é s✐♠♣❧❡s ❡ ❝♦♥s✐st❡ ♥♦s s❡❣✉✐♥t❡s ♣❛ss♦s✿✭✐✮ ❝♦♠ ✉♠❛ tr❡♥❛ ❡st✐❝❛❞❛✱ s♦❧t❛✲s❡ ❛ ❡s❢❡r❛ ❞❛ ❛❧t✉r❛ ❞❡ ✉♠ ♠❡tr♦❀✭✐✐✮ ❛♦ s♦❧t❛r ❛ ❡s❢❡r❛✱ ❞✐s♣❛r❛✲s❡ ♦ ❝r♦♥ô♠❡tr♦❀✭✐✐✐✮ ♦ ❝r♦♥ô♠❡tr♦ é tr❛✈❛❞♦ q✉❛♥❞♦ ❛ ❡s❢❡r❛ t♦❝❛r ♦ ❝❤ã♦❀✭✐✈✮ ❛♥♦t❛✲s❡ ♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ♠❡❞✐❞♦ ❞✐r❡t❛♠❡♥t❡ ♥♦ ❝r♦♥ô♠❡tr♦❀✭✈✮ ③❡r❛✲s❡ ♦ ❝r♦♥ô♠❡tr♦ ❡ ♦ ♣r♦❝❡ss♦ é r❡♣❡t✐❞♦ ✈ár✐❛s ✈❡③❡s✳❊st❡ s✐♠♣❧❡s ♣r♦❝❡ss♦ ❞❡ ♠❡❞✐❞❛ ❞❡ t❡♠♣♦ ❞❡ q✉❡❞❛ ✭❡ t❛♠❜é♠ ❞❡ ❛❧t✉r❛ ❞❡ q✉❡❞❛✱ ♣♦✐s

♣r❡❝✐s❛♠♦s ♠❛r❝❛r ✉♠ ♠❡tr♦ ❞❡ ❛❧t✉r❛ ❡♠ r❡❧❛çã♦ ❛♦ ❝❤ã♦✮✱ ❝♦♠♦ ❞❡s❝r✐t♦✱ ✈❡♠ ❛❝♦♠♣❛♥❤❛❞♦❞❡ ❡rr♦s✳ ❊st❡s ❡rr♦s r❡❧❛❝✐♦♥❛♠✲s❡ ❛s ❡s❝❛❧❛s ❞❛ tr❡♥❛ ❡ ❞♦ ❝r♦♥ô♠❡tr♦ ✉t✐❧✐③❛❞♦s✳ ◆♦ ❡♥t❛♥t♦✱♦ ❡rr♦ ♣r✐♥❝✐♣❛❧ q✉❡ ❞❡s❡❥❛♠♦s ❛♥❛❧✐s❛r é ♦ ❡rr♦ ❝♦♠❡t✐❞♦ ♣❡❧♦ ❡①♣❡r✐♠❡♥t❛❞♦r✱ q✉❡ ♣♦❞❡ s❡r♠✐♥✐♠✐③❛❞♦ ❝♦♠ ❛ ♣rát✐❝❛ ❞❡ ♠❡❞✐❞❛✳ ❊st❡s ❡rr♦s ❡①✐st❡♠ ♣♦rq✉❡ ♦ ❡①♣❡r✐♠❡♥t❛❞♦r✱ ❛♦ s❡❣✉r❛r❛ ❡s❢❡r❛ ♣❛r❛ s♦❧tá✲❧❛✱ ♥✉♥❝❛ ❝♦♥s❡❣✉❡ s♦❧tá✲❧❛ ❡①❛t❛♠❡♥t❡ ❞❛ ❛❧t✉r❛ ❞❡ ✉♠ ♠❡tr♦❀ s❡♠♣r❡❡①✐st❡ ✉♠❛ ✢✉t✉❛çã♦✱ ❡♠❜♦r❛ ♣❡q✉❡♥❛✱ ♥❛ ♣♦s✐çã♦ ✐♥✐❝✐❛❧ ❞❛ ❡s❢❡r❛ ❡♠ t♦r♥♦ ❞❛ ❛❧t✉r❛ ✐♥✐❝✐❛❧❡s❝♦❧❤✐❞❛✳ ❆❧é♠ ❞❡st❡ ❡rr♦✱ t❛♠❜é♠ ❤á ♦ ❡rr♦ ❝♦♠❡t✐❞♦ ♥❛ t♦♠❛❞❛ ❞♦s t❡♠♣♦s✱ ♦✉ s❡❥❛✱ ♦❡①♣❡r✐♠❡♥t❛❞♦r ♥✉♥❝❛ ❞✐s♣❛r❛ ♦ ❝r♦♥ô♠❡tr♦ ❡①❛t❛♠❡♥t❡ ♥♦ ♠❡s♠♦ ✐♥st❛♥t❡ ❡♠ q✉❡ ❡❧❡ s♦❧t❛❛ ❡s❢❡r❛✱ tã♦ ♣♦✉❝♦ tr❛✈❛ ♦ ❝r♦♥ô♠❡tr♦ ❡①❛t❛♠❡♥t❡ ♥♦ ♠❡s♠♦ ✐♥st❛♥t❡ ❡♠ q✉❡ ❡st❛ t♦❝❛ ♦❝❤ã♦✳ ❆ss✐♠✱ ❤á t❛♠❜é♠ ✉♠❛ ✢✉t✉❛çã♦ ♥♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ❞❡ q✉❡❞❛✳ ❊st❡s ❡rr♦s✱ ❝♦♠♦

✸✳✶✳ ❱❆▲❖❘ ▼➱❉■❖ ❊ ❱❆▲❖❘ ▼❆■❙ P❘❖❱➪❱❊▲ ✷✶

❥á ♠❡♥❝✐♦♥❛❞♦✱ ♣♦❞❡♠ s❡r r❡❞✉③✐❞♦s ♣❡❧❛ ♣rát✐❝❛ ❞♦ ❡①♣❡r✐♠❡♥t❛❞♦r✱ ♣❡❧❛ ✉t✐❧✐③❛çã♦ ❞❡ ✉♠❛♣❛r❛t♦ ❡①♣❡r✐♠❡♥t❛❧ q✉❡ ♣❡r♠✐t❛ q✉❡ ❛ ❜♦❧❛ s❡❥❛ ❧❛♥ç❛❞❛ s❡♠♣r❡ ❞❛ ♠❡s♠❛ ❛❧t✉r❛✱ ❡ t❛♠❜é♠♣♦r ✉♠ ♥ú♠❡r♦ ❣r❛♥❞❡ ♠❡❞✐❞❛s✳

❆ t❛❜❡❧❛ ❛♣r❡s❡♥t❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛s ❞❡ t❡♠♣♦ ❞❡ q✉❡❞❛ ❞❡ ✉♠❛ ❡s❢❡r❛✳ ❆❧❣✉♠❛s♣❡r❣✉♥t❛s sã♦ ✐♠❡❞✐❛t❛s✿ q✉❛❧ ♦ ✈❛❧♦r ❝♦rr❡t♦ ❞❡st❡ t❡♠♣♦ ❞❡ q✉❡❞❛❄ ◗✉❛❧ ♦ ✈❛❧♦r ❝♦rr❡t♦ ❞❛❛❝❡❧❡r❛çã♦ ❞❡ q✉❡❞❛ ❞❛ ❡s❢❡r✐♥❤❛❄ ❊①✐st❡♠ ♦✉tr♦s ❡rr♦s ❛ss♦❝✐❛❞♦s ❛♦ ♣r♦❝❡ss♦ ❞❡ ♠❡❞✐❞❛ ❡✱❝❛s♦ ❡①✐st❛♠✱ é ♣♦ssí✈❡❧ ❡❧✐♠✐♥á✲❧♦s✱ ❛♦ ♠❡♥♦s r❡❞✉③✐✲❧♦s❄ ❆té q✉❡ ♣♦♥t♦ ♣♦❞❡r❡♠♦s ❝♦♥✜❛r♥♦s r❡s✉❧t❛❞♦s✱ ❛❝❡✐t❛♥❞♦✲♦s ❝♦♠♦ ✈á❧✐❞♦s✱ ♦✉ s❡❥❛✱ ❛té q✉❡ ♣♦♥t♦ ♦ ❡①♣❡r✐♠❡♥t♦ ♣♦❞❡ s❡r❝♦♥s✐❞❡r❛❞♦ ❝♦rr❡t♦✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ❛❞♦t❛❞♦ é ✈á❧✐❞♦❄ P♦❞❡rí❛♠♦s ❢❛③❡r♦✉tr❛s ✐♥ú♠❡r❛s ♣❡r❣✉♥t❛s ♠❛s✱ ♣♦r ❡♥q✉❛♥t♦✱ ✈❛♠♦s ❡♠ ❜✉s❝❛ ❞❛s r❡s♣♦st❛s✳

Pr✐♠❡✐r❛♠❡♥t❡✱ ❞❡✈❡♠♦s ♦❜s❡r✈❛r q✉❡ ❞✐✈❡rs♦s ✈❛❧♦r❡s sã♦ r❡♣❡t✐❞♦s✱ ❞❡♥tr♦ ❞♦ t♦t❛❧ ❞❛s ✹✽♠❡❞✐❞❛s r❡❛❧✐③❛❞❛s✱ ♦ q✉❡ ♥♦s ♣❡r♠✐t❡ ❞❡✜♥✐r ❛ ❢r❡q✉ê♥❝✐❛ ❞❛ ♠❡❞✐❞❛ ❝♦♠♦ ♦ ♥ú♠❡r♦ ❞❡ ✈❡③❡sq✉❡ ♦ ✈❛❧♦r ❢♦✐ ♦❜t✐❞♦ ❞❡♥tr♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ N ♠❡❞✐❞❛s✳ P♦r ❡①❡♠♣❧♦✱ ❛ ❢r❡q✉ê♥❝✐❛ ❞♦ ✈❛❧♦r✵✱✹✶ é ✶ ❡ ❛ ❢r❡q✉ê♥❝✐❛ ❞♦ ✈❛❧♦r ✵✱✹✹ é ✶✶✳ P♦❞❡♠♦s✱ ❡♥tã♦✱ ❝♦♥str✉✐r ✉♠❛ ♥♦✈❛ t❛❜❡❧❛ ♥❛ q✉❛❧sã♦ ❝♦❧♦❝❛❞❛s ♦ ✈❛❧♦r ❞❛ ♠❡❞✐❞❛ ❡ s✉❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❢r❡q✉ê♥❝✐❛✳ ❆ t❛❜❡❧❛ ✸✳✷ ❛❜❛✐①♦ ✐❧✉str❛♦ ♣r❡s❡♥t❡ ❝❛s♦✳

t❡♠♣♦ ❞❡ q✉❡❞❛ ❢r❡q✉ê♥❝✐❛✵✱✹✶ ✶✵✱✹✷ ✹✵✱✹✸ ✻✵✱✹✹ ✶✶✵✱✹✺ ✶✽✵✱✹✻ ✻✵✱✹✼ ✷

❚❛❜❡❧❛ ✸✳✷✿ ❚❛❜❡❧❛ ❝♦♠ ♦s ✈❛❧♦r❡s ❞♦ t❡♠♣♦ ❞❡ q✉❡❞❛ ❡ ❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❢r❡q✉ê♥❝✐❛✳

❈♦♠♦ é ❢❛❝✐❧♠❡♥t❡ ♦❜s❡r✈❛❞♦✱ ❡①✐st❡ ✉♠❛ ♠❡❞✐❞❛ q✉❡ ❛♣r❡s❡♥t❛ ❛ ♠❛✐♦r ❢r❡q✉ê♥❝✐❛✱ ♦✉s❡❥❛✱ ♦❝♦rr❡ ✉♠ ♠❛✐♦r ♥ú♠❡r♦ ❞❡ ✈❡③❡s✱ ❡st❛♥❞♦ ❛s ❞❡♠❛✐s ♠❡❞✐❞❛s ❡s♣❛❧❤❛❞❛s ❡♠ t♦r♥♦ ❞❡st❛✳P♦❞❡♠♦s ❛té ✐♥❢❡r✐r q✉❡✱ ♥✉♠ ♣ró①✐♠♦ ❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛s r❡❛❧✐③❛❞❛s ❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛ ❡s♦❜ ❛s ♠❡s♠❛s ❝♦♥❞✐çõ❡s ❢ís✐❝❛s✱ ❛ ♠❡❞✐❞❛ ❞❡ ✈❛❧♦r 0, 45 s ♦❝♦rr❛ ❝♦♠ ♠❛✐♦r ❢r❡q✉ê♥❝✐❛✳ ❊st❡s✐♠♣❧❡s ❡①♣❡r✐♠❡♥t♦ ❢♦✐ r❡♣❡t✐❞♦ ♦✉tr❛s ✹✽ ✈❡③❡s✱ ♦❜t❡♥❞♦✲s❡ ❛ t❛❜❡❧❛ ❛❜❛✐①♦✱ ❝♦♠ ❛s r❡s♣❡❝t✐✈❛s❢r❡q✉ê♥❝✐❛s✳

t✭s✮ ✵✱✹✺ ✵✱✹✼ ✵✱✹✹ ✵✱✹✺ ✵✱✹✸ ✵✱✹✻ ✵✱✹✺ ✵✱✹✺ ✵✱✹✹ ✵✱✹✺ ✵✱✹✷ ✵✱✹✺✵✱✹✻ ✵✱✹✶ ✵✱✹✶ ✵✱✹✸ ✵✱✹✺ ✵✱✹✻ ✵✱✹✺ ✵✱✹✺ ✵✱✹✷ ✵✱✹✹ ✵✱✹✺ ✵✱✹✹✵✱✹✸ ✵✱✹✺ ✵✱✹✺ ✵✱✹✹ ✵✱✹✺ ✵✱✹✻ ✵✱✹✺ ✵✱✹✹ ✵✱✹✹ ✵✱✹✸ ✵✱✹✹ ✵✱✹✺✵✱✹✹ ✵✱✹✹ ✵✱✹✺ ✵✱✹✻ ✵✱✹✺ ✵✱✹✺ ✵✱✹✸ ✵✱✹✹ ✵✱✹✹ ✵✱✹✺ ✵✱✹✺ ✵✱✹✸

t❡♠♣♦ ❞❡ q✉❡❞❛ ✵✱✹✶ ✵✱✹✷ ✵✱✹✸ ✵✱✹✹ ✵✱✹✺ ✵✱✹✻ ✵✱✹✼❢r❡q✉ê♥❝✐❛ ✷ ✷ ✻ ✶✷ ✷✵ ✺ ✶

❚❛❜❡❧❛ ✸✳✸✿ ◆♦✈❛ t❛❜❡❧❛ ❞❡ ✈❛❧♦r❡s ❞❡ t❡♠♣♦ ❞❡ q✉❡❞❛ ❞❡ ✉♠❛ ❡s❢❡r✐♥❤❛✱ ❛ ♣❛rt✐r ❞❛ ❛❧t✉r❛ ❞❡1, 00m ❛❝✐♠❛ ❞♦ ❝❤ã♦✳ ❆ t❛❜❡❧❛ s✉♣❡r✐♦r ♠♦str❛ ♦s ✈❛❧♦r❡s ♠❡❞✐❞♦s✱ ❡ ❛ t❛❜❡❧❛ ✐♥❢❡r✐♦r✱ ♦s✈❛❧♦r❡s ❞❛s ❢r❡q✉ê♥❝✐❛s ❞❡ ❝❛❞❛ ♠❡❞✐❞❛✱ ❝❛❧❝✉❧❛❞❛s s♦❜r❡ t♦❞❛ ❛ t❛❜❡❧❛✳

✷✷ ❈❆P❮❚❯▲❖ ✸✳ ❊❘❘❖❙ ❊ ■◆❈❊❘❚❊❩❆❙

◆♦✈❛♠❡♥t❡✱ ♦❜s❡r✈❛✲s❡ q✉❡ ♦ ✈❛❧♦r 0, 45 s✱ ♣❛r❛ ♦ t❡♠♣♦ ❞❡ q✉❡❞❛✱ ❝♦rr❡s♣♦♥❞❡ ❛♦ ✈❛❧♦r❞❡ ♠❛✐♦r ❢r❡q✉ê♥❝✐❛ ❡ ♦s ❞❡♠❛✐s ✈❛❧♦r❡s ❡♥❝♦♥tr❛♠✲s❡ ❡s♣❛❧❤❛❞♦s ❡♠ t♦r♥♦ ❞❡st❡ ♦✉tr♦✳ ❆♣r✐♥❝í♣✐♦✱ ♣♦❞❡rí❛♠♦s ❞❡❞✉③✐r q✉❡ ♦ ✈❛❧♦r ♣❛r❛ ♦ t❡♠♣♦ ❞❡ q✉❡❞❛ ❞❡ 0, 45 s s❡r✐❛ ♦ ✈❛❧♦r ♠❛✐s♣r♦✈á✈❡❧ ❞❡ s❡r ♦❜t✐❞♦ ♥✉♠ ♥♦✈♦ ❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛s✳ ◆♦ ❡♥t❛♥t♦✱ ♥ã♦ ❤á ♥❛❞❛ q✉❡ ♥♦s ❣❛r❛♥t❛q✉❡ ❡st❡ s❡❥❛ r❡❛❧♠❡♥t❡ ♦ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧ ❛ s❡r ♦❜t✐❞♦ ❡♠ ♥♦✈♦ ❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛s✳ ❉❡❢❛t♦✱ s❡ ✜③❡r♠♦s ✉♠ ♥ú♠❡r♦ ♠❡♥♦r ❞❡ ♠❡❞✐❞❛s✱ ❞✐❣❛♠♦s ✸ ♦✉ ✶✵✱ ♣♦❞❡rá ❛❝♦♥t❡❝❡r ❞❡ ♦ ✈❛❧♦r0, 45 s ♥ã♦ s❡r ♦❜t✐❞♦ ♦✉✱ s❡r ♦❜t✐❞♦ ❝♦♠ ✉♠ ♥ú♠❡r♦ ♠❡♥♦r ❞❡ ✈❡③❡s✳ ❚❡♥❞♦ ❡♠ ✈✐st❛ q✉❡ ♦ ❡rr♦✐♥trí♥s❡❝♦ ❛♦ ♣r♦❝❡ss♦ ❞❡ ♠❡❞✐❞❛ ❢❛③ ❝♦♠ q✉❡ ♦s ✈❛❧♦r❡s s❡❥❛♠ ♠❡❞✐❞♦s ❞❡ ♠❛♥❡✐r❛ ❛❧❡❛tór✐❛✱♣♦❞❡♠♦s ♥♦s ✈❛❧❡r ❞♦s r❡s✉❧t❛❞♦s ❝♦♥❤❡❝✐❞♦s ❞❛ t❡♦r✐❛ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❛✜♠ ❞❡ ♦❜t❡r♠♦s❛❧❣✉♠❛s r❡❧❛çõ❡s ❞❡ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦ ♥♦ss♦ ❡st✉❞♦✳

▲❡♠❜r❛♥❞♦ ❛ t❡♦r✐❛ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s✱ s❛❜❡♠♦s q✉❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❜t❡r✲s❡ ❝❛r❛ ♥✉♠❧❛♥ç❛♠❡♥t♦ ❞❡ ✉♠❛ ♠♦❡❞❛ é 0, 50 ✭♦✉ 50%✮✳ ▼❛s ✐st♦ ♥ã♦ é s✐♠♣❧❡s♠❡♥t❡ ♣♦rq✉❡ ♥♦ ❧❛♥ç❛✲♠❡♥t♦ ♣♦❞❡♠♦s ♦❜t❡r ❞♦✐s r❡s✉❧t❛❞♦s ❡ q✉❡ ♥ã♦ s❛❜❡♠♦s✶ ❛♦ ❝❡rt♦ q✉❛❧ ✈❛❧♦r s❡rá ♦❜t✐❞♦✳ ➱❛❧❣♦ ❛ ♠❛✐s✳ ◆❛ r❡❛❧✐❞❛❞❡✱ ♣❛r❛ ✉♠ ♥ú♠❡r♦ r❡❧❛t✐✈❛♠❡♥t❡ ♣❡q✉❡♥♦ ❞❡ ❧❛♥ç❛♠❡♥t♦s t❡r❡♠♦s✱❡♠ ❣❡r❛❧✱ q✉❛♥t✐❞❛❞❡s ❞✐❢❡r❡♥t❡s ❞❡ ❝❛r❛ ❡ ❞❡ ❝♦r♦❛✱ ♦✉ s❡❥❛✱ ♣♦❞❡♠♦s ❞✐③❡r ❛ ❢r❡q✉ê♥❝✐❛ ❝♦♠ ❛q✉❛❧ ♦❜t❡♠♦s ❝❛r❛ é ❞✐❢❡r❡♥t❡ ❞❛ ❢r❡q✉ê♥❝✐❛ ❝♦♠ ❛ q✉❛❧ ♦❜t❡♠♦s ❝♦r♦❛✳ ◆♦ ❡♥t❛♥t♦✱ ❡①♣❡r✐♠❡♥✲t❛❧♠❡♥t❡ é ✈❡r✐✜❝❛❞♦ q✉❡✱ ❛✉♠❡♥t❛♥❞♦✲s❡ ♦ ♥ú♠❡r♦ ❞❡ ❧❛♥ç❛♠❡♥t♦s ❝♦♥s✐❞❡r❛✈❡❧♠❡♥t❡✱ ❡st❛s❢r❡q✉ê♥❝✐❛s t♦r♥❛♠✲s❡ ❝❛❞❛ ✈❡③ ♠❛✐s ♣ró①✐♠❛s ✉♠❛ ❞❛ ♦✉tr❛ ❡ ❛ ❢r❡q✉ê♥❝✐❛ r❡❧❛t✐✈❛ ✭♥ú♠❡r♦ ❞❡✈❡③❡s q✉❡ ✉♠ r❡s✉❧t❛❞♦ é ♦❜t✐❞♦ ❞✐✈✐❞✐❞♦ ♣❡❧♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ r❡s✉❧t❛❞♦s ♦❜t✐❞♦s✮ ❛♣r♦①✐♠❛✲s❡❞❡ 1/2✳ ❈♦♥s✐❞❡r❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣♦st♦ ♣♦r 30 ❧❛♥ç❛♠❡♥t♦s ❞❡ ✉♠❛ ♠♦❡❞❛✳ P♦❞❡ ❛❝♦♥t❡❝❡r❞❡ ♦❜t❡r✲s❡ 17 r❡s✉❧t❛❞♦s ♣❛r❛ ❝❛r❛ ❝♦♠ 13 r❡s✉❧t❛❞♦s ♣❛r❛ ❝♦r♦❛✳ ❙❡ ✉♠ ♥♦✈♦ ❝♦♥❥✉♥t♦ ❞❡ 30❧❛♥ç❛♠❡♥t♦s ❢♦r ❢❡✐t♦✱ s♦❜ ❛s ♠❡s♠❛s ❝♦♥❞✐çõ❡s ❡ ❝♦♠ ❛ ♠❡s♠❛ ♠♦❡❞❛ ♣♦❞❡r✲s❡✲á ♦❜t❡r 16✱ ♦✉15✱ ♦✉ 14 ♦✉ 13 r❡s✉❧t❛❞♦s ♣❛r❛ ❝❛r❛ ❝♦♠ ♦ ❝♦rr❡s♣♦♥❞❡♥t❡ r❡s✉❧t❛❞♦ ❞❡ ❝♦r♦❛✳ ➱ ✈❡r✐✜❝❛❞♦❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ q✉❡ ❛ ❢r❡q✉ê♥❝✐❛ ✢✉t✉❛ ❡♠ t♦r♥♦ ❞❡ 15✱ q✉❡ é ❥✉st❛♠❡♥t❡ ❛ ♠❡t❛❞❡ ❞♦✈❛❧♦r t♦t❛❧ ❞❡ ❧❛♥ç❛♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦✱ q✉❡ ✈❛❧❡ 30✳ ❱♦❝ê ♣♦❞❡ ❝♦♥t❛r ♦ ♥ú♠❡r♦ ❞❡ ✈❡③❡s q✉❡❝❛❞❛ ✉♠❛ ❞❛s ♠❡❞✐❞❛s 0, 41 s✱ 0, 42 s✱· · · ✱ 0, 47 s ❢♦✐ ♦❜t✐❞❛ ✉♥✐♥❞♦ ❛s t❛❜❡❧❛s ✸✳✶ ❡ ✸✳✸✳ ❈♦♥t❡♣r✐♠❡✐r❛♠❡♥t❡ s♦❜r❡ ❛ ♣r✐♠❡✐r❛ ❧✐♥❤❛✱ ❞❡♣♦✐s ❝♦♥s✐❞❡r❛♥❞♦ ❛s ❞✉❛s ♣r✐♠❡✐r❛s ❧✐♥❤❛s✱ s♦❜r❡ ❛strês ❧✐♥❤❛s✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✱ ❛té ✜♥❛❧✐③❛r ❛ ❝♦♥t❛❣❡♠ s♦❜r❡ t♦❞❛ ❛ t❛❜❡❧❛✳ ➱ ❢á❝✐❧ ✈❡r✐✜❝❛rq✉❡ ❛s ❢r❡q✉ê♥❝✐❛s r❡❧❛t✐✈❛s ♣❛r❛ ❛ ♠❡❞✐❞❛ 0, 45 s sã♦ ❡st❛s ❛♣r❡s❡♥t❛❞❛s ❛❜❛✐①♦✳

✻✴✶✷ ✶✶✴✷✹ ✶✻✴✸✻ ✶✾✴✹✽ ✷✺✴✻✵ ✷✾✴✼✷ ✸✹✴✽✹ ✸✾✴✾✻✵✱✺✵✵ ✵✱✹✺✽ ✵✱✹✹✹ ✵✱✸✾✻ ✵✱✹✶✼ ✵✱✹✵✸ ✵✱✹✵✺ ✵✱✹✵✻

❚❛❜❡❧❛ ✸✳✹✿ ❚❛❜❡❧❛ ❝♦♥t❡♥❞♦ ❛ ❢r❡q✉ê♥❝✐❛ r❡❧❛t✐✈❛ ❞♦ ✈❛❧♦r 0, 45 s ♣❛r❛ ♦ t❡♠♣♦ ❞❡ q✉❡❞❛ ❞❡✉♠❛ ❡s❢❡r❛✱ ❝♦♥s✐❞❡r❛♥❞♦✲s❡ ❛ ✶➟ ❧✐♥❤❛✱ ✶➟ ❡ ✷➟ ❧✐♥❤❛s✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✱ ❥✉♥t❛♥❞♦✲s❡ ❛st❛❜❡❧❛s ✸✳✶ ❡ ✸✳✸✳ ❆ ❧✐♥❤❛ s✉♣❡r✐♦r ♠♦str❛ ❛ ❢r❡q✉ê♥❝✐❛ r❡❧❛t✐✈❛ s♦❜ ❛ ❢♦r♠❛ ❞❡ ❢r❛çã♦✳ ❆ ❧✐♥❤❛✐♥❢❡r✐♦r ♠♦str❛ ♦ ✈❛❧♦r ♥✉♠ér✐❝♦✳ ❆ ❢r❛çã♦ 11/24 ✐♥❞✐❝❛ q✉❡ ♦ r❡s✉❧t❛❞♦ 0, 45 s ❢♦✐ ♦❜t✐❞♦ 11✈❡③❡s ♥✉♠ t♦t❛❧ ❞❡ 24 ♠❡❞✐❞❛s✳

❈♦♠♦ ♣♦❞❡ s❡r ♦❜s❡r✈❛❞♦ ❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ✉♠ ✈❛❧♦r ♠❡❞✐❞♦ ❛✉♠❡♥t❛ q✉❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡♠❡❞✐❞❛s é ❛✉♠❡♥t❛❞♦✱ ♠❛s ❛ ❢r❡q✉ê♥❝✐❛ r❡❧❛t✐✈❛ t❡♥❞❡ ❛ ✉♠ ✈❛❧♦r ❝♦♥st❛♥t❡✱ ❞❛ ♦r❞❡♠ ❞❡ 0, 405✳P♦rt❛♥t♦✱ s❡♥❞♦ 0, 45 s ❛ ♠❡❞✐❞❛ ❝♦♠ ♠❛✐♦r ❢r❡q✉ê♥❝✐❛✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ ❡st❡ é ♦ ✈❛❧♦r ♠❛✐s♣r♦✈á✈❡❧ ♣❛r❛ ❛ ♠❡❞✐❞❛ ❡♠ q✉❡stã♦✱ ❛ s❛❜❡r✱ ♦ t❡♠♣♦ ❞❡ q✉❡❞❛ ❞❡ ✉♠❛ ❡s❢❡r❛ q✉❛♥❞♦ s♦❧t❛

✶◆ã♦ ♣♦❞❡♠♦s ♣r❡✈❡r ♦ r❡s✉❧t❛❞♦ ❝♦rr❡t♦ ♣♦rq✉❡ ♥ã♦ ❝♦♥s❡❣✉✐♠♦s ❧❛♥ç❛r ❛ ♠♦❡❞❛ ❝♦♠ ❛ ♠❡s♠❛ ❢♦rç❛ ❡❛♣❧✐❝❛♥❞♦ ♦ ♠❡s♠♦ t♦rq✉❡✳ ■st♦ ❢❛③ ❝♦♠ q✉❡ ❤❛❥❛ ✢✉t✉❛çõ❡s ♥❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ❞♦ ❧❛♥ç❛♠❡♥t♦✳ ❯♠ ♦✉tr♦❛r❣✉♠❡♥t♦ ♣❧❛✉sí✈❡❧ é q✉❡ ♥ã♦ t❡♠♦s ✐♥❢♦r♠❛çõ❡s s✉✜❝✐❡♥t❡s ♣❛r❛ ❞❡s❝r❡✈❡r ❝♦♠♣❧❡t❛♠❡♥t❡ ♦ s✐st❡♠❛✱ ♦✉ s❡❥❛✱♥♦ss❛ ✐❣♥♦râ♥❝✐❛ ❝♦♠ r❡❧❛çã♦ ❛s ❝❛r❛❝t❡ríst✐❝❛s ❞♦ s✐st❡♠❛ ♥ã♦ ♣❡r♠✐t❡ ♣r❡❞✐③❡r ♦ s❡✉ ❝♦♠♣♦rt❛♠❡♥t♦ ❢✉t✉r♦✳

✸✳✷✳ ❉❊❙❱■❖ ❊ ❉❊❙❱■❖ ▼➱❉■❖ ✷✸

❞❛ ❛❧t✉r❛ ❞❡ 1, 00m ❛❝✐♠❛ ❞♦ ❝❤ã♦✳ ❖ q✉❡ ♣r❡❝✐s❛♠♦s ❢❛③❡r✱ ❡♥tã♦✱ é ✈❡r✐✜❝❛r s❡ ❡st❛ é ✉♠❛❜♦❛ ❞❡✜♥✐çã♦ ❞❡ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧ ❞❡ ✉♠❛ ♠❡❞✐❞❛✱ ♦✉ s❡❥❛✱ s❡ ♣♦❞❡♠♦s r❡❛❧♠❡♥t❡ ❝♦♥s✐❞❡r❛r❝♦♠♦ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧ ❞❡ ✉♠❛ ♠❡❞✐❞❛ ❛q✉❡❧❛ q✉❡ ❛♣r❡s❡♥t❛r ❛ ♠❛✐♦r ❢r❡q✉ê♥❝✐❛ r❡❧❛t✐✈❛♥✉♠ ❝♦♥❥✉♥t♦ ❞❡ N ♠❡❞✐❞❛s✳

❖ ❝á❧❝✉❧♦ ♠❛✐s s✐♠♣❧❡s q✉❡ ♣♦❞❡♠♦s ❢❛③❡r é ♦ ❞♦ ✈❛❧♦r ♠é❞✐♦ ❞❛ ♠❡❞✐❞❛✱ ❞❡✜♥✐❞♦ s✐♠✲♣❧❡s♠❡♥t❡ ❝♦♠♦ ❛ s♦♠❛s ❞♦s ✈❛❧♦r❡s ❞❛ ♠❡❞✐❞❛✱ ❞✐✈✐❞✐❞♦ ♣❡❧♦ ♥ú♠❡r♦ ❞❡ ♠❡❞✐❞❛s✳ ❙❡❥❛ {xi}✱i = 1, 2, · · · , N ✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s N ✈❛❧♦r❡s ♠❡❞✐❞♦s ♣❛r❛ ❛ ❣r❛♥❞❡③❛ x✳ ❖ ✈❛❧♦r ♠é❞✐♦ 〈x〉 ❞❛❣r❛♥❞❡③❛ x é✱ ♣♦r ❞❡✜♥✐çã♦

〈x〉 = 1

N

N∑

i=1

xi ✭✸✳✶✳✶✮

❡ ♣♦❞❡♠♦s ♣r❡s✉♠✐r q✉❡ ♦ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧ ❞❡ ✉♠❛ ♠❡❞✐❞❛ s❡❥❛ ♦ ❧✐♠✐t❡ ❞♦ ✈❛❧♦r ♠é❞✐♦ ♣❛r❛✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✐♥✜♥✐t❛s ♠❡❞✐❞❛s✳ ❊✈✐❞❡♥t❡♠❡♥t❡✱ ♥ã♦ ♣♦❞❡♠♦s r❡❛❧✐③❛r ✐♥✜♥✐t❛s ♠❡❞✐❞❛s✱♠❛s ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ✉♠ ♥ú♠❡r♦ ❣r❛♥❞❡ ♦ s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡ ❛ ❢r❡q✉ê♥❝✐❛ r❡❧❛t✐✈❛ ♣♦ss❛s❡r tr❛t❛❞❛ ❝♦♠♦ ✉♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡✳ ❊♥tã♦ ❞❡✜♥✐♠♦s

℘(xi) =Ni

N✭✸✳✶✳✷✮

é ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ q✉❡ ♦ ✈❛❧♦r xi s❡❥❛ ♦❜t✐❞♦ Ni ✈❡③❡s ♥✉♠ ❝♦♥❥✉♥t♦ ❞❡ N ♠❡❞✐❞❛s✱ ❝♦♠ Ni

❡ N s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡s✳ ❉❛s ❞❡✜♥✐çõ❡s ✭✸✳✶✳✶✮ ❡ ✭✸✳✶✳✷✮ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

〈x〉 =N ′

∑

j=1

℘(xj)xj

❡♠ q✉❡ N ′ é ♦ ♥ú♠❡r♦ ❞❡ ✈❛❧♦r❡s ❞✐❢❡r❡♥t❡s q✉❡ ❢♦r❛♠ ♦❜t✐❞♦s ♣❛r❛ ❛ ❣r❛♥❞❡③❛ ♠❡❞✐❞❛✱ ❞❡♥tr♦❞♦ ❝♦♥❥✉♥t♦ ❞❡ N ♠❡❞✐❞❛s✳ P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ♦ ❝❛s♦ ❞❛ q✉❡❞❛ ❞❛ ❡s❢❡r❛ ❛♥❛❧✐s❛❞♦ ♥❛ t❛❜❡❧❛✸✳✸ ❛❝✐♠❛✱ t❡♠♦s N = 48 ❡ N ′ = 7✳ P♦rt❛♥t♦

〈x〉 = (0, 45 + 0, 47 + 0, 44 + 0, 45 + 0, 43 + 0, 46 + 0, 45 + · · · )/48 = 0, 443541666

♦✉

〈x〉 = 2

48× 0, 41 +

2

48× 0, 42 +

6

48× 0, 43 +

12

48× 0, 44 +

20

48× 0, 45 +

5

48× 0, 46 +

1

48× 0, 47 = 0, 443541666

❞❛♥❞♦ ♦ ♠❡s♠♦ r❡s✉❧t❛❞♦✱ ❝♦♠♦ ❡r❛ ❡s♣❡r❛❞♦✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ♦s ❝r✐tér✐♦s ❞❡ ❛rr❡❞♦♥❞❛♠❡♥t♦✈✐st♦s ❛♥t❡r✐♦r♠❡♥t❡ ♦ ✈❛❧♦r ♠é❞✐♦✱ ♦✉ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧✱ ❞❛ ❣r❛♥❞❡③❛ x é 0, 44✳ P♦rt❛♥t♦✱♥ã♦ é ♥❡❝❡ssár✐♦ q✉❡ ♦ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧ ❞❡ ✉♠❛ ♠❡❞✐❞❛ s❡❥❛ ❛q✉❡❧❡ q✉❡ ❛♣r❡s❡♥t❡ ❛ ♠❛✐♦r❢r❡q✉ê♥❝✐❛ r❡❧❛t✐✈❛ ❡ ♦❜s❡r✈❡ q✉❡ ♣❛r❛ ♦ ♣r❡s❡♥t❡ ❝❛s♦ ♦ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧ ❡stá ❡♥tr❡ 0, 44 s ❡0, 45 s✳ ❆❧é♠ ❞✐ss♦ é ❢❛❝✐❧♠❡♥t❡ ♦❜s❡r✈á✈❡❧ q✉❡ ❛❧❣✉♥s ✈❛❧♦r❡s ❡stã♦ ✏❛❝✐♠❛✑ ❞❛ ♠é❞✐❛✱ ♦✉tr♦s❡stã♦ ✏❛❜❛✐①♦✑ ❞❛ ♠é❞✐❛ ❡ ❛❧❣✉♥s ❝♦✐♥❝✐❞❡♠ ❝♦♠ ♦ ✈❛❧♦r ♠é❞✐♦✳

P♦❞❡♠♦s ❝❛❧❝✉❧❛r ♦ q✉❛♥t♦ ❝❛❞❛ ♠❡❞✐❞❛ ❛♣r❡s❡♥t❛✲s❡ ✏❢♦r❛✑ ♦✉ ❞❡s❧♦❝❛❞❛ ❞❛ ♠é❞✐❛✳

✸✳✷ ❉❡s✈✐♦ ❡ ❞❡s✈✐♦ ♠é❞✐♦

❈♦♠♦ ✈✐st♦✱ ❝❛❞❛ ♠❡❞✐❞❛ ♣♦❞❡ ❡st❛r ❛❝✐♠❛ ♦✉ ❛❜❛✐①♦ ❞♦ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧✱ ❛❝✐♠❛ ♦✉ ❛❜❛✐①♦❞❛ ♠é❞✐❛ ❝♦♠♦ s❡ ❝♦st✉♠❛ ❞✐③❡r✱ ❞❡ ♠♦❞♦ q✉❡ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ ❞❡s✈✐♦ ❞❛ ♠❡❞✐❞❛ i ♣❡❧❛❡①♣r❡ssã♦

∆xi = xi − 〈x〉

✷✹ ❈❆P❮❚❯▲❖ ✸✳ ❊❘❘❖❙ ❊ ■◆❈❊❘❚❊❩❆❙

❙❡❣✉❡ q✉❡ ❛s ♠❡❞✐❞❛s ❡st❛rã♦ ❡s♣❛❧❤❛❞❛s ❡♠ t♦r♥♦ ❞❛ ♠é❞✐❛✱ ❝❛❞❛ ✉♠❛ ❝♦♠ s❡✉ ❞❡s✈✐♦✱ ❡♣♦❞❡♠♦s ❝♦♥❝❡❜❡r ♦ ❞❡s✈✐♦ ♠é❞✐♦✳ ◆♦ ❡♥t❛♥t♦✱ ❛ss✉♠✐♥❞♦ q✉❡ ♦s ✈❛❧♦r❡s ♦❜t✐❞♦s sã♦ ✐❣✉❛❧♠❡♥t❡♣r♦✈á✈❡✐s✱ t❡r❡♠♦s ✈❛❧♦r❡s ❞❡ ❞❡s✈✐♦s t❛♠❜é♠ ✐❣✉❛❧♠❡♥t❡ ♣r♦✈á✈❡✐s❀ ❛ss✐♠✱ ♥✉♠ ❝♦♥❥✉♥t♦ ❝♦♠✉♠ ♥ú♠❡r♦ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ❞❡ ♠❡❞✐❞❛s✱ ❡s♣❡r❛♠♦s q✉❡ ♦ ❞❡s✈✐♦ ♠é❞✐♦ s❡❥❛ ♥✉❧♦✳ ❉❡❢❛t♦✱ 〈∆x〉 = 〈xi − 〈x〉〉 = 0✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ♠❡❞✐❞❛ ❞♦ ❞❡s✈✐♦ ♠é❞✐♦ ♣♦❞❡ ♥♦s ❞❛r ✐♥❢♦r♠❛çõ❡s❛ r❡s♣❡✐t♦ ❞♦ ❡rr♦ ❞❡ ✉♠❛ ♠❡❞✐❞❛ ❡ ❝♦♠♦ ❞❡s❡❥❛♠♦s ❡st✐♠❛r ♦ ❡rr♦ ♠á①✐♠♦ q✉❡ ♣♦❞❡ s❡r❝♦♠❡t✐❞♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ ❞❡s✈✐♦ ❛❜s♦❧✉t♦ ❡✱ ❡♠ t❡r♠♦s ❞❡st❡✱ ♦ ❞❡s✈✐♦ ❛❜s♦❧✉t♦ ♠é❞✐♦✳ ❖❞❡s✈✐♦ ❛❜s♦❧✉t♦ é s✐♠♣❧❡s♠❡♥t❡ |∆xi| = | 〈xi − 〈x〉〉 | ❡ 〈|∆x|〉 = 〈|xi − 〈x〉 |〉✱ ♦✉ s❡❥❛

〈|∆x|〉 = 1

N

N∑

i=1

|xi − 〈x〉 | ✭✸✳✷✳✶✮

❖ ❞❡s✈✐♦ ❞❡ ✉♠❛ ♠❡❞✐❞❛ ❛✐♥❞❛ ♣♦❞❡ s❡r ❡st❛❜❡❧❡❝✐❞♦ ❞❡ ✉♠❛ ♦✉tr❛ ♠❛♥❡✐r❛✳ ❉✐❣❛♠♦sq✉❡ ✉♠❛✱ ❡ ❛♣❡♥❛s ✉♠❛✱ ♠❡❞✐❞❛ ❢♦ss❡ r❡❛❧✐③❛❞❛✳ ◆❡st❡ ❝❛s♦✱ ♥ã♦ ♣♦❞❡♠♦s ❡st❛❜❡❧❡❝❡r ✉♠ ✈❛❧♦r♠é❞✐♦✱ tã♦ ♣♦✉❝♦✱ ✉♠ ❞❡s✈✐♦ ❛❜s♦❧✉t♦ ♠é❞✐♦✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ♠❡❞✐❞❛ r❡❛❧✐③❛❞❛ ❡①✐❣✐✉ ❛ ✉t✐❧✐③❛çã♦❞❡ ❞❡t❡r♠✐♥❛❞♦s ❛♣❛r❡❧❤♦s ❡ ✐♥str✉♠❡♥t♦s ❞❡ ♠❡❞✐❞❛✱ s❡♥❞♦ q✉❡ ❡st❡s ❛♣❛r❡❧❤♦s ❡ ✐♥str✉♠❡♥t♦❛♣r❡s❡♥t❡♠ ❡rr♦s ✐♥trí♥s❡❝♦s✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❡st❛❜❡❧❡❝❡r ♦ ❞❡s✈✐♦ ❞❡ ✉♠❛ ♠❡❞✐❞❛ ❡♠ t❡r♠♦s❞♦s ❡rr♦s ❡①✐st❡♥t❡s ♥❛ ❡s❝❛❧❛ ❞♦ ❛♣❛r❡❧❤♦s✳ ❈♦♥s✐❞❡r❡ ♦ s✐♠♣❧❡s ❝❛s♦ ❞❡ ♠❡❞✐r ♦ ❝♦♠♣r✐♠❡♥t♦❞❡ ✉♠❛ ♠❡s❛✳ ❙❡ ❢♦r ✉t✐❧✐③❛❞❛ ✉♠❛ ré❣✉❛ ❝❡♥t✐♠❡tr❛❞❛ ❡♥tã♦ ❡st❛ ♠❡❞✐❞❛✱ q✉❛♥❞♦ ❡①♣r❡ss❛ ❡♠❝❡♥tí♠❡tr♦s✱ ❞❡✈❡ ❛♣r❡s❡♥t❛r ✉♠❛ ❝❛s❛ ❞❡❝✐♠❛❧ ♣♦rq✉❡ ♣♦❞❡♠♦s ❡st✐♠❛r ❡st❛ ❝❛s❛ ❞❡❝✐♠❛❧ ❡ ✐st♦é ❢❡✐t♦ ❝♦♠♣❛r❛♥❞♦✲s❡ ❛ ❢r❛çã♦ ❞♦ ❝❡♥tí♠❡tr♦ ❝♦♠ ♦ ❝❡♥tí♠❡tr♦ ❡ ❡st❛ ❢r❛çã♦ ♣♦❞❡ s❡r t♦♠❛❞❛❝♦♠ ❛ ❞é❝✐♠❛ ♣❛rt❡ ❞♦ ❝❡♥tí♠❡tr♦✳ P♦rt❛♥t♦✱ ♦ ❡rr♦ ♥❛ ♠❡❞✐❞❛ ❞❡✈❡ ❛♣r❡s❡♥t❛r t❛♠❜é♠ ✉♠❛❝❛s❛ ❞❡❝✐♠❛❧✱ q✉❛♥❞♦ ❡①♣r❡ss♦ ❡♠ ❝❡♥tí♠❡tr♦s✳ ❚❛❧ ♠❡❞✐❞❛ ♣♦❞❡r✐❛ s❡r✱ ❞✐❣❛♠♦s✱ 152, 7 cm✱✈✐st♦ q✉❡ ❢❛❝✐❧♠❡♥t❡ é ✐❞❡♥t✐✜❝❛❞♦ q✉❡ ❛ ♠❡❞✐❞❛ é ✐♥❢❡r✐♦r ❛ 153, 0 cm ❡ s✉♣❡r✐♦r ❛ 152, 0 cm✳❆❧é♠ ❞✐ss♦✱ ♦✉tr♦ ❡①♣❡r✐♠❡♥t❛❞♦r ♣♦❞❡r✐❛✱ s❡♠ ♥❡♥❤✉♠ ♣r♦❜❧❡♠❛✱ t❡r ♠❡❞✐❞♦ ♦ ✈❛❧♦r 152, 6 cm✳P♦rt❛♥t♦✱ t❡♥❞♦ ✈❡r✐✜❝❛❞♦ q✉❡ ❛ ♠❡♥♦r ❞✐✈✐sã♦ ❞❡ ❡s❝❛❧❛ ✈❛❧❡ ✉♠ ❝❡♥tí♠❡tr♦✱ ♣♦❞❡♠♦s ❛ss✉♠✐rq✉❡ ❛ ✐♥❝❡rt❡③❛ ❡♠ t❛❧ ♠❡❞✐❞❛ s❡❥❛ ❛ ♠❡t❛❞❡ ❞❡st❛ ♠❡♥♦r ❞✐✈✐sã♦ ❞❡ ❡s❝❛❧❛✱ ♦✉ s❡❥❛✱ 0, 5 cm✳■st♦ ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❛♦ t✐♣♦ ❞❡ ✐♥str✉♠❡♥t♦ ✉t✐❧✐③❛❞♦ ♥♦ ♣r♦❝❡ss♦ ❞❡ ♠❡❞✐❞❛✱ ✈✐st♦q✉❡ ❢♦✐ ♥❡❝❡ssár✐♦ ❛✈❛❧✐❛r ♦ ú❧t✐♠♦ ❛❧❣❛r✐s♠♦ s✐❣♥✐✜❝❛t✐✈♦✱ ♣♦rt❛♥t♦ ♦ ❛❧❣❛r✐s♠♦ ❞✉✈✐❞♦s♦✱ ❞❛♠❡❞✐❞❛✳ ◆♦ ❡♥t❛♥t♦✱ ❞❡♣❡♥❞❡♥❞♦ ❞♦ ✐♥str✉♠❡♥t♦ ❞❡ ♠❡❞✐❞❛ ✉t✐❧✐③❛❞♦✱ ♣♦❞❡rá ❛❝♦♥t❡❝❡r ❞❡ ♥ã♦❡①✐st✐r ♦ ❛❧❣❛r✐s♠♦ ❡st✐♠❛❞♦✱ ♦✉ s❡❥❛✱ ♦ ❛♣❛r❡❧❤♦ ♣♦❞❡ s❡r ❝♦♥str✉í❞♦ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ t♦❞♦s ♦s❛❧❣❛r✐s♠♦s ❞❛ ♠❡❞✐❞❛ ♣♦ss❛♠ s❡r ❧✐❞♦s✳ ❯♠ ❡①❡♠♣❧♦ ❜❡♠ ❝♦♠✉♠ é ❛ ❧❡✐t✉r❛ ❞❡ ✉♠ ✐♥t❡r✈❛❧♦❞❡ t❡♠♣♦ ♥✉♠ ❝r♦♥ô♠❡tr♦ ❞✐❣✐t❛❧✱ ❝♦♠♦ ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✳

■◆❙❊❘■❘ ❋■●❯❘❆ ❈❖▼ ▲❊■❚❯❘❆ ❊▼ ❈❘❖◆Ô▼❊❚❘❖❙✳

❋✐❣✉r❛ ✸✳✷✳✶✿ ▲❡✐t✉r❛ ♥✉♠ ❝r♦♥ô♠❡tr♦ ❞✐❣✐t❛❧✳ ❚♦❞♦s ♦s ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s ❞❛ ♠❡❞✐❞❛♣♦❞❡♠ s❡r ❧✐❞♦s ❞✐r❡t❛♠❡♥t❡ ♥♦ ❝r♦♥ô♠❡tr♦✳

▼❡s♠♦ q✉❡ t♦❞♦s ♦s ❛❧❣❛r✐s♠♦s ♣♦ss❛♠ s❡r ❧✐❞♦s✱ é ✐♠♣♦rt❛♥t❡ ♥♦t❛r q✉❡ ♦ ú❧t✐♠♦ ❛❧❣❛r✐s♠♦❝♦rr❡s♣♦♥❞❡ ❛♦ ❛❧❣❛r✐s♠♦ ❞✉✈✐❞♦s♦ ♣♦rq✉❡ ❞❡ ❛❧❣✉♠❛ ♠❛♥❡✐r❛ ♦ ♠❡❝❛♥✐s♠♦ ✐♥t❡r♥♦ ❞❡ ❢✉♥❝✐♦♥❛✲♠❡♥t♦ ❞♦ r❡❧ó❣✐♦ ♣r❡❝✐s❛ ❞❡❝✐❞✐r q✉❛❧ ✈❛❧♦r ❞❡✈❛ s❡r ❛♣r❡s❡♥t❛❞♦✳ P♦rt❛♥t♦✱ ✐♥t❡r♥❛♠❡♥t❡✱ ❡❧❡❞❡✈❡ t❡r ❛❧❣✉♠ ❝r✐tér✐♦ ❞❡ ❛rr❡❞♦♥❞❛♠❡♥t♦ ❡ ♣♦r ✐ss♦✱ ♦ ú❧t✐♠♦ ❛❧❣❛r✐s♠♦ ❝♦rr❡s♣♦♥❞❡✱ ❞❡ ❢❛t♦✱❛♦ ❛❧❣❛r✐s♠♦ ❞✉✈✐❞♦s♦✳ ❖ ❢❛t♦ ❞❡ ❛❧❣✉♥s ❛♣❛r❡❧❤♦s ❢♦r♥❡❝❡r❡♠ ❛ ❧❡✐t✉r❛ ❞❡ t♦❞♦s ♦s ❛❧❣❛r✐s♠♦s❞❛ ♠❡❞✐❞❛ ❡ ♦✉tr♦s ❛♣❛r❡❧❤♦s ♣❡r♠✐t✐r❡♠ ❛ ❡st✐♠❛t✐✈❛ ❞♦ ú❧t✐♠♦ ❛❧❣❛r✐s♠♦ ❞❛ ♠❡❞✐❞❛ ❢❛③ ❝♦♠q✉❡ ❡❧❡s s❡❥❛♠ ❝❧❛ss✐✜❝❛❞♦s ❡♠ ❞♦✐s t✐♣♦s✱ ❛ s❛❜❡r✿

✸✳✸✳ ❉❊❙❱■❖ P❆❉❘➹❖ ✷✺

✐♥str✉♠❡♥t♦s ❛♥❛❧ó❣✐❝♦s✿ sã♦ ♦s ✐♥str✉♠❡♥t♦s q✉❡ ♥ã♦ ❢♦r♥❡❝❡♠ t♦❞♦s ♦s ❛❧❣❛r✐s♠♦s ❞❡ ✉♠❛♠❡❞✐❞❛✱ s❡♥❞♦ q✉❡ ♦ ú❧t✐♠♦ ❛❧❣❛r✐s♠♦ ❞❡✈❡ s❡r ❛✈❛❧✐❛❞♦✳ ❙ã♦ ❡①❡♠♣❧♦s ❛ tr❡♥❛ ❡ ♦t❡r♠ô♠❡tr♦ ❞❡ ♠❡r❝úr✐♦✳ P❛r❛ ❡st❡s ❛♣❛r❡❧❤♦s ❛ss✉♠❡✲s❡ q✉❡ ♦ ❡rr♦ ❞❡ ❡s❝❛❧❛ s❡❥❛ ❛♠❡t❛❞❡ ❞❛ ♠❡♥♦r ❞✐✈✐sã♦ ❞❡ ❡s❝❛❧❛✳

✐♥str✉♠❡♥t♦s ♥ã♦ ❛♥❛❧ó❣✐❝♦s✿ sã♦ ♦s ✐♥str✉♠❡♥t♦s q✉❡ ❢♦r♥❡❝❡♠ t♦❞♦s ♦s ❛❧❣❛r✐s♠♦s ❞❛♠❡❞✐❞❛✱ ♦✉ s❡❥❛✱ ❡❧❡ ♥ã♦ ♣❡r♠✐t❡ ❛ ❛✈❛❧✐❛çã♦ ❞❡ ♥❡♥❤✉♠ ❛❧❣❛r✐s♠♦✳ ❙ã♦ ❡①❡♠♣❧♦s✷ ✉♠r❡❧ó❣✐♦ ❞✐❣✐t❛❧ ❡ ✉♠❛ ❜❛❧❛♥ç❛ ❞✐❣✐t❛❧✳ ❆ss✉♠❡✲s❡✱ ♣❛r❛ t❛✐s ❛♣❛r❡❧❤♦s✱ q✉❡ ♦ ❡rr♦ ❞❡ ❡s❝❛❧❛s❡❥❛ ❛ ♠❡♥♦r ❞✐✈✐sã♦ ❞❡ ❡s❝❛❧❛✳

◆♦ ❡①❡♠♣❧♦ ❛❝✐♠❛ s♦❜r❡ ❛ ♠❡❞✐❞❛ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♠❡s❛✱ ❛ ♠❡♥♦r ❞✐✈✐sã♦ ❞❡ ❡s❝❛❧❛ é ♦❝❡♥tí♠❡tr♦✳ P♦rt❛♥t♦ ♦ ❡rr♦ ❞❡ ❡s❝❛❧❛ ✈❛❧❡ 0, 5 cm ❡ ❛ ♠❡❞✐❞❛ ♠❛✐s ♣r♦✈á✈❡❧ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛♠❡s❛ ❞❡✈❡ ❡st❛r ❡♥tr❡ (152, 7− 0, 5) cm ❡ (152, 7 + 0, 5) cm✱ ♦✉ s❡❥❛✱ ❡♥tr❡ 152, 2 cm ❡ 153, 2 cm✳

P♦rt❛♥t♦✱ s❡♠♣r❡ q✉❡ ❡s♣❡❝✐✜❝❛r♠♦s ❛ ♠❡❞✐❞❛ ❞❡ ✉♠❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛✱ ❞❡✈❡r❡♠♦s ❢❛③ê✲❧♦ ❞❛s❡❣✉✐♥t❡ ♠❛♥❡✐r❛

〈x〉 ± δx ✭✸✳✷✳✷✮

s❡♥❞♦ q✉❡ ♦ ✈❛❧♦r ❞❛ ✐♥❝❡rt❡③❛ δx ♣♦❞❡ r❡♣r❡s❡♥t❛r ♦ ❞❡s✈✐♦ ❛❜s♦❧✉t♦ ♠é❞✐♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡♠❡❞✐❞❛s✱ ♦✉ ♠❡s♠♦ ♦ ✈❛❧♦r ❞♦ ❡rr♦ ❞❡ ❡s❝❛❧❛ ❛ss♦❝✐❛❞♦ ❛♦ ✐♥str✉♠❡♥t♦ ✉t✐❧✐③❛❞♦ ♥♦ ♣r♦❝❡ss♦ ❞❡♠❡❞✐❞❛✳ ❊st❛ ♠❡s♠❛ ✐♥❝❡rt❡③❛ ♣❡r♠✐t❡ ❛✐♥❞❛ ♦✉tr❛ ❞❡✜♥✐çã♦✳ ❉❡✜♥✐♠♦s ❛ ✐♥❝❡rt❡③❛ ❢r❛❝✐♦♥ár✐❛❝♦♠♦ ❛ r❛③ã♦ ❡♥tr❡ ❛ ✐♥❝❡rt❡③❛ ❞❛ ♠❡❞✐❞❛ ❡ ♦ s❡✉ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧✱ δx/ 〈x〉✱ ♣♦❞❡♥❞♦ s❡r❡s❝r✐t❛ ❝♦♠♦ ✉♠❛ ♣❡r❝❡♥t❛❣❡♠✳ P❛r❛ ♦ ❡①❡♠♣❧♦ ❞✐s❝✉t✐❞♦ ❛❝✐♠❛✱ t❡♠♦s 0, 5/152, 7 = 0, 3%✳❆ss✐♠ ♦ ✈❛❧♦r ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♠❡s❛ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦

(152, 7± 0, 5) cm

♦✉(152, 7± 0, 3%) cm.

✸✳✸ ❉❡s✈✐♦ ♣❛❞rã♦

❱✐♠♦s q✉❡ ✉♠❛ ♠❛♥❡✐r❛ ❞❡ s❡ ❡st✐♠❛r ❛ ✐♥❝❡rt❡③❛ ♥✉♠❛ ♠❡❞✐❞❛ é ❛tr❛✈és ❞♦ ❝á❧❝✉❧♦ ❞♦ ❞❡s✈✐♦❛❜s♦❧✉t♦ ♠é❞✐♦✱ ❡♠ ❣❡r❛❧ ❝❤❛♠❛❞♦ s✐♠♣❧❡s♠❡♥t❡ ❞❡ ❞❡s✈✐♦ ♠é❞✐♦✳ ◆♦ ❡♥t❛♥t♦✱ ❤á ✉♠❛ ♦✉tr❛♠❛♥❡✐r❛ ❞❡ s❡ ❞❡✜♥✐r ❛ ✐♥❝❡rt❡③❛ ❞❡ ✉♠❛ ♠❡❞✐❞❛✱ ✉s❛♥❞♦✲s❡ ♦ ❝♦♥❝❡✐t♦ ❞♦ ❞❡s✈✐♦ ♣❛❞rã♦✳ ❉❡❢❛t♦✱ ♦ ❞❡s✈✐♦ ♣❛❞rã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛s x1, x2, · · · , xN é ✉♠❛ ❡st✐♠❛t✐✈❛ ❞❛ ✐♥❝❡rt❡③❛♠é❞✐❛ ❞❛s ♠❡❞✐❞❛s x1, x2, · · · , xN ✳ ❖ ❞❡s✈✐♦ ♣❛❞rã♦ é ❞❡✜♥✐❞♦ ❡♠ t❡r♠♦s ❞❛ r❛✐③ q✉❛❞rát✐❝❛♠é❞✐❛ ❞♦s ❞❡s✈✐♦✱ ❝♦♠♦ s❡❣✉❡✳ ❯♠❛ ✈❡③ q✉❡ t❡♥❤❛♠♦s ♦s ✈❛❧♦r❡s ❞❛ ❣r❛♥❞❡③❛ x✱ ❝❛❧❝✉❧❛♠♦s ♦s❡✉ ✈❛❧♦r ♠é❞✐♦ ❡✱ ❛ ♣❛rt✐r ❞❡st❡✱ ♦ ❞❡s✈✐♦ ❞❡ ❝❛❞❛ ♠❡❞✐❞❛✳ ❊❧❡✈❛♠♦s ❝❛❞❛ ❞❡s✈✐♦ ❛♦ q✉❛❞r❛❞♦❡ t♦♠❛♠♦s ♦ ✈❛❧♦r ♠é❞✐♦ ♦❜t❡♥❞♦ ❛ q✉❛♥t✐❞❛❞❡

1

N

N∑

i=1

(xi − 〈x〉)2

❝✉❥❛ r❛✐③ q✉❛❞r❛❞❛ ♣♦ss✉✐ ❛ ♠❡s♠❛ ❞✐♠❡♥sã♦ ❞❛ ❣r❛♥❞❡③❛ x✱ ❛ss✐♠ ❞❡✜♥✐♥❞♦ ♦ ❞❡s✈✐♦ ♣❛❞rã♦❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛s x1, x2, · · · , xN ✳ ❊♠ sí♠❜♦❧♦s

σx =

√

√

√

√

1

N

N∑

i=1

(xi − 〈x〉)2 . ✭✸✳✸✳✶✮

✷❖s ❛♣❛r❡❧❤♦s ❞♦t❛❞♦s ❞❡ ♥ô♥✐♦ ✭♦✉ ❱❡r♥✐❡r✮ t❛♠❜é♠ s❡ ❡♥q✉❛❞r❛♠ ♥❡st❛ ❝❛t❡❣♦r✐❛✱ ❡ s❡rã♦ ❛♥❛❧✐s❛❞♦s♣♦st❡r✐♦r♠❡♥t❡✳

✷✻ ❈❆P❮❚❯▲❖ ✸✳ ❊❘❘❖❙ ❊ ■◆❈❊❘❚❊❩❆❙

❊st❡ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡ ❡❧❡✈❛r ✉♠❛ q✉❛♥t✐❞❛❞❡ ❛♦ q✉❛❞r❛❞♦✱ ❞❡♣♦✐s ❝❛❧❝✉❧❛r s❡✉ ✈❛❧♦r ♠é❞✐♦❡ ♣♦r ✜♠✱ ❡①tr❛✐r ❛ r❛✐③ q✉❛❞r❛❞❛ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ❝á❧❝✉❧♦ ❞♦ ✈❛❧♦r q✉❛❞rát✐❝♦ ♠é❞✐♦ ♦✉❘▼❙ ✭r♦♦t ♠❡❛♥ sq✉❛r❡✮✳ ❊①✐st❡♠ ❛r❣✉♠❡♥t♦s t❡ór✐❝♦s q✉❡ ❥✉st✐✜❝❛♠ q✉❡ ♦ ❞❡♥♦♠✐♥❛❞♦r ♥❛❡q✳✭✸✳✸✳✶✮ s❡❥❛ N − 1 ❛♦ ✐♥✈és ❞❡ N ✳ ❯♠ ❛r❣✉♠❡♥t♦ q✉❡ s❡ ♣♦❞❡r✐❛ ✉s❛r é q✉❡ ♣❛r❛ ♦ ❝❛s♦♣❛rt✐❝✉❧❛r ❞❡ ❛♣❡♥❛s ✉♠❛ ♠❡❞✐❞❛✱ ♦ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧ ❞❡st❛ s❡r✐❛ ❛ ♣ró♣r✐❛ ♠❡❞✐❞❛✱ q✉❡ t❡r✐❛✉♠ ❞❡s✈✐♦ ♣❛❞rã♦ ♥✉❧♦✳ P♦rt❛♥t♦ s❡r✐❛ ✉♠❛ ♠❡❞✐❞❛ ❡①❛t❛✱ ♦ q✉❡ ♥ã♦ é ✈❡r❞❛❞❡✳ P♦r ♦✉tr♦ ❧❛❞♦✱❡st❡ ♠❡s♠♦ ❝❛s♦ ❞❡ ❛♣❡♥❛s ✉♠❛ ♠❡❞✐❞❛ ❛♣r❡s❡♥t❛r✐❛ ✉♠ ❞❡s✈✐♦ ♣❛❞rã♦✱ s❡❣✉♥❞♦ ❡q✳✭✸✳✸✳✶✮ ❝♦♠N − 1 ♥♦ ❧✉❣❛r ❞❡ N ✱ ❝♦♠♦ s❡♥❞♦ 0

0q✉❡ é ✉♠❛ ✐♥❞❡t❡r♠✐♥❛çã♦ ♠❛t❡♠át✐❝❛✳ ❖✉ s❡❥❛✱ ♦ ❞❡s✈✐♦

♣❛❞rã♦ ♥ã♦ ❡st❛r✐❛ ❞❡✜♥✐❞♦✳ ▼❛s ✐st♦ é ❝♦♠♣❛tí✈❡❧ ❝♦♠ ♥♦ss❛ ❢❛❧t❛ ❞❡ ✐♥❢♦r♠❛çã♦ ❛❝❡r❝❛ ❞❛✐♥❝❡rt❡③❛ ❞❛ ♠❡❞✐❞❛ ❛♣ós t❡r♠♦s r❡❛❧✐③❛❞♦ s♦♠❡♥t❡ ✉♠❛ ♠❡❞✐❞❛✳ P♦rt❛♥t♦✱ ❛❞♦t❛r❡♠♦s ❝♦♠♦❞❡✜♥✐çã♦ ❞♦ ❞❡s✈✐♦ ♣❛❞rã♦ ❛ r❡❧❛çã♦

σx =

√

√

√

√

1

N − 1

N∑

i=1

(xi − 〈x〉)2 . ✭✸✳✸✳✷✮

❖ ❞❡s✈✐♦ ♣❛❞rã♦ t❛♠❜é♠ ♣♦❞❡ s❡r✈✐r ❝♦♠♦ ❛ ✐♥❝❡rt❡③❛ ❞❡ ✉♠❛ ú♥✐❝❛ ♠❡❞✐❞❛✳ ■st♦ ♣♦rq✉❡s❡ ✉♠❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ❢♦r ♠❡♥s✉r❛❞❛ r❡♣❡t✐❞❛s ✈❡③❡s✱ s❡❣✉✐♥❞♦ ♦s ♠❡s♠♦s ❝r✐tér✐♦s ❡ ♠ét♦❞♦s✱❡ s❡ ❛s ❢♦♥t❡s ❞❡ ❡rr♦s ❝♦♥tr✐❜✉ír❡♠ ❝♦♠ ❡rr♦s ♣❡q✉❡♥♦s ❡ ❛❧❡❛tór✐♦s✱ ❡♥tã♦ ♦s ✈❛❧♦r❡s ♠❡❞✐❞♦s❡st❛rã♦ ❡s♣❛❧❤❛❞♦s ❡♠ t♦r♥♦ ❞♦ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧ ❞❡ t❛❧ ♠♦❞♦ q✉❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ 68% ❞❡❧❡s❡st❛rã♦ ♥♦ ✐♥t❡r✈❛❧♦ (〈x〉 − σx, 〈x〉+ σx)✳ P♦rt❛♥t♦ ♣♦❞❡♠♦s ✐♥❢❡r✐r q✉❡✱ ❛♦ r❡❛❧✐③❛r♠♦s ❛♣❡♥❛s✉♠❛ ♠❡❞✐❞❛ ❞❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ❡♠ q✉❡stã♦✱ s❡❣✉✐♥❞♦✲s❡ ♦s ♠ét♦❞♦s ❡ ❝r✐tér✐♦s ♣ré✲❡st❛❜❡❧❡❝✐❞♦s✱❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ q✉❡ ♦ ✈❛❧♦r ❡st❡❥❛ ♥♦ ✐♥t❡r✈❛❧♦ (〈x〉 − σx, 〈x〉+ σx) ✈❛❧❡✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✱68%. ❆ss✐♠✱ ♦ ✈❛❧♦r ❞❡ ✉♠❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ❞❡✈❡ s❡r ❡s❝r✐t♦ ❝♦♠♦

〈x〉 ± σx

✸✳✹ ❊rr♦ s✐st❡♠át✐❝♦

❚♦❞♦ ♣r♦❝❡ss♦ ❞❡ ♠❡❞✐❞❛ ✉t✐❧✐③❛ ❛♦ ♠❡♥♦s ✉♠ ✐♥str✉♠❡♥t♦ ❞❡ ♠❡❞✐❞❛✱ q✉❡ ♣♦❞❡ s❡r ✉♠❛s✐♠♣❧❡s ré❣✉❛✱ ♦✉ ✉♠ ❝r♦♥ô♠❡tr♦✱ ✉♠❛ ❜❛❧❛♥ç❛✱ ♦✉ ✉♠ ❞✐♥❛♠ô♠❡tr♦✱ ❡♥tr❡ ♦✉tr♦s✳ ❈♦♥s✐❞❡r❡❛s ♠❡❞✐❞❛s ❞♦ ♣❡s♦ ❞❡ ❞✐❢❡r❡♥t❡s ♦❜❥❡t♦s✱ q✉❡ ♣♦❞❡ s❡r ❢❡✐t❛ ♣❡♥❞✉r❛♥❞♦ t❛✐s ♦❜❥❡t♦s ♥✉♠❞✐♥❛♠ô♠❡tr♦✳ ❙❡ ♦ ❞✐♥❛♠ô♠❡tr♦ ♥ã♦ ❡st✐✈❡r ❝♦rr❡t❛♠❡♥t❡ ❝❛❧✐❜r❛❞♦✱ ♦✉ s❡❥❛✱ s❡ ❛ ♣♦s✐çã♦❞❡ ❡q✉✐❧í❜r✐♦ ❞❛ ♠♦❧❛ ♥ã♦ ❡st✐✈❡r ❝♦rr❡t❛♠❡♥t❡ ❛❧✐♥❤❛❞❛ ❝♦♠ ❛ ♠❛r❝❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ③❡r♦❞❡ ❢♦rç❛✱ ❡♥tã♦ t♦❞♦s ♦s ✈❛❧♦r❡s ❞❡ ♣❡s♦ ♠❡❞✐❞♦s ❡st❛rã♦ ❞❡s❧♦❝❛❞♦s ❞♦ ✈❛❧♦r r❡❛❧✱ ♣♦❞❡♥❞♦❛♣r❡s❡♥t❛r ✉♠ ✈❛❧♦r ♠❛✐♦r ♦✉ ♠❡♥♦r✳ ❆ss✐♠✱ ♦s ✈❛❧♦r❡s ♠❡❞✐❞♦s ❡st❛rã♦ s✐st❡♠❛t✐❝❛♠❡♥t❡❛❢❛st❛❞♦s ❞♦ ✈❛❧♦r r❡❛❧✳ ◆♦ ❡♥t❛♥t♦✱ ✉♠❛ ✈❡③ ✐❞❡♥t✐✜❝❛❞♦ ❡st❡ ❡rr♦ ✐♥tr♦❞✉③✐❞♦ ♣♦r ❝♦♥t❛ ❞❛ ♠á❝❛❧✐❜r❛çã♦ ❞♦ ❞✐♥❛♠ô♠❡tr♦✱ ♣♦❞❡✲s❡ ❡❧✐♠✐♥❛r ❡st❡ t✐♣♦ ❞❡ ❡rr♦✳ ❊♠ s✉♠❛✱ ♦ ❡rr♦ s✐st❡♠át✐❝♦ é❛q✉❡❧❡ q✉❡ ❛t✉❛ s❡♠♣r❡ ❞❛ ♠❡s♠❛ ❢♦r♠❛ ❡✱ ✉♠❛ ✈❡③ ✐❞❡♥t✐✜❝❛❞♦✱ ♣♦❞❡ s❡r ❡❧✐♠✐♥❛❞♦✳

✸✳✺ Pr♦♣❛❣❛çã♦ ❞❡ ❡rr♦s

❱✐♠♦s q✉❡ t♦❞❛s ❛s ♠❡❞✐❞❛s ❛♣r❡s❡♥t❛♠ ❡rr♦s✱ s❡❥❛♠ ❡❧❡s ❡rr♦s ❞❡ ❡s❝❛❧❛✱ ❡rr♦s s✐st❡♠át✐❝♦s♦✉ ❡rr♦s ❛❧❡❛tór✐♦s ❡ q✉❡ ♠✉✐t❛s ❣r❛♥❞❡③❛s ❢ís✐❝❛s sã♦ ❞❡t❡r♠✐♥❛❞❛s ❛tr❛✈és ❞❡ ♠❡❞✐❞❛s ✐♥✲❞✐r❡t❛s✱ ♦✉ s❡❥❛✱ ❛tr❛✈és ❞❡ ♦♣❡r❛çõ❡s ♠❛t❡♠át✐❝❛s r❡❛❧✐③❛❞❛s ❝♦♠ ❣r❛♥❞❡③❛s ❢ís✐❝❛s ♠❡❞✐❞❛s❞✐r❡t❛♠❡♥t❡✳ P♦rt❛♥t♦✱ ❡s♣❡r❛♠♦s q✉❡ ♦s ❡rr♦s ❞❡st❛s ❣r❛♥❞❡③❛s s❡❥❛♠ ✏♣r♦♣❛❣❛❞♦s✑ ♣❛r❛ ❛s❣r❛♥❞❡③❛s ♠❡❞✐❞❛s ✐♥❞✐r❡t❛♠❡♥t❡✳ ◗✉❛♥❞♦ ❡st❛❜❡❧❡❝❡♠♦s q✉❡ ❛ ❛❝❡❧❡r❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡ ♣♦❞❡s❡r ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦ t❡♠♣♦ ❞❡ q✉❡❞❛ ❞❡ ✉♠❛ ♣❡q✉❡♥❛ ❡s❢❡r❛✱ ❛tr❛✈és ❞❛ r❡❧❛çã♦ g = 2d/t2✱❡♥tã♦ ✜❝❛ ❡✈✐❞❡♥t❡ q✉❡ t❛♥t♦ ❛ ♠❡❞✐❞❛ ❞♦ t❡♠♣♦ ❞❡ q✉❡❞❛ q✉❛♥t♦ ❛ ♠❡❞✐❞❛ ❞❛ ❛❧t✉r❛ ❞❡ q✉❡❞❛❛♣r❡s❡♥t❛♠ ❡rr♦s✱ ♥♦ ♠í♥✐♠♦ ❡rr♦s ❞❡ ❡s❝❛❧❛✳ ❊♥tã♦ q✉❛❧ ♦ ❡rr♦ ❛ss♦❝✐❛❞♦ ❛♦ ✈❛❧♦r ❞❡ g❄

✸✳✺✳ P❘❖P❆●❆➬➹❖ ❉❊ ❊❘❘❖❙ ✷✼

❈♦♠❡❝❡♠♦s ❝♦♠ ❝á❧❝✉❧♦s ♠❛✐s s✐♠♣❧❡s✳ ❱❛♠♦s s✉♣♦r q✉❡ ✉♠❛ ❞❛❞❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ s❡❥❛❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛ s♦♠❛ ❞❡ ❞✉❛s ♦✉tr❛s ❣r❛♥❞❡③❛s✱ ♦❜✈✐❛♠❡♥t❡ ❞❡ ♠❡s♠❛ ❞✐♠❡♥sã♦✱ ❝♦♠♦ u =v + w ❡ s❡❥❛♠ δv ❡ δw ♦s ❡rr♦s ❛ss♦❝✐❛❞♦s ❛ v ❡ w✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ✐♥❞✐❝❛♥❞♦ q✉❡ ♦ ✈❛❧♦r♠❛✐s ♣r♦✈á✈❡❧ ❞❡ v ❡stá ❡♥tr❡ v − δv ❡ v + δv✳ ❉♦ ♠❡s♠♦ ♠♦❞♦✱ ♦ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧ ❞❡ w❡stá ❡♥tr❡ w − δw ❡ w + δw✳ ❊s♣❡r❛♠♦s q✉❡ ♦ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧ ❞❡ u ❡st❡❥❛ ❡♥tr❡ u − δu❡ u + δu✳ ▼❛s ♦ ♠❛✐♦r ✈❛❧♦r ❞❡ u é umax = (v + δv) + (w + δw) ❡♥q✉❛♥t♦ ♦ ♠❡♥♦r ✈❛❧♦r éumın = (v − δv) + (w − δw)✳ ❊♥tã♦ ♦ ✐♥t❡r✈❛❧♦ ❞❡ ✈❛❧♦r❡s ♣❛r❛ u é

(v + w)− (δv + δw) < u < (v + w) + (δv + δw)

♥♦s r❡✈❡❧❛♥❞♦ q✉❡

δu = δv + δw .

❆❣♦r❛ ✈❛♠♦s s✉♣♦r q✉❡ ♥♦ss❛ ✈❛r✐á✈❡❧ s❡❥❛ ❞❛❞❛ ♣❡❧♦ ♣r♦❞✉t♦ ❞❛s ♦✉tr❛s ❞✉❛s✱ ♦✉ s❡❥❛✱ u = vw✳❈♦♠♦ ♥♦ ❝❛s♦ ❛♥t❡r✐♦r✱ ❞❡s❡❥❛♠♦s s❛❜❡r q✉❛❧ ♦ ❡rr♦ ♥❛ ❣r❛♥❞❡③❛ u q✉❛♥❞♦ sã♦ ♠❡❞✐❞❛s ❛s❣r❛♥❞❡③❛s v ❡ w✳ ❙❡❣✉✐♥❞♦ ❛s ❧✐♥❤❛s ❛♥t❡r✐♦r❡s✱ ✈❡♠♦s q✉❡ umax = (v + δv) × (w + δw) ❡q✉❡ umın = (v − δv) × (w − δw)✳ ❙❡❣✉❡ q✉❡ umax = vw + vδw + wδv + δvδw✳ ▼❛s ♣❡❧♦❢❛t♦ ❞❡ ❡st❛r♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ❡rr♦s ♣❡q✉❡♥♦s✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ ♦s ❡rr♦s ❡♠ ❣❡r❛❧ sã♦ ✉♠❛♦r❞❡♠ ❞❡ ❣r❛♥❞❡③❛ ♠❡♥♦r❡s q✉❡ ❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛✱ ♣♦❞❡♠♦s ❞❡s♣r❡③❛r ♦ t❡r♠♦ δvδw ❢r❡♥t❡❛♦s ❞❡♠❛✐s✱ ♦❜t❡♥❞♦ umax ≈ vw + (vδw + wδv)✳ ❉❡ ♠♦❞♦ s❡♠❡❧❤❛♥t❡✱ ♦❜té♠✲s❡ q✉❡ umın ≈vw − (vδw + wδv)✳ ❈♦♥❝❧✉í♠♦s q✉❡

δu = vδw + wδv .

❊ ♣❛r❛ ❛ s✐t✉❛çã♦ ❡♠ q✉❡ u = f(v) é ✉♠❛ ❢✉♥çã♦ q✉❛❧q✉❡r ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ♦✉ u = f(v, w)é ✉♠❛ ❢✉♥çã♦ q✉❛❧q✉❡r ❞❡ ❞✉❛s ✈❛r✐á✈❡✐s❄ ❈♦♥s✐❞❡r❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ ♦ ❝❛s♦ ❞❡ ✉♠❛ ❢✉♥çã♦❞❡ ❛♣❡♥❛s ✉♠❛ ✈❛r✐á✈❡❧✳ P♦❞❡♠♦s ❛❞♠✐t✐r q✉❡ umax = f(vmax)✱ ♦✉ s❡❥❛✱ u + δu = f(v + δv) ❡✱♣♦rt❛♥t♦✱ δu = f(v + δv) − f(v)✳ ❆❣♦r❛ ♣♦❞❡♠♦s ❢❛③❡r ✉♠❛ ❡①♣❛♥sã♦ ❡♠ sér✐❡ ❞❡ ❚❛②❧♦r ❡♠

t♦r♥♦ ❞♦ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧ ♦❜t❡♥❞♦ f(v + δv) ≈ f(v) +df

dvδv✳ ▲♦❣♦✱

δu ≈ df

dvδv

P❡❧♦ ❢❛t♦ ❞❡ ❛ ❢✉♥çã♦ f ♣♦❞❡r s❡r ❞❡❝r❡s❝❡♥t❡ ❡ q✉❡r❡r♠♦s ❡st✐♠❛r ♦ ♠❛✐♦r ❡rr♦✱ t♦♠❛♠♦s♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❛ ❞❡r✐✈❛❞❛ q✉❡ ❛♣❛r❡❝❡ ♥❛ ❡q✉❛çã♦ ❛❝✐♠❛ ♣❛r❛ ❛ss❡❣✉r❛r q✉❡ ❡st❛♠♦s✱ ❞❡❢❛t♦✱ ❡st✐♠❛♥❞♦ ♦ ♠❛✐♦r ❡rr♦ ♣r♦♣❛❣❛❞♦ ♣♦ssí✈❡❧ ❡ ♣♦❞❡♠♦s t❛♠❜é♠ tr♦❝❛r ♦ sí♠❜♦❧♦ ❞❡ ✈❛❧♦r❛♣r♦①✐♠❛❞♦ ♣❡❧♦ sí♠❜♦❧♦ ❞❛ ✐❣✉❛❧❞❛❞❡✳ P❛r❛ ♦ ❝❛s♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❞❡ ❞✉❛s ♦✉ ♠❛✐s ✈❛r✐á✈❡✐s♣r♦❝❡❞❡✲s❡ ❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛✱ ♦❜t❡♥❞♦✲s❡

δu =

∣

∣

∣

∣

∂f

∂v

∣

∣

∣

∣

δv +

∣

∣

∣

∣

∂f

∂w

∣

∣

∣

∣

δw , ✭✸✳✺✳✶✮

❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❡q✉❛çã♦ ❞♦ ❡rr♦ ♣r♦♣❛❣❛❞♦✳ ❯♠❛ ✈❡③ ❝♦♥❤❡❝✐❞♦s ♦s ❡rr♦s ♥❛s ♠❡❞✐❞❛s ❞♦t❡♠♣♦ ❡ ❞❛ ❛❧t✉r❛ ❞❡ q✉❡❞❛ ❞❛ ❡s❢❡r❛ ❝♦♠❡♥t❛❞❛ ❛❝✐♠❛✱ ❝♦♥❝❧✉✐✲s❡ δg = 2

t2δd+ 4d

t3δt✳

❈♦♥❝❧✉í♠♦s ❡st❛ s❡çã♦ ♦❜s❡r✈❛♥❞♦ q✉❡ ♦ ❡rr♦ t♦t❛❧ ♥❛ ♠❡❞✐❞❛ ❞❡ ✉♠❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ é ❛s♦♠❛ ❞♦s ❡rr♦s s✐st❡♠át✐❝♦✱ ❞❡ ❡s❝❛❧❛ ❡ ❛❧❡❛tór✐♦✱ ♦✉ s❡❥❛

δx = (δx)sis + (δx)esc + (δx)ale

✷✽ ❈❆P❮❚❯▲❖ ✸✳ ❊❘❘❖❙ ❊ ■◆❈❊❘❚❊❩❆❙

✸✳✻ ❈♦♥✜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ♠❡❞✐❞❛

P❡❧♦ q✉❡ ❛♥❛❧✐s♦✉✲s❡ ❛té ♦ ♣r❡s❡♥t❡ ♠♦♠❡♥t♦✱ é ♣r❡❝✐s♦ ✉♠ ❝r✐tér✐♦ ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❡♠ ✉♠❛♠❡❞✐❞❛✳ ❆té q✉❡ ♣♦♥t♦ ♣♦❞❡♠♦s ❣❛r❛♥t✐r q✉❡ ♦ ✈❛❧♦r ❞❛ ♠❡❞✐❞❛ ❞❡ ✉♠❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ❡stá❝♦rr❡t♦❄ ❙❡ ♥♦s ❛♣r❡s❡♥t❛r❡♠ ♦ ✈❛❧♦r ❞❡ ✉♠❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛✱ ♦❜t✐❞❛ ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡✱ ❝♦♠❞❡t❡r♠✐♥❛❞♦ ♥ú♠❡r♦ ❞❡ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s ❡ ❝♦♠ ❛ r❡s♣❡❝t✐✈❛ ✐♥❝❡rt❡③❛✱ ❡①♣r❡ss❛ ❡♠t❡r♠♦s ❞♦ ❞❡s✈✐♦ ♠é❞✐♦ ♦✉ ❞❛ ❞✐s♣❡rsã♦✱ ♣♦❞❡r❡♠♦s s✐♠♣❧❡s♠❡♥t❡ ❛❝❡✐t❛r ♦ ✈❛❧♦r ❛♣r❡s❡♥t❛❞♦❄◗✉❡ ❝♦♥✜❛♥ç❛ t❡r❡♠♦s ♥♦ ✈❛❧♦r✱ ♦ q✉ã♦ ✧✈❡r❞❛❞❡✐r❛✧ é ❡st❛ ♠❡❞✐❞❛❄

❈♦♥s✐❞❡r❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛s ❞♦ ✈❛❧♦r ❞❛ ❛❝❡❧❡r❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡ g✳ ❉✐❣❛♠♦s q✉❡❞♦✐s ❡①♣❡r✐♠❡♥t❛❞♦r❡s r❡❛❧✐③❛r❛♠ ❛ ♠❡❞✐❞❛ ❞❡ g ❛tr❛✈és ❞❡ ❡①♣❡r✐♠❡♥t♦s ❞✐❢❡r❡♥t❡s✸✳ ❉✐❣❛♠♦sq✉❡ ❛s ♠❡❞✐❞❛s ❞♦s ❞♦✐s ❡①♣❡r✐♠❡♥t❛❞♦r❡s A ❡ B t❡♥❤❛♠ s✐❞♦

gA = (9, 7± 0, 4)m/s2

❡gB = (9, 5± 0, 2)m/s2

r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ➱ ❡✈✐❞❡♥t❡ q✉❡ ❛ ♠❡❞✐❞❛ ❛♣r❡s❡♥t❛❞❛ ♣♦r B ♣♦ss✉✐ ✉♠❛ ✐♥❝❡rt❡③❛ ♠❡♥♦r ❞♦q✉❡ ❛ ♠❡❞✐❞❛ ❛♣r❡s❡♥t❛❞❛ ♣♦r A✳ ◆♦ ❡♥t❛♥t♦✱ ♦ ✐♥t❡r✈❛❧♦ ❡st❛❜❡❧❡❝✐❞♦ ♣❡❧❛ ♠❡❞✐❞❛ ❞❡ A✱ ♦✉s❡❥❛✱ ♦ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧ ❞❡ gA ❡stá ♥♦ ✐♥t❡r✈❛❧♦ 10, 1 < gA < 9, 3✱ ♥♦ q✉❛❧ ❡stá ✐♥❝❧✉í❞♦ ♦ ✈❛❧♦r❞❡ r❡❢❡rê♥❝✐❛ gref = 9, 8m/s2✳ ❖ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧ ❞❡ gB ❡stá ♥♦ ✐♥t❡r✈❛❧♦ 9, 7 < gB < 9, 3 ♦q✉❛❧ ❡①❝❧✉✐ ♦ ✈❛❧♦r ❞❡ r❡❢❡rê♥❝✐❛✳ P♦rt❛♥t♦✱ ♦ ❣r❛✉ ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ♠❡❞✐❞❛ ❡♥✈♦❧✈❡ ❛❝♦♠♣❛r❛çã♦ ❞❡st❛ ❝♦♠ ✉♠ ✈❛❧♦r ❛❞♦t❛❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛✳ P♦❞❡♠♦s ❞❡✜♥✐r♠♦s ♦ ❡rr♦ ♣❡r❝❡♥t✉❛❧❞❡ ✉♠❛ ♠❡❞✐❞❛ ❛tr❛✈és ❞❛ r❡❧❛çã♦

ε% =|x− xref |

xref

× 100% ✭✸✳✻✳✶✮

❡♠ q✉❡ x é ♦ ✈❛❧♦r ❞❡t❡r♠✐♥❛❞♦ ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ♣❛r❛ ❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ❡♠ q✉❡stã♦ ❡ xref

é ♦ ✈❛❧♦r ❞❛ ♠❡s♠❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ q✉❡ s❡ t❡♠ ❝♦♠♦ r❡❢❡rê♥❝✐❛✳ ◆♦ ❝❛s♦ ❞❛ ❛❝❡❧❡r❛çã♦ ❞❛❣r❛✈✐❞❛❞❡✱ ♦ ✈❛❧♦r q✉❡ s❡ ❝♦st✉♠❛ ❛❞♦t❛r é 9, 8m/s2✱ ❞❡ ♠♦❞♦ q✉❡ ♦ ❡rr♦ ♣❡r❝❡♥t✉❛❧ ♥♦s ✈❛❧♦r❡s❞❡ A ❡ B sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ εA = 1, 0% ❡ εB = 3, 1%✳ ❖ q✉❡ ❞❡✈❡ ✜❝❛r ❝❧❛r♦ é q✉❡ ✐♥❝❡rt❡③❛♣❡q✉❡♥❛ ♥✉♠❛ ♠❡❞✐❞❛ ♥ã♦ é ♦ ú♥✐❝♦ ❝r✐tér✐♦ ❛ s❡r s❡❣✉✐❞♦ ♣❛r❛ ✐♥❞✐❝❛r q✉❡ ✉♠❛ ♠❡❞✐❞❛ é ❜♦❛♦✉ q✉❡ ❡stá ❝♦rr❡t❛✱ ♠❛s t❛♠❜é♠ ♦ q✉❛♥t♦ ❡❧❛ ❡stá ♣ró①✐♠❛ ❞❡ ✉♠ ✈❛❧♦r ❞❡ r❡❢❡rê♥❝✐❛✱ ❛tr❛✈és❞♦ ❝á❧❝✉❧♦ ❞❡ s❡✉ ❡rr♦ ♣❡r❝❡♥t✉❛❧✳ ❯♠ ♦✉tr♦ ❡①♣❡r✐♠❡♥t❛❞♦r C ♣♦❞❡r✐❛ r❡❛❧✐③❛r ❛s ♠❡❞✐❞❛s ❞❡g ✉s❛♥❞♦ ❡q✉✐♣❛♠❡♥t♦s t❛✐s q✉❡ ❧❤♦ ♣❡r♠✐t✐ss❡ ♦❜t❡r ♦ ✈❛❧♦r ❞❛ ❛❝❡❧❡r❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡ ❝♦♠✉♠ ♥ú♠❡r♦ ♠❛✐♦r ❞❡ ❝❛s❛s ❞❡❝✐♠❛✐s ❡ ❝♦♠ ✉♠❛ ✐♥❝❡rt❡③❛ ♠❡♥♦r✱ ❝♦♠♦

gC = (9, 672± 0, 007)m/s2

♠❛s ❝♦♠ ❛❧❣✉♠ ❡rr♦ s✐st❡♠át✐❝♦ q✉❡ ❢❡③ ❝♦♠ q✉❡ t♦❞❛s ❛s s✉❛s ♠❡❞✐❞❛s ✜❝❛ss❡♠ ❞❡s✈✐❛❞❛s ♣❛r❛❜❛✐①♦ ❞♦ ✈❛❧♦r ❞❡ r❡❢❡rê♥❝✐❛✳ ▼❡s♠♦ t❡♥❞♦ ✉♠ ❣r❛✉ ❞❡ ♣r❡❝✐sã♦ ♠❛✐♦r ❞♦ q✉❡ ♦ ❛♣r❡s❡♥t❛❞♦ ♣❡❧♦s♦✉tr♦s ❞♦✐s ❡①♣❡r✐♠❡♥t❛❞♦r❡s✱ ❡st❛ ♠❡❞✐❞❛ ❛♣r❡s❡♥t❛ ✉♠ ❣r❛✉ ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡ r❡❧❛t✐✈❛♠❡♥t❡❜❛✐①♦✳

❯♠ r❡s✉❧t❛❞♦ ❝♦♠♦ ❡st❡ ❡①✐❣❡✱ ❡♥tã♦✱ q✉❡ ♦ ❡①♣❡r✐♠❡♥t♦ s❡❥❛ r❡❢❡✐t♦✱ q✉❡ ❛s ♠❡❞✐❞❛s s❡❥❛♠❛♥❛❧✐s❛❞❛s ♣❛r❛ ❛ ♣r♦❝✉r❛ ❞♦ ❡rr♦ s✐st❡♠át✐❝♦ q✉❡ ❝❛✉s♦✉ ♦ ❞❡s✈✐♦ ♣❛r❛ ❛❜❛✐①♦ ❞♦ ✈❛❧♦r ❞❡r❡❢❡rê♥❝✐❛✳

✸❯♠ ❞♦s ❡①♣❡r✐♠❡♥t♦s ♣♦❞❡ s❡r ♦ ❞❡ q✉❡❞❛ ❧✐✈r❡ ❞❡ ✉♠❛ ♣❡q✉❡♥❛ ❡s❢❡r❛✱ ❝♦♠♦ ❞✐s❝✉t✐❞♦ ♥♦ ✐♥í❝✐♦ ❞❡st❛s❡çã♦❀ ♦ ♦✉tr♦ ❡①♣❡r✐♠❡♥t♦ ♣♦❞❡ t❡r s✐❞♦ ♦ ❞❡ ♠❡❞✐❞❛s ❞♦ ♣❡rí♦❞♦ ❞❡ ✉♠ ♣ê♥❞✉❧♦ s✐♠♣❧❡s✳

❈❛♣ít✉❧♦ ✹

●❘➪❋■❈❖❙

❊❞✐♦ ❈✉♥❤❛ ❞❛ ❈♦st❛

✹✳✶ ❱❛r✐á✈❡✐s ❞❡♣❡♥❞❡♥t❡s ❡ ✐♥❞❡♣❡♥❞❡♥t❡s

❈♦♠♦ ❥á ✈✐♠♦s✱ t♦❞♦ ♠♦❞❡❧♦ t❡ór✐❝♦ ❜✉s❝❛ ❡①♣❧✐❝❛r ❛ r❡❧❛çã♦ ❡①♣❡r✐♠❡♥t❛❧ ❡♥❝♦♥tr❛❞❛ ❡♥tr❡❞✉❛s ❣r❛♥❞❡③❛s ❢ís✐❝❛s ❞✐❢❡r❡♥t❡s✳ ❊st❡s ♠♦❞❡❧♦s ♣❛rt❡♠ ❞❡ ♣r✐♥❝í♣✐♦s ❢ís✐❝♦s ❡ ❞❡ ▲❡✐s ❞❡❈♦♥s❡r✈❛çã♦✱ ❝♦♠♦ ❛ ❝♦♥s❡r✈❛çã♦ ❞❛ ❡♥❡r❣✐❛ ❡ ❞♦ ♠♦♠❡♥t♦ ❧✐♥❡❛r✱ q✉❡ ❞❡✈❡♠✱ ❛ ♣r✐♥❝í♣✐♦✱✐♥❞✐❝❛r q✉❛❧ é ❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ ❡ q✉❛❧ é ❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡✳ P♦r ❡①❡♠♣❧♦✱ ❞❛❞♦ q✉❡ ♦t❡♠♣♦ ✏s✐♠♣❧❡s♠❡♥t❡ ✢✉✐✑✱ ❡st❡ é s❡♠♣r❡ t♦♠❛❞♦ ❝♦♠♦ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡✱ s❡♥❞♦ q✉❛❧q✉❡r♦✉tr❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ❡s❝r✐t❛ ❝♦♠♦ ❢✉♥çã♦ ❞❡st❡✳ ❙ã♦ ♦s ❝❛s♦s ❞❛ ♣♦s✐çã♦ ❡ ✈❡❧♦❝✐❞❛❞❡ ❞❡✉♠❛ ♣❛rtí❝✉❧❛✱ ❢♦rç❛✱ ❡♥❡r❣✐❛✱ ❝♦rr❡♥t❡ ❡❧étr✐❝❛✱ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦✱ t❡♠♣❡r❛t✉r❛✱ ❡t❝✳ ❚❛♠❜é♠♣♦rq✉❡✱ ❡♠ ❣❡r❛❧✱ ❡st❛s ❣r❛♥❞❡③❛s ❢ís✐❝❛s sã♦ ♠❡❞✐❞❛s ❡♠ ❞✐❢❡r❡♥t❡s ✐♥st❛♥t❡s ❞❡ t❡♠♣♦✱ ✈✐st♦q✉❡ s♦♠❡♥t❡ ♣♦❞❡♠♦s ✐♥❞✐❝❛r s❡✉s ✈❛❧♦r❡s ♥✉♠ ❞❛❞♦ ✐♥st❛♥t❡ ❞❡ t❡♠♣♦✳

❈♦♥s✐❞❡r❡ ❛❣♦r❛ ♦ s❡❣✉✐♥t❡ ❡①♣❡r✐♠❡♥t♦✿ ✉♠❛ ♣❡q✉❡♥❛ ❡s❢❡r❛ é s♦❧t❛ ❞♦ r❡♣♦✉s♦✱ ❞❡ ✉♠❛❛❧t✉r❛ ho ❡s❝♦❧❤✐❞❛ ❛r❜✐tr❛r✐❛♠❡♥t❡✱ ❡ ♦ t❡♠♣♦ ❞❡ q✉❡❞❛✶ ❞❛ ❡s❢❡r❛ é ♠❡❞✐❞♦✳ ❈♦♥s✐❞❡r❛♥❞♦❡st❡ ❝♦♠♦ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ q✉❡❞❛ ❧✐✈r❡✱ s❛❜❡♠♦s q✉❡ ❛ ❛❧t✉r❛ ❞❛ ❡s❢❡r❛✱ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ♥í✈❡❧❞❡ r❡❢❡rê♥❝✐❛ é y = ho − gt2/2 ♦✉ s❡❥❛✱ ❛ ❛❧t✉r❛ é ❢✉♥çã♦ ❞♦ t❡♠♣♦✱ ♣♦✐s ♣♦❞❡♠♦s ❡s❝♦❧❤❡r✉♠ ✐♥st❛♥t❡ ❡♥tr❡ ♦ ❧❛♥ç❛♠❡♥t♦ ❡ ❛ q✉❡❞❛✱ ❡ ♠❡❞✐r ❛ ❛❧t✉r❛ ❞❛ ❡s❢❡r❛ ♥❡st❡ ✐♥st❛♥t❡✳ ❊♥tã♦ é❡✈✐❞❡♥t❡ q✉❡ y = y(t) ❡ ✉♠ ❣rá✜❝♦ y × t ♣♦❞❡ s❡r ❝♦♥str✉í❞♦✳ ◆♦ ❡♥t❛♥t♦✱ ♣♦❞❡♠♦s ♠❡❞✐r ♦t❡♠♣♦ t♦t❛❧ ❞❡ q✉❡❞❛ ❞❛ ❡s❢❡r❛✱ ♦✉ s❡❥❛✱ ♠❡❞✐r ♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ♥❡❝❡ssár✐♦ ♣❛r❛ q✉❡ ❡❧❛♣❡r❝♦rr❛ ❛ ❞✐stâ♥❝✐❛ ✈❡rt✐❝❛❧ ho✳ ❊st❡ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦✱ ♦✉ ✐♥st❛♥t❡ ♥♦ q✉❛❧ ❛ ❡s❢❡r❛ ❛t✐♥❣❡ ❛

❛❧t✉r❛ y = 0✱ é tq =

√

2ho

g❡ s✉❝❡ss✐✈❛s ♠❡❞✐❞❛s ♣❛r❛ ❛❧t✉r❛s ❞✐❢❡r❡♥t❡s ❞❡✈❡♠ ❢♦r♥❡❝❡r ✈❛❧♦r❡s

❞✐❢❡r❡♥t❡s ♣❛r❛ ♦ t❡♠♣♦ ❞❡ q✉❡❞❛✳ ❊♥tã♦✱ t❡♥❞♦ ❡s❝♦❧❤✐❞♦ ✉♠❛ ❛❧t✉r❛ ♣❛r❛ ❛ q✉❡❞❛✱ ♦ t❡♠♣♦ ❞❡q✉❡❞❛ ♣❛ss❛ ❛ s❡r ✉♠❛ ❢✉♥çã♦ ❞❛ ❛❧t✉r❛ ❞❡ ❧❛♥ç❛♠❡♥t♦ ❡✱ ❝❧❛r❛♠❡♥t❡✱ t❡♠♦s t❡♠♣♦ ✭✐♥t❡r✈❛❧♦❞❡ t❡♠♣♦✮ ❝♦♠♦ ❢✉♥çã♦ ❞♦ ❞❡s❧♦❝❛♠❡♥t♦✳ ❖❜s❡r✈❡♠♦s q✉❡ ❡st❛ r❡❧❛çã♦ ❞❡ ❞❡♣❡♥❞ê♥❝✐❛ ✜❝❛❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛s ❝♦♥❞✐çõ❡s ❞♦ ❡①♣❡r✐♠❡♥t♦✿ ❡s❝♦❧❤❡♠♦s ❛ ❛❧t✉r❛✱ ✐st♦ é✱ ♣♦❞❡♠♦s s♦❧t❛r ❛♣❡q✉❡♥❛ ❡s❢❡r❛ ❞❡ q✉❛❧q✉❡r ❛❧t✉r❛ ❡✱ ❡♠ ❝♦♥s❡q✉ê♥❝✐❛✱ ♠❡❞✐r ♦ t❡♠♣♦ ❞❡ q✉❡❞❛ ❞❛ ♠❡s♠❛✳❊♥tã♦✱ tq = tq(ho) ❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞❡✈❡ s❡r tq × ho ❝♦♠ tq ♥♦ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s❡ ho ♥♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s✳ ▼❡s♠♦ ♠❛♥t❡♥❞♦ ❛ ❛❧t✉r❛ ❞❡ ❧❛♥ç❛♠❡♥t♦ ✜①❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

t = t(y) s♦❜ ❛ ❢♦r♠❛ t =

√

2

g(ho − y) ❞❡s❞❡ q✉❡ ♦ ❡①♣❡r✐♠❡♥t♦ s❡❥❛ ♦ ❞❡ s♦❧t❛r ❛ ❡s❢❡r❛ ❞❛

❛❧t✉r❛ ho ❡ ♠❡❞✐r ♦ ✐♥st❛♥t❡ ♥♦ q✉❛❧ ❡❧❛ s❡ ❡♥❝♦♥tr❛ ♥✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❛❧t✉r❛ y❀ ❡st❡ ✐♥st❛♥t❡✈❛r✐❛ ❝♦♥❢♦r♠❡ ❛ ❛❧t✉r❛ ❡s❝♦❧❤✐❞❛✳ ❘❡s✉♠✐♥❞♦✱ sã♦ s✐t✉❛çõ❡s ❡①♣❡r✐♠❡♥t❛✐s ❞✐❢❡r❡♥t❡s✿ ✜①❛r♠♦s

✶◆❛ r❡❛❧✐❞❛❞❡ é ♠❡❞✐❞♦ ♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ❞❡ q✉❡❞❛ ❞❛ ❡s❢❡r❛✳

✷✾

✸✵ ❈❆P❮❚❯▲❖ ✹✳ ●❘➪❋■❈❖❙

✉♠ ✐♥st❛♥t❡ ❞❡ t❡♠♣♦ ♣❛r❛ ♠❡❞✐r♠♦s ❛ ❛❧t✉r❛ ❞❛ ❡s❢❡r❛ ♥❡st❡ ✐♥st❛♥t❡ ❡ ✜①❛r♠♦s ✉♠❛ ❛❧t✉r❛♣❛r❛ ❛ ❡s❢❡r❛ ❡ ♠❡❞✐r♠♦s ♦ ✐♥st❛♥t❡ ♥♦ q✉❛❧ ❡❧❛ ♣❛ss❛ ♣♦r ❡st❛ ❛❧t✉r❛✳ ➱ ♦ ❡①♣❡r✐♠❡♥t❛❞♦rq✉❡♠ ❛❝❛❜❛ ❞❡✜♥✐♥❞♦ ❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ ❡ ❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡✳ ❱❡❥❛ ❛s t❛❜❡❧❛s ❛❜❛✐①♦✿

t(s)

to = 0, 0t1 = 1, 0t2 = 2, 0t3 = 5, 0t4 = 10, 0t5 = 25, 0

y(cm)

yoy1y2y3y4y5

y(cm)

yo = 100, 0y1 = 90, 0y2 = 75, 0y3 = 50, 0y4 = 25, 0y5 = 10, 0

t(s)

tot1t2t3t4t5

❚❛❜❡❧❛ ✹✳✶✿ ❊sq✉❡r❞❛✿ ♥❡st❡ ❝❛s♦ ❛ ♣♦s✐çã♦ ❞❛ ❡s❢❡r❛ é ✉♠❛ ❢✉♥çã♦ ❞♦ t❡♠♣♦✱ ♦✉ s❡❥❛ y = y(t)✱♣♦✐s ♣♦❞❡♠♦s ♣❡r❣✉♥t❛r ✏q✉❛❧ ❛ ❛❧t✉r❛ ❞❛ ❡s❢❡r❛ ♥♦ ✐♥st❛♥t❡ t = 2, 5 s❄✑ ❉✐r❡✐t❛✿ ♥❡st❡ ❝❛s♦ ♦✭✐♥t❡r✈❛❧♦ ❞❡✮ t❡♠♣♦ ❞❡ q✉❡❞❛ é ❢✉♥çã♦ ❞❛ ❛❧t✉r❛✱ ♦✉ s❡❥❛ t = t(y)✱ ♣♦✐s ♣♦❞❡♠♦s ♣❡r❣✉♥t❛r✏❡♠ q✉❡ ✐♥st❛♥t❡ ❛ ❡s❢❡r❛ ❡stá ♥❛ ❛❧t✉r❛ y = 30, 0 cm❄✑

❙✐t✉❛çã♦ s❡♠❡❧❤❛♥t❡ ♦❝♦rr❡ q✉❛♥❞♦ ✉t✐❧✐③❛♠♦s ✉♠❛ ♠♦❧❛ ❡ q✉❡r❡♠♦s ❞❡t❡r♠✐♥❛r s✉❛ ❝♦♥✲st❛♥t❡ ❡❧ást✐❝❛ ✭♦✉ ❝♦♥st❛♥t❡ ❞❡ ❢♦rç❛✮✳ P❛r❛ ♣❡q✉❡♥❛s ✈❛r✐❛çõ❡s ♥♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♠♦❧❛✭x = lo − l ❝♦♠ l ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♠♦❧❛ ❡ lo s❡✉ ❝♦♠♣r✐♠❡♥t♦ ♥❛t✉r❛❧✮ ❛ ▲❡✐ ❞❡ ❍♦♦❦❡ ❡st❛❜✲❡❧❡❝❡ q✉❡ ~F = −kx i✳ ❊s❝r✐t❛ ❞❡st❛ ❢♦r♠❛✱ ❡st❛ r❡❧❛çã♦ ♥♦s ❞✐③ q✉❡ ✉♠❛ ♣❛rtí❝✉❧❛ ♣r❡s❛ ❛ ✉♠❛♠♦❧❛✱ ♦✉ s✉❥❡✐t❛ ❛ ✉♠❛ ❢♦rç❛ r❡st❛✉r❛❞♦r❛✱ s♦❢r❡ ✉♠❛ ❢♦rç❛ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ s❡✉ ❞❡s❧♦❝❛♠❡♥t♦♣❛r❛ ❢♦r❛ ❞❡ s✉❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦✱ s❡♥❞♦ ❡✈✐❞❡♥t❡ q✉❡ F = F (x)✳ ◆♦ ❡♥t❛♥t♦✱ ❝♦♥s✐❞✲❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ❡①♣❡r✐♠❡♥t♦✿ ✉♠❛ ♠♦❧❛ ❞❡ ❝♦♥st❛♥t❡ ❡❧ást✐❝❛ k é ♣❡♥❞✉r❛❞❛ ♥❛ ✈❡rt✐❝❛❧ ❡s❡✉ ❝♦♠♣r✐♠❡♥t♦ ♥❛t✉r❛❧ lo é ♠❡❞✐❞♦✳ ❆♦ ♣❡♥❞✉r❛r♠♦s ✉♠ ❝♦r♣♦ ❞❡ ♠❛ss❛ m à s✉❛ ❡①tr❡♠✲✐❞❛❞❡ ❧✐✈r❡✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♠♦❧❛ ♣❛ss❛rá ❛ s❡r l✱ ♦✉ s❡❥❛✱ ♦ ♣❡s♦ mg ❞♦ ♦❜❥❡t♦ ♣❡♥❞✉r❛❞♦♣r♦❞✉③ ✉♠❛ ❞❡❢♦r♠❛çã♦ x = l − lo ♥❛ ♠♦❧❛ ❡✱ ♥❛ ♥♦✈❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ ❝♦r♣♦ ✈❛❧❡ ❛✐❣✉❛❧❞❛❞❡ mg = kx✳ ❱❡♠♦s q✉❡ ❞✐❢❡r❡♥t❡s ❢♦rç❛s ♣r♦❞✉③❡♠ ❞✐❢❡r❡♥t❡s ❞❡❢♦r♠❛çõ❡s ♥❛ ♠♦❧❛ ❡✱♣❛r❛ ♣r♦❞✉③✐r♠♦s ❞✐❢❡r❡♥t❡s ❞❡❢♦r♠❛çõ❡s ♥❛ ♠♦❧❛ ♥❡❝❡ss✐t❛♠♦s ❞❡ ❞✐❢❡r❡♥t❡s ❢♦rç❛s ❛♣❧✐❝❛❞❛s ♥❛❡①tr❡♠✐❞❛❞❡ ❧✐✈r❡ ❞❛ ♠♦❧❛✳ ❊♠❜♦r❛ ❡st❛ ❛✜r♠❛çã♦ ♣❛r❡ç❛ ❛ ♠❡s♠❛ ❝♦✐s❛✱ ❡❧❛s ♣♦ss✉❡♠ s✐❣♥✐✜✲❝❛❞♦ ❞✐❢❡r❡♥t❡✱ ♣♦✐s ❞❡t❡r♠✐♥❛♠ q✉❛❧ é ❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ ❡ q✉❛❧ é ❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡✳❱❛♠♦s s✉♣♦r q✉❡ ❞✐❢❡r❡♥t❡s ♠❛ss❛s s❡❥❛♠ ❡s❝♦❧❤✐❞❛s ♣❛r❛ ♣❡♥❞✉r❛r♠♦s à ❡①tr❡♠✐❞❛❞❡ ❧✐✈r❡❞❛ ♠♦❧❛✳ ❈♦♠ ✐st♦✱ ❞✐❢❡r❡♥t❡s ❞❡❢♦r♠❛çõ❡s s❡rã♦ ♣r♦❞✉③✐❞❛s ❡ ❞❡✈❡♠♦s t❡r x = x(m)✱ ♦✉ ❞❡♠♦❞♦ ❡q✉✐✈❛❧❡♥t❡ x = x(F )✱ ✐♥❢♦r♠❛♥❞♦ q✉❡ ❛ ❞❡❢♦r♠❛çã♦ ♣r♦❞✉③✐❞❛ ♥❛ ♠♦❧❛ é ❢✉♥çã♦ ❞❛ ❢♦rç❛❛♣❧✐❝❛❞❛ à ♠❡s♠❛✳ ▼❛s ♣♦❞❡rí❛♠♦s t❡r ♣ré✲❡st❛❜❡❧❡❝✐❞♦ ✈❛❧♦r❡s ❞❡ x ❡ ❞❡s❝♦❜r✐r q✉❡ ✈❛❧♦r❡s ❞❡m✱ ❡ ♣♦rt❛♥t♦ ❞❡ ❢♦rç❛✱ ♣r♦❞✉③ ❡st❛ ❞❡❢♦r♠❛çã♦✳ ◆❡st❡ ❝❛s♦✱ m = m(x) ♦✉ F = F (x)✳ ❱❡♠♦s♥❡st❡ ❡①❡♠♣❧♦ q✉❡ ❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ ❡ ❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡ t❛♠❜é♠ ✜❝❛♠ ❞❡t❡r♠✐♥❛❞❛s♣❡❧♦ ❡①♣❡r✐♠❡♥t❛❞♦r✳

❉❡✜♥✐çã♦ ✸✳ ❯♠❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ s❡rá ❝♦♥s✐❞❡r❛❞❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡ q✉❛♥❞♦ s❡✉ ✈❛❧♦r ❢♦r❡s❝♦❧❤✐❞♦✱ ❛r❜✐tr❛r✐❛♠❡♥t❡✱ ♣❡❧♦ ❡①♣❡r✐♠❡♥t❛❞♦r✳

❉❡✜♥✐çã♦ ✹✳ ❯♠❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ s❡rá ❝♦♥s✐❞❡r❛❞❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ q✉❛♥❞♦ s❡✉ ✈❛❧♦r ❢♦r❞❡t❡r♠✐♥❛❞♦ ❡♠ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ❛❧t❡r❛çã♦ ♥♦ ✈❛❧♦r ❞❡ ♦✉tr❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛✱ t♦♠❛❞❛ ❝♦♠♦✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡✳

P♦rt❛♥t♦✱ ❡♠ q✉❛❧q✉❡r ❡①♣❡r✐♠❡♥t♦ ❞❡st✐♥❛❞♦ ❛ ❜✉s❝❛r ❛ r❡❧❛çã♦ ♠❛t❡♠át✐❝❛ ❡♥tr❡ ❣r❛♥❞❡③❛s❢ís✐❝❛s✱ ❞❡✈❡♠♦s ✈❡r✐✜❝❛r q✉❛❧ é ❛ ❣r❛♥❞❡③❛ ❞❡♣❡♥❞❡♥t❡ ❡ q✉❛❧ é ❛ ❣r❛♥❞❡③❛ ✐♥❞❡♣❡♥❞❡♥t❡✱ ♣❛r❛♦ ❡①♣❡r✐♠❡♥t♦ ❝♦♥s✐❞❡r❛❞♦✳

✹✳✷✳ ❈❖◆❙❚❆◆❚❊❙ ❊ P❆❘➶▼❊❚❘❖❙ ✸✶

✹✳✷ ❈♦♥st❛♥t❡s ❡ P❛râ♠❡tr♦s

❚❡♥❞♦ ❞❡✜♥✐❞❛s ❛s ✈❛r✐á✈❡✐s ❞❡♣❡♥❞❡♥t❡ ❡ ✐♥❞❡♣❡♥❞❡♥t❡ ♣♦❞❡♠♦s ❜✉s❝❛r ✉♠❛ r❡❧❛çã♦ ♠❛t❡♠át✐❝❛❡♥tr❡ ❡❧❛s ❛tr❛✈és ❞❡ ❛❧❣✉♠ ♠♦❞❡❧♦ t❡ór✐❝♦ ♦✉ ❛tr❛✈és ❞♦s ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s✳ ❆s r❡❧❛çõ❡ssã♦ ❛s ♠❛✐s ✈❛r✐❛❞❛s ♣♦ssí✈❡✐s✱ ❞❡s❞❡ ❛ ♠❛✐s s✐♠♣❧❡s ❝♦♠♦ ❛ r❡t❛✱ ♦✉ s❡❥❛✱ r❡❧❛çã♦ ❧✐♥❡❛r❝♦♠♦ y(x) = Ax + B ♦♥❞❡ A ❡ B sã♦ ❝♦♥st❛♥t❡s✱ ❛té ✉♠❛ r❡❧❛çã♦ ♠❛✐s ❝♦♠♣❧✐❝❛❞❛✱ ❝♦♠♦y(x) = A ln [cosh (Bx)] ❝♦♠ A ❡ B t❛♠❜é♠ ❝♦♥st❛♥t❡s✳ ❙❡❥❛ ❝♦♠♦ ❢♦r✱ ❛s r❡❧❛çõ❡s ♠❛t❡♠át✐❝❛sq✉❡ ❞❡✜♥❡♠ ❛ ❢✉♥çã♦ y = y(x) ❡♥✈♦❧✈❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❝♦♥st❛♥t❡ q✉❡ ♣♦ss✉✐ ✉♠ s✐❣♥✐✜✲❝❛❞♦ ❢ís✐❝♦✱ s❡♥❞♦ ♥❡❝❡ssár✐❛ ❛ s✉❛ ❞❡t❡r♠✐♥❛çã♦✳ ◆♦ ❡♥t❛♥t♦✱ ♣♦❞❡rí❛♠♦s ♣❡r❣✉♥t❛r✿ ♦ q✉❡ é❝♦♥st❛♥t❡ ❡ ♦ q✉❡ é ♣❛râ♠❡tr♦❄ ❖✉ sã♦ ❛ ♠❡s♠❛ ❝♦✐s❛❄

P❛r❛ r❡s♣♦♥❞❡r♠♦s ❛ ❡st❛ ♣❡r❣✉♥t❛ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦✿ ✉♠❛ ❜♦❧❛ ❞❡♠❛ss❛ m s❡♥❞♦ ❧❛♥ç❛❞❛ ✈❡rt✐❝❛❧♠❡♥t❡ ♣❛r❛ ❝✐♠❛✳ ❊st❛ ❜♦❧❛✱ ✉♠❛ ✈❡③ ❧❛♥ç❛❞❛✱ ❡stá s✉❥❡✐t❛ ❛❛çã♦ ❞❛s ❢♦rç❛s ♣❡s♦ ❡ r❡s✐stê♥❝✐❛ ❞♦ ❛r✳ ❙❛❜❡♠♦s q✉❡ ❛ ❢♦rç❛ ♣❡s♦ é ❞❛❞❛ ♣♦r mg✱ s❡♥❞♦ g ♦✈❛❧♦r ❞❛ ❛❝❡❧❡r❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡ ♥♦ ♣♦♥t♦ ♦♥❞❡ ❡stá ♦ ❝❡♥tr♦ ❞❡ ❣r❛✈✐❞❛❞❡ ❞❛ ❜♦❧❛✳ ❆ ❢♦rç❛ ❞❡r❡s✐stê♥❝✐❛ ❞♦ ❛r é ♣r♦♣♦r❝✐♦♥❛❧ ❛ ✉♠❛ ♣♦tê♥❝✐❛ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❡ ❛ ❝♦♥st❛♥t❡ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡❞❡♣❡♥❞❡✱ ❡♥tr❡ ♦✉tr❛s ❝♦✐s❛s✱ ❞❛ ❢♦r♠❛ ❣❡♦♠étr✐❝❛ ❞♦ ❝♦r♣♦ ❡♠ q✉❡❞❛ ✭♥♦ ❝❛s♦ ✉♠❛ ❜♦❧❛✮ ❡❞❛ ❞❡♥s✐❞❛❞❡ ❞♦ ❛r✳ ❖ ♠♦✈✐♠❡♥t♦ ❞❛ ❜♦❧❛ é ❞❡s❝r✐t♦ ❡♠ t❡r♠♦s ❞❡ s✉❛ ✈❡❧♦❝✐❞❛❞❡ ❡ ♣♦s✐çã♦❝♦♠♦ ❢✉♥çã♦ ❞♦ t❡♠♣♦✳ ❉❡ ✉♠ ♠♦❞♦ ❣❡r❛❧✱ ♦ ✈❛❧♦r ❞❡ g ✈❛r✐❛ ❝♦♠ ❛ ❛❧t✉r❛ ❞❛ ❜♦❧❛ ❡♠ r❡❧❛çã♦❛♦ ♣♦♥t♦ ❞❡ ❧❛♥ç❛♠❡♥t♦✳ ❆❧é♠ ❞✐ss♦✱ s❛té❧✐t❡s sã♦ ❧❛♥ç❛❞♦s ❡ r❡❝❡❜✐❞♦s✱ ❛ ❚❡rr❛ ❡stá s✉❥❡✐t❛ ❛q✉❡❞❛ ❞❡ ♠❡t❡♦r✐t♦s✱ ❞❡ ♠♦❞♦ q✉❡ s✉❛ ♠❛ss❛ s♦❢r❡ ✢✉t✉❛çõ❡s❀ ♣♦rt❛♥t♦ ✈❛r✐❛ ❝♦♠ ♦ t❡♠♣♦ ♦q✉❡ ❧❡✈❛ ❛ ✉♠❛ ✈❛r✐❛çã♦ ❞❡ g ❝♦♠ ♦ t❡♠♣♦✳ P♦❞❡rí❛♠♦s ❛✐♥❞❛ s✉♣♦r q✉❡✱ ❛ ♠❡❞✐❞❛ ❡♠ q✉❡ ❛❜♦❧❛ s♦❜❡✱ ♣❛rtí❝✉❧❛s ❞❡ ♣♦❡✐r❛ ❡ ✈❛♣♦r s❡ ❛❞❡r❡♠ à ♠❡s♠❛✱ ❛✉♠❡♥t❛♥❞♦ ❛ ♠❛ss❛ ❞❛ ❜♦❧❛✱ ♦✉s❡❥❛✱ m t❛♠❜é♠ ❞❡✈❡ s❡r ❢✉♥çã♦ ❞♦ t❡♠♣♦✳ ■st♦ ❧❡✈❛ ❛ ✉♠❛ ❞❡s❝r✐çã♦ t❡ór✐❝❛ ❡①tr❡♠❛♠❡♥t❡❝♦♠♣❧❡①❛✱ ♣r❛ ♥ã♦ ❞✐③❡r ✐♠♣♦ssí✈❡❧✱ ❞❡ s❡r ❢❡✐t❛ ❜❡♠ ❝♦♠♦ ❛ ♦❜t❡♥çã♦ ❞♦ ✈❛❧♦r ❞❛s ❣r❛♥❞❡③❛s❢ís✐❝❛s ❡♥✈♦❧✈✐❞❛s ❛ ❝❛❞❛ ✐♥st❛♥t❡✳ ◆♦ ❡♥t❛♥t♦✱ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ❧❛♥ç❛♠❡♥t♦ ❞❛❜♦❧❛ ♥ã♦ s❡❥❛ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ♣❛r❛ q✉❡ ❡❧❛ s✉❜❛ s❡♠ ❡①♣❡r✐♠❡♥t❛r ✈❛r✐❛çõ❡s ❛♣r❡❝✐á✈❡✐s❡♠ g✱ ♣♦❞❡r❡♠♦s ❝♦♥s✐❞❡r❛r g ❝♦♠♦ ✉♠ ✈❛❧♦r ✜①♦ ❞✉r❛♥t❡ t♦❞♦ ♦ ♠♦✈✐♠❡♥t♦ ❞❡ s✉❜✐❞❛ ❡ ❞❡❞❡s❝✐❞❛ ❞❛ ❜♦❧❛✱ ♦✉ s❡❥❛✱ ❝♦♥s✐❞❡r❛r g ❝♦♠♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ♣♦s✐çã♦ ❞❛ ❜♦❧❛✳ ❱✐st♦ q✉❡ ❡st❡♠❡s♠♦ ❡①♣❡r✐♠❡♥t♦ ♣♦❞❡ s❡r r❡♣❡t✐❞♦ ❡♠ ♦✉tr❛s ♣❛rt❡s ❞❡ ♥♦ss♦ ♣❧❛♥❡t❛✱ ♦♥❞❡ g ♣♦ss✉✐ ♦✉tr♦✈❛❧♦r✷✱ ♣♦ré♠ ❝♦♥st❛♥t❡✱ ♣❛r❛ ❛❧t✉r❛s ♥ã♦ ♠✉✐t♦ ❣r❛♥❞❡s✱ t♦♠❛r❡♠♦s g ❝♦♠♦ ✉♠ ♣❛râ♠❡tr♦✳❙❡ ♠❡❞✐r♠♦s ❛ ♠❛ss❛ ❞❛ ❜♦❧❛ ❛♥t❡s ❡ ❛♣ós ♦ ❧❛♥ç❛♠❡♥t♦✱ ✈❡r❡♠♦s q✉❡ ❛ ♠❛ss❛ ❞❡ ♣♦❡✐r❛♣♦ss✐✈❡❧♠❡♥t❡ ❛❞❡r✐❞❛ ♥❛ ❜♦❧❛ é ♠✉✐t♦✱ ♠✉✐t♦ ♠❡♥♦r ❞♦ q✉❡ ❛ ♠❛ss❛ ❞❛ ❜♦❧❛✱ ❞❡ ♠♦❞♦ q✉❡t❛♠❜é♠ ♣♦❞❡r❡♠♦s t♦♠❛r ❛ ♠❛ss❛ ❞❛ ❜♦❧❛ ❝♦♠♦ ✉♠ ♣❛râ♠❡tr♦✳ ❙❡ ♣❡❣❛r♠♦s ♦✉tr❛ ❜♦❧❛✱ ❡❧❛t❡rá ♠❛ss❛ ❞✐❢❡r❡♥t❡✳ ❚❛♠❜é♠ ♣♦❞❡♠♦s ❞❡s♣r❡③❛r ❛s ✈❛r✐❛çõ❡s ♥❛ ❞❡♥s✐❞❛❞❡ ❞♦ ❛r✱ ♦✉ s❡❥❛✱❞❛ ❛t♠♦s❢❡r❛✱ ❞✉r❛♥t❡ ♦ tr❛❥❡t♦ ❞❛ ❜♦❧❛✳ P♦r ✜♠✱ ♣♦❞❡♠♦s ❞❡s♣r❡③❛r ❛s ♣♦ssí✈❡✐s ✈❛r✐❛çõ❡s ♥❛♠❛ss❛ ❞❛ ❚❡rr❛ ❞✉r❛♥t❡ ♦ ♠♦✈✐♠❡♥t♦ ❞❛ ❜♦❧❛ ♣♦r ❝❛✉s❛ ❞❛s ❡s❝❛❧❛s ❞❡ t❡♠♣♦ ❡♥✈♦❧✈✐❞❛s✱ ♦✉s❡❥❛✱ ♦ t❡♠♣♦ ❞❡ ♠♦✈✐♠❡♥t♦ ❞❛ ❜♦❧❛ é ♠✉✐t♦ ♠❡♥♦r ❞♦ q✉❡ ♦ t❡♠♣♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ♦❜s❡r✈❛r♠♦s✈❛r✐❛çõ❡s ✭❛♣r❡❝✐á✈❡✐s✮ ♥❛ ♠❛ss❛ ❞❛ ❚❡rr❛✳ ❆ss✐♠✱ ♦ ♠♦✈✐♠❡♥t♦ ❞❛ ❜♦❧❛ r❡s✉♠✐✲s❡ ❛♣❡♥❛s ❡♠♦❜t❡r s✉❛ ✈❡❧♦❝✐❞❛❞❡ ❡ ♣♦s✐çã♦ s❡♠ ❧❡✈❛r♠♦s ❡♠ ❝♦♥t❛ ✈❛r✐❛çõ❡s ♥♦s ♣❛râ♠❡tr♦s q✉❡ ♣♦❞❡♠❛❢❡t❛r ♦ ♠♦✈✐♠❡♥t♦ ❞❛ ❜♦❧❛✳ P♦rt❛♥t♦✱ ❛❞♦t❛r❡♠♦s ❛s s❡❣✉✐♥t❡s ❞❡✜♥✐çõ❡s✿

❉❡✜♥✐çã♦ ✺✳ P❛râ♠❡tr♦ é t♦❞❛ ❡ q✉❛❧q✉❡r ❣r❛♥❞❡③❛ ❢ís✐❝❛ q✉❡ ♥ã♦ ✈❛r✐❛ ❛♣r❡❝✐❛✈❡❧♠❡♥t❡ ♥✉♠❝❡rt♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦✱ ❡♠ ❝♦♠♣❛r❛çã♦ ❝♦♠ ❛ ✈❛r✐❛çã♦ ❞❡ ♦✉tr❛s ❣r❛♥❞❡③❛s ❢ís✐❝❛s ♥♦ ♠❡s♠♦✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦✱ ♦✉ ❝✉❥❛ ✈❛r✐❛çã♦ ❡s♣❛❝✐❛❧ ♣♦❞❡ s❡r ❞❡s❝♦♥s✐❞❡r❛❞❛ ❞❡♥tr♦ ❞❛ ❡s❝❛❧❛ ❞❡❞✐stâ♥❝✐❛s ✉t✐❧✐③❛❞❛s ♥♦ ❡①♣❡r✐♠❡♥t♦✳

❉❡✜♥✐çã♦ ✻✳ ❈♦♥st❛♥t❡ é t♦❞❛ ❡ q✉❛❧q✉❡r ❣r❛♥❞❡③❛ ❢ís✐❝❛ ❝✉❥♦ ✈❛❧♦r s❡♠♣r❡ ♣❡r♠❛♥❡❝❡ ♦

✷❆ ❛❝❡❧❡r❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡✱ ❛❧é♠ ❞❡ ❞❡♣❡♥❞❡r ❞❛ ❛❧t✉r❛✱ ❞❡♣❡♥❞❡ ❞❛ ❧❛t✐t✉❞❡ ❞♦ ❧♦❝❛❧✳

✸✷ ❈❆P❮❚❯▲❖ ✹✳ ●❘➪❋■❈❖❙

♠❡s♠♦✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ ❧♦❝❛❧ ❡ ❞♦ ✐♥st❛♥t❡ ❡♠ q✉❡ ❢♦r ♠❡♥s✉r❛❞❛✸✳

❆ ❞❡t❡r♠✐♥❛çã♦ ❡①♣❡r✐♠❡♥t❛❧ ❞♦s ♣❛râ♠❡tr♦s ❡♥✈♦❧✈✐❞♦s r❡q✉❡r ♠ét♦❞♦s ♥✉♠ér✐❝♦s ♠❛s✱♠✉✐t❛s ✈❡③❡s✱ é ♣♦ssí✈❡❧ s✐♠♣❧✐✜❝❛r s✉❛ ❞❡t❡r♠✐♥❛çã♦ ❛tr❛✈és ❞❡ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ q✉❡ ❝♦♥s✐st❡❡♠ ✏tr❛♥s❢♦r♠❛r✑ ❛ ❞❡♣❡♥❞ê♥❝✐❛ y = y(x) ♥✉♠ ❞❡♣❡♥❞ê♥❝✐❛ ❧✐♥❡❛r✱ ♦✉ s❡❥❛✱ tr❛♥s❢♦r♠❛♠♦s❛ ❡q✉❛çã♦ y = y(x) ♥❛ ❡q✉❛çã♦ ❞❡ ✉♠❛ r❡t❛✱ ❝♦♠♦ y(x) = Ax + B✳ ❆❣♦r❛ ♦s ♣❛râ♠❡tr♦s❡♥✈♦❧✈✐❞♦s r❡❧❛❝✐♦♥❛♠✲s❡✱ ❛tr❛✈és ❞❛ tr❛♥s❢♦r♠❛çã♦✱ ❛♦s ❝♦❡✜❝✐❡♥t❡s ❛♥❣✉❧❛r ✭A✮ ❡ ❧✐♥❡❛r ✭B✮❞❛ r❡t❛✱ q✉❡ sã♦ ❢❛❝✐❧♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦s ♣♦r ❞♦✐s ♣♦♥t♦s ❞❛ r❡t❛✳

✹✳✸ ❈♦♥str✉çã♦ ❞❡ ●rá✜❝♦s

❆ ❝♦♥str✉çã♦ ❞❡ ✉♠ ❣rá✜❝♦ r❡q✉❡r ❝❡rt❛s r❡❣r❛s ♣❛r❛ q✉❡ ♦ ♠❡s♠♦ s❡❥❛ ❞❡ ❢á❝✐❧ ❧❡✐t✉r❛ ❡✐♥t❡r♣r❡t❛çã♦✳ ❆ s❡❣✉✐r✱ ❧✐st❛♠♦s ♦s ♣r✐♥❝✐♣❛✐s ✐t❡♥s ♣❛r❛ ❛ ❝♦♥❢❡❝çã♦ ❞❡ ✉♠ ❣rá✜❝♦✳

✹✳✸✳✶ ❊✐①♦s

❆♦ r❡♣r❡s❡♥t❛r ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❡①♣❡r✐♠❡♥t❛✐s ♥✉♠ ❣rá✜❝♦✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ❛ ♦r✐❡♥t❛çã♦❞♦ ♣❛♣❡❧ ❝♦♠ ✏r❡tr❛t♦✑ ♦✉ ❝♦♠♦ ✏♣❛✐s❛❣❡♠✑✳ ❊♠ q✉❛❧q✉❡r ❞♦s ❝❛s♦s✱ s❡♠♣r❡ ❝♦❧♦❝❛r❡♠♦s ♥♦❡✐①♦ ✈❡rt✐❝❛❧ ✭❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s✮ ♦s ✈❛❧♦r❡s ❞❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡✱ ❡♥q✉❛♥t♦ q✉❡ ♥♦ ❡✐①♦❤♦r✐③♦♥t❛❧ ✭❡✐①♦ ❞❛s ❛❜s❝✐ss❛s✮ sã♦ ♠❛r❝❛❞♦s ♦s ✈❛❧♦r❡s ❞❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡✳

❆♦ ❧❛❞♦ ❞♦ ❡✐①♦ ✈❡rt✐❝❛❧✱ ♥❛ ♣❛rt❡ s✉♣❡r✐♦r✱ ✐♥❞✐❝❛♠♦s ❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ♠❡❞✐❞❛ ❝♦♠ s✉❛✉♥✐❞❛❞❡✳ ❈❛s♦ ♥❡❝❡ssár✐♦✱ t❛♠❜é♠ ❞❡✈❡ s❡r ✐♥❞✐❝❛❞❛ ❛ ♣♦tê♥❝✐❛ ❞❡ 10 ✉t✐❧✐③❛❞❛✳ ❆❜❛✐①♦ ❞♦❡✐①♦ ❤♦r✐③♦♥t❛❧✱ ♥♦ ❝❛♥t♦ ✐♥❢❡r✐♦r ❞✐r❡✐t♦✱ ✐♥❞✐❝❛♠♦s ❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❣r❛♥❞❡③❛ t❛♠❜é♠ ❝♦♠s✉❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ❡ ♣♦tê♥❝✐❛ ❞❡ 10✳ ➱ ❡①tr❡♠❛♠❡♥t❡ ♥❡❝❡ssár✐♦ ❡ ✐♠♣♦rt❛♥t❡ ❛ ❝♦rr❡t❛✐♥❢♦r♠❛çã♦ ❛❝❡r❝❛ ❞❛ ❣r❛♥❞❡③❛ ❡ s✉❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛✳

✹✳✸✳✷ ❊s❝❛❧❛

❆♦ s❡ ❝♦♥str✉✐r ✉♠ ❣rá✜❝♦ t❛♠❜é♠ é ♥❡❝❡ssár✐❛ ❛ ❡s❝♦❧❤❛ ❛❞❡q✉❛❞❛ ❞❛s ❡s❝❛❧❛s✱ ✈✐st♦ q✉❡❛ ❡s❝❛❧❛ ❞❛ ❢♦❧❤❛ ✉t✐❧✐③❛❞❛ ♣❛r❛ ❛ ❝♦♥❢❡❝çã♦ ❞♦ ❣rá✜❝♦ ♣♦❞❡ ♥ã♦ ❝♦♠♣♦rt❛r ♦ ✐♥t❡r✈❛❧♦ ❞❡✈❛❧♦r❡s ❞❛s ❣r❛♥❞❡③❛s ❢ís✐❝❛s ♠❡❞✐❞❛s✳ ❆ ❡s❝♦❧❤❛ ❞❛ ❡s❝❛❧❛ t❛♠❜é♠ ❞❡✈❡ ❢❛❝✐❧✐t❛r ❛ ❧❡✐t✉r❛♣♦r ♣❛rt❡ ❞❡ q✉❛❧q✉❡r ♦✉tr❛ ♣❡ss♦❛✳ ❆ss✐♠✱ ❡①✐st❡♠ ❝❡rt❛s ❡s❝❛❧❛s q✉❡ sã♦ ♠❛✐s ❛♣r♦♣r✐❛❞❛s❡♠ r❡❧❛çã♦ ❛ ♦✉tr❛s✳ P♦r ❡①❡♠♣❧♦✱ ✉♠❛ ❡s❝❛❧❛ ♥❛ q✉❛❧ ❝❛❞❛ ❝❡♥tí♠❡tr♦ ❞❡ ♣❛♣❡❧ ♠✐❧✐♠❡tr❛❞♦❝♦rr❡s♣♦♥❞❛ ❛ 2 ✉♥✐❞❛❞❡s ❞❡ ♠❡❞✐❞❛ ✭2 u.m.✮ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠❛ ❡s❝❛❧❛✱ ❡♠ ♣❛♣❡❧ ♠✐❧✐♠❡tr❛❞♦✱❞❡ 100mm/2u.m. = 5mm/u.m. ♦✉ 1 u.m./5mm ♦✉ ❛✐♥❞❛ 0, 2 u.m./mm✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡ ❝❛❞❛♠✐❧í♠❡tr♦ ❞♦ ♣❛♣❡❧ ❝♦rr❡s♣♦♥❞❡ ❛ 0, 2 u.m. ❞❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❣r❛♥❞❡③❛✳ ❊st❛ é ✉♠❛ ❡s❝❛❧❛ ❞❡❢á❝✐❧ ❧❡✐t✉r❛ ♣♦✐s é ❡q✉✐✈❛❧❡♥t❡ ❛ 2 u.m./cm✳ ■♠❛❣✐♥❡ ❛❣♦r❛ q✉❡ ♦ ✐♥t❡r✈❛❧♦ ❞❡ ✈❛❧♦r❡s ♠❡❞✐❞♦s❢♦ss❡ t❛❧ q✉❡ t✐✈éss❡♠♦s 9 u.m./7mm✱ ♦✉ s❡❥❛✱ 1, 28571 · · · u.m./cm✱ s❡♥❞♦ q✉❡ ❞❡✈❡♠♦s ✐♥❞✐❝❛r✈ár✐❛s ❝❛s❛s ❞❡❝✐♠❛✐s✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♣r❡❝✐sã♦ ❞❛s ♠❡❞✐❞❛s✳ P♦r ❡①❡♠♣❧♦✱ 1, 286 u.m.✳ ❖♥❞❡❡st❛rá ♦ ♣♦♥t♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ✈❛❧♦r 6, 503 u.m. ❞❛ ❣r❛♥❞❡③❛❄ ❉✐❢í❝✐❧ ❞❡ ❡♥❝♦♥tr❛r✦ ❆ss✐♠✱❞❛♠♦s ♣r❡❢❡rê♥❝✐❛ ❛ ❡s❝❛❧❛s q✉❡ s❡❥❛♠ ♠ú❧t✐♣❧❛s ❞❡ 1, 2, 5 ♦✉ 10 ♣♦r s❡r ❞❡ ♠❛✐s ❢á❝✐❧ ❧❡✐t✉r❛✳❚❛♠❜é♠ ♣♦❞❡♠ s❡r 2, 5 ❡ 4✳

❖❜s❡r✈❡♠♦s t❛♠❜é♠ q✉❡ ♥❡♠ s❡♠♣r❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❛♣r❡s❡♥t❛ ♦ ♣♦♥t♦✭0, 0✮ ❞❡ ♠♦❞♦ q✉❡ ❛s ❡s❝❛❧❛s ♥ã♦ ♣r❡❝✐s❛♠ ❝♦♠❡ç❛r ♥❛ ♦r✐❣❡♠✳ ❈❛❞❛ ❡s❝❛❧❛ x ♦✉ y ♣♦❞❡❝♦♠❡ç❛r ❝♦♠ ✉♠ ✈❛❧♦r ♠❛✐s ❛❞❡q✉❛❞♦ ❡ ❛s ❡s❝❛❧❛s ❡s❝♦❧❤✐❞❛s ♣❛r❛ ♦s ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s sã♦

✸❆t✉❛❧♠❡♥t❡ ❡①✐st❡♠ t❡♦r✐❛s q✉❡ ✈❡r✐✜❝❛♠ s❡ ❛s ❝♦♥st❛♥t❡s ❞❛ ❋ís✐❝❛✱ ❝♦♠♦ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③ ❡ ❛ ❝♦♥st❛♥t❡❞❛ ❣r❛✈✐t❛çã♦✱ sã♦ r❡❛❧♠❡♥t❡ ❝♦♥st❛♥t❡s ♦✉ s❡ ♣♦❞❡♠ ✈❛r✐❛r ❞❡♥tr♦ ❞❡ ✉♠❛ ❝❡rt❛ ❡s❝❛❧❛ ❞❡ t❡♠♣♦✳ P❛r❛ ♠❛✐s❞❡t❛❧❤❡s ❝♦♥s✉❧t❛r ♦ tr❛❜❛❧❤♦ ❙❡❛r❝❤ ❢♦r ❱❛r✐❛t✐♦♥s ♦❢ ❋✉♥❞❛♠❡♥t❛❧ ❈♦♥st❛♥ts ✉s✐♥❣ ❆t♦♠✐❝ ❋♦✉♥t❛✐♥❈❧♦❝❦s❬✸❪✳

✹✳✸✳ ❈❖◆❙❚❘❯➬➹❖ ❉❊ ●❘➪❋■❈❖❙ ✸✸

✐♥❞❡♣❡♥❞❡♥t❡s✱ ♦✉ s❡❥❛✱ ♦s ❡✐①♦s ♣♦❞❡♠ ❛♣r❡s❡♥t❛r ❡s❝❛❧❛s ❞✐❢❡r❡♥t❡s✳ ❙❡♠♣r❡ q✉❡ ♣♦ssí✈❡❧♦✉ ♥❡❝❡ssár✐♦✱ ♣♦❞❡♠♦s ✉t✐❧✐③❛r ✉♠❛ ❡s❝❛❧❛ ♠ú❧t✐♣❧❛ ❞❡ 10✱ ❝♦♠♦ 5 × 10−3 u.m./mm ♦✉ 4 ×104 u.m./mm✳ P❛r❛ ❡s❝♦❧❤❡r♠♦s ❛ ❡s❝❛❧❛ ♣r♦❝❡❞❡♠♦s ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✳ ❖❜s❡r✈❛♠♦s ♥❛t❛❜❡❧❛ ❞❡ ❞❛❞♦s ♦s ✈❛❧♦r❡s ♠á①✐♠♦s ❡ ♠í♥✐♠♦s ❞❛ ❣r❛♥❞❡③❛ ♠❡❞✐❞❛ ✭xmın, x1, x2, · · · , xmax✮ ❡❝❛❧❝✉❧❛♠♦s ❛ r❛③ã♦

escx =xmax − xmın

Tpx

, ✭✹✳✸✳✶✮

❡♠ q✉❡ escx é ♦ ✈❛❧♦r ❞❛ ❡s❝❛❧❛ ❛ s❡r ✉s❛❞❛ ❡ Tpx é ♦ t❛♠❛♥❤♦ ❞♦ ♣❛♣❡❧✱ ❡♠ ❝❡♥tí♠❡tr♦s✱ ❛s❡r ✉s❛❞♦ ♣❛r❛ ❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡ x✳ ◆♦ ❝❛s♦ ❞♦ ♣❛♣❡❧ ♠✐❧✐♠❡tr❛❞♦ ❆✹✱ ♣❛❞r♦♥✐③❛❞♦ ♥♦❇r❛s✐❧✱ ✉♠❛ ❡s❝❛❧❛ ❛♣r❡s❡♥t❛ ♦ t❛♠❛♥❤♦ ❞❡ 18 cm ❡ ❛ ♦✉tr❛ ❡s❝❛❧❛✱ 28 cm✳ ❆ss✐♠✱ é ❞❡s❡❥á✈❡❧q✉❡ ❛ r❡❧❛çã♦ ✭✹✳✸✳✶✮ s❡❥❛ 1, 2, 5 ♦✉ 10✱ ❡✈❡♥t✉❛❧♠❡♥t❡ 4✳ ❆❧❣✉♥s ✈❡③❡s ♣♦❞❡ s❡r út✐❧✱ ♣❛r❛ ✜♥s❞❡ ❝á❧❝✉❧♦ ❞❛ ❡s❝❛❧❛✱ ✉t✐❧✐③❛r ♥ã♦ ♦s 18 cm ❞♦ ♣❛♣❡❧ ♠❛s ✉t✐❧✐③❛r 15 cm✳ ❖ ✐♠♣♦rt❛♥t❡ é ✉t✐❧✐③❛r♦ ♠❛✐♦r ❡s♣❛ç♦ ♣♦ssí✈❡❧ ❞♦ ♣❛♣❡❧ ♣❛r❛ s❡ t❡r ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❝❧❛r❛ ❞❛ ❡s❝❛❧❛✱ ❞♦s ♣♦♥t♦s ❡❛ ❧❡✐t✉r❛ ❞❡ ♦✉tr♦s ♣♦♥t♦s✱ q✉❡ ♥ã♦ ♦s ♣♦♥t♦s ❡①♣❡r✐♠❡♥t❛✐s✳ ❊st❡ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡✈❡ s❡r ❢❡✐t♦❡♠ ❛♠❜♦s ♦s ❡✐①♦s✳

✹✳✸✳✸ ▼❛r❝❛çã♦ ❞♦s ♣♦♥t♦s ❡ tr❛ç❛❞♦ ❞❛ ❝✉r✈❛

❖s ♣♦♥t♦s ❡①♣❡r✐♠❡♥t❛✐s ❞❡✈❡♠ s❡r ❝♦rr❡t❛♠❡♥t❡ ♠❛r❝❛❞♦s ♣❛r❛ q✉❡ s❡❥❛♠ ❢❛❝✐❧♠❡♥t❡ r❡❝♦♥✲❤❡❝✐❞♦s✳ P❛r❛ ✐st♦✱ ❝♦st✉♠❛✲s❡ r❡♣r❡s❡♥tá✲❧♦s ♣♦r sí♠❜♦❧♦s✱ ❛♦ ✐♥✈és ❞❡ ♣♦♥t♦s✳ ❚❛♠❜é♠ ♣♦❞❡♠s❡r ✉s❛❞♦s ♣♦♥t♦s ❝♦♠ sí♠❜♦❧♦s✳ ❆❜❛✐①♦ ♠♦str❛♠♦s ❛❧❣✉♥s ❞♦s ♠❛✐s ✉t✐❧✐③❛❞♦s✳

⊗ ⊡ ⊙ + × ♦ ⊠ ⊞ ⊛ ⊕

❈❛s♦ s❡❥❛ ♣r❡❝✐s♦ ❝♦❧♦❝❛r ♥✉♠ ♠❡s♠♦ ❣rá✜❝♦ ❞♦✐s ♦✉ ♠❛✐s ❝♦♥❥✉♥t♦s ❞❡ ♣♦♥t♦s ❡①♣❡r✐♠❡♥t❛✐s✱❞❡✈❡✲s❡ ✉s❛r ✉♠ t✐♣♦ ❞❡ sí♠❜♦❧♦ ♣❛r❛ ❝❛❞❛ ❝♦♥❥✉♥t♦ ❡ ✐♥s❡r✐r ✉♠❛ ❧❡❣❡♥❞❛ ♥♦ ❣rá✜❝♦✳ ❆ ✜❣✉r❛❛❜❛✐①♦ ✐❧✉str❛✳

✸✹ ❈❆P❮❚❯▲❖ ✹✳ ●❘➪❋■❈❖❙

variável depende

0,20

0,40

0,60

0,80

1,00

1,20

variável dependente

2,0 3,0 4,0 5,0 6,0 7,0 8,0 9,0 10,0

Dados 1Dados 2

❋✐❣✉r❛ ✹✳✸✳✶✿ ❉♦✐s ❝♦♥❥✉♥t♦s ❞❡ ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ♥✉♠ ♠❡s♠♦ ❣rá✜❝♦✳

❈♦♠♦ ❥á ❝♦♠❡♥t❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ♦s ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❛♣r❡s❡♥t❛♠ ❡rr♦s ✐♥❡r❡♥t❡s ❛♦♣r♦❝❡ss♦ ❞❡ ♠❡❞✐❞❛ ♦ q✉❡ ❢❛③ ❝♦♠ q✉❡ ♦s ♣♦♥t♦s ❡①♣❡r✐♠❡♥t❛✐s ♥ã♦ ❝❛✐❛♠ ❡①❛t❛♠❡♥t❡ s♦❜r❡✉♠❛ r❡t❛ ♦✉ ❝✉r✈❛✱ ♠❛s q✉❡ s✐❣❛♠ ✉♠❛ ❝❡rt❛ t❡♥❞ê♥❝✐❛✳ ❆ ✜❣✉r❛ ❛❜❛✐①♦ ✐❧✉str❛ ❛ s✐t✉❛çã♦✳

x(cm)

0,0

5,0

10,0

15,0

20,0

t(s)

1,0 2,0 3,0 4,0 5,0 6,0 7,0 8,0 9,0

dados experimentais ajuste linear

x(cm)

0,00

10,00

20,00

30,00

40,00

50,00

60,00

70,00

t(s)

1,00 2,00 3,00 4,00 5,00 6,00 7,00 8,00 9,00

Dados experimentais ajuste quadrático

❋✐❣✉r❛ ✹✳✸✳✷✿ ❚❡♥❞ê♥❝✐❛ ❞♦s ♣♦♥t♦s ❡①♣❡r✐♠❡♥t❛✐s✳ ◆❛ ✜❣✉r❛ ❞❛ ❡sq✉❡r❞❛ ♦s ♣♦♥t♦s t❡♥❞❡♠ ❛s❡❣✉✐r ✉♠❛ r❡t❛✱ ❡♥q✉❛♥t♦ ♦s ♣♦♥t♦s ❞❛ ❞✐r❡✐t❛ t❡♥❞❡♠ ❛ s❡❣✉✐r ✉♠❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛✱ ❝♦♠♦ ✉♠❛♣❛rá❜♦❧❛✱ ♣♦r ❡①❡♠♣❧♦✳

❏❛♠❛✐s ♣♦❞❡r❡♠♦s ✉♥✐r ♦s ♣♦♥t♦s ❡①♣❡r✐♠❡♥t❛✐s ♣♦r ✉♠❛ ♣♦❧✐❣♦♥❛❧✱ ♠❛s s✐♠✱ tr❛ç❛r s✉❛✈❡✲♠❡♥t❡ ❛ t❡♥❞ê♥❝✐❛ ❞♦s ♣♦♥t♦s✳ ◆❡st❡ ♣r♦❝❡ss♦ ❞❡ tr❛ç❛❞♦ s✉❛✈❡ ❡ ❝♦♥tí♥✉♦ ❞❡✈❡ s❡r ♦❜s❡r✈❛❞♦

✹✳✸✳ ❈❖◆❙❚❘❯➬➹❖ ❉❊ ●❘➪❋■❈❖❙ ✸✺

q✉❡ ❛ ❝✉r✈❛ tr❛ç❛❞❛ ♥ã♦ ♣r❡❝✐s❛ ♣❛ss❛r ♣❡❧♦s ♣♦♥t♦s ❡①♣❡r✐♠❡♥t❛✐s✱ ♣♦ré♠✱ ♥ã♦ ❞❡✈❡ ❡st❛r tã♦❞✐st❛♥t❡ ❞♦s ♠❡s♠♦s✳ P♦❞❡ ❛té ♦❝♦rr❡r ❞❡❧❛ ♣❛ss❛r ♣♦r ✉♠ ♦✉ ♦✉tr♦ ♣♦♥t♦✳ ❖ tr❛ç❛❞♦ ❞❛ ❝✉r✈❛✱♦✉ ❞❛ r❡t❛✱ ♣♦❞❡ s❡r ❢❡✐t♦ ❛ ♠ã♦ ❧✐✈r❡ ♦✉ ❛tr❛✈és ❞♦ ✉s♦ ❞❡ ré❣✉❛s ❝♦♠✉♥s ❡ ❞❡ ré❣✉❛ ❢r❛♥❝❡s❛✳❙❡♥❞♦ ♦ tr❛ç❛❞♦ ❢❡✐t♦ ❞❡ ♠♦❞♦ ❛ s❡ ❡s❝♦❧❤❡r✱ ✈✐s✉❛❧♠❡♥t❡✱ ❛ ♠❡❧❤♦r ❝✉r✈❛ q✉❡ ❞❡s❝r❡✈❡ ♦s ♣♦♥✲t♦s✱ ✜❝❛ ❛ ♣❡r❣✉♥t❛✿ ❝♦♠♦ s❛❜❡r s❡ ❛ ❝✉r✈❛ tr❛ç❛❞❛ ❡stá ❝♦rr❡t❛❄ ❙❡ t✐✈❡r♠♦s ✉♠❛ t❡♥❞ê♥❝✐❛❧✐♥❡❛r ♣❛r❛ ♦s ♣♦♥t♦s ❡①♣❡r✐♠❡♥t❛✐s✱ ❝♦♠♦ s❛❜❡r ❛ ♠❡❧❤♦r r❡t❛ q✉❡ ❞❡s❝r❡✈❡ t❛✐s ♣♦♥t♦s❄ ❊st❛s❡ ♦✉tr❛s ♣❡r❣✉♥t❛s ♣♦❞❡♠ s❡r r❡s♣♦♥❞✐❞❛s ✉s❛♥❞♦✲s❡ ♠ét♦❞♦s ♥✉♠ér✐❝♦s✳ ❯♠ ❡①❡♠♣❧♦ ❜❡♠ ❢❛✲♠✐❧✐❛r é ♦ ♠ét♦❞♦ ❞♦s ♠í♥✐♠♦s q✉❛❞r❛❞♦s q✉❡✱ ❝♦♠ ❜❛s❡ ♥♦s ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s✱ ♥♦s ♣❡r♠✐t❡❝❛❧❝✉❧❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❛♥❣✉❧❛r ❡ ❧✐♥❡❛r ❞❛ ♠❡❧❤♦r r❡t❛ q✉❡ r❡♣r❡s❡♥t❛ ♦s ♣♦♥t♦s ❞♦ ❣rá✜❝♦✳

❉❡st❛ ❢♦r♠❛✱ ❡♠❜♦r❛ ♦s ♣♦♥t♦s ❡①♣❡r✐♠❡♥t❛✐s ♥ã♦ ❝❛✐❛♠ s♦❜r❡ ✉♠❛ r❡t❛✱ ♣♦❞❡♠♦s ❜✉s❝❛r✉♠❛ ✏tr❛♥s❢♦r♠❛çã♦✑ q✉❡ tr❛♥s❢♦r♠❡ ❛ r❡❧❛çã♦ ♥ã♦✲❧✐♥❡❛r ❡♥tr❡ y ❡ x ♥✉♠❛ r❡❧❛çã♦ ❧✐♥❡❛r ❡♥tr❡❡❧❡s✳ ◆❡st❛ tr❛♥s❢♦r♠❛çã♦✱ ❛s ❝♦♥st❛♥t❡s ❡ ♣❛râ♠❡tr♦s ❡♥✈♦❧✈✐❞♦s r❡❧❛❝✐♦♥❛♠✲s❡ ❛♦s ❝♦❡✜❝✐❡♥t❡s❞❛ r❡t❛✳

✹✳✸✳✹ ▲✐♥❡❛r✐③❛çã♦

▼✉✐t❛s ✈❡③❡s ❛ r❡❧❛çã♦ ❡♥tr❡ ❞✉❛s ❣r❛♥❞❡③❛s ❢ís✐❝❛s é ♥ã♦✲❧✐♥❡❛r ❡ ❡♥✈♦❧✈❡ ❝♦♥st❛♥t❡s ❡ ♣❛râ♠❡t✲r♦s ❞❡s❝♦♥❤❡❝✐❞♦s✳ ❯♠ ♠♦❞❡❧♦ t❡ór✐❝♦ ♣♦❞❡ ❞❛r ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ♦ t✐♣♦ ❞❡ r❡❧❛çã♦ ❡♥tr❡❛s ❣r❛♥❞❡③❛s ♠❡❞✐❞❛s✱ ❜❡♠ ❝♦♠♦✱ ❛s ❝♦♥st❛♥t❡s ❡♥✈♦❧✈✐❞❛s✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛s❝♦♥st❛♥t❡s ♣♦❞❡ s❡r ✉♠ ♣r♦❝❡ss♦ ❧♦♥❣♦ ❡ tr❛❜❛❧❤♦s♦ q✉❡ ♣♦❞❡ s❡r s✐♠♣❧✐✜❝❛❞♦✱ ❝♦♠♦ ♠❡♥✲❝✐♦♥❛❞♦ ❛❝✐♠❛✱ ✉s❛♥❞♦✲s❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❧✐♥❡❛r✐③❛çã♦✱ q✉❡ ❝♦♥s✐st❡♠ ❡♠ tr❛♥s❢♦r♠❛çã♦ ❛ r❡❧❛çã♦y = y(x) ♥❛ ❡q✉❛çã♦ ❞✉♠❛ r❡t❛✳ ❖s ❝♦❡✜❝✐❡♥t❡s ❞❛ r❡t❛ ❡st❛rã♦ r❡❧❛❝✐♦♥❛❞♦s às ❝♦♥st❛♥t❡s ❛s❡r❡♠ ❞❡t❡r♠✐♥❛❞❛s✳

❊♠ ❣❡r❛❧ ♥ã♦ ❤á ✉♠❛ r❡❣r❛ ❞❡✜♥✐❞❛ ♣❛r❛ ♦ ♣r♦❝❡ss♦ ❞❡ ❧✐♥❡❛r✐③❛çã♦✱ t❛♥t♦ q✉❡ ✉♠❛ ♠❡s♠❛❡q✉❛çã♦ ♣♦❞❡ s❡r ❧✐♥❡❛r✐③❛❞❛ ❞❡ ❞✉❛s ♦✉ ♠❛✐s ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s✳ ❊st❡ ♣r♦❝❡❞✐♠❡♥t♦ ❡♥✈♦❧✈❡❛ ❡s❝♦❧❤❛ ❛❞❡q✉❛❞❛ ❞❡ ♥♦✈❛s ✈❛r✐á✈❡✐s y′ ❡ x′ t❛✐s q✉❡ ✈❛❧❤❛ ❛ r❡❧❛çã♦ y′(x′) = A′x′ + B′✳ ❯♠❛❞❛s ♠❛♥❡✐r❛s ❞❡ ❧✐♥❡❛r✐③❛çã♦ ✉♠❛ ❢✉♥çã♦ é ❝♦♠♣❛rá✲❧❛ ❞✐r❡t❛♠❡♥t❡ ❝♦♠ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ ❛✜♠❞❡ ❡♥❝♦♥tr❛r y′✱ x′ ❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❛♥❣✉❧❛r ✭A′✮ ❡ ❧✐♥❡❛r ✭B′✮✳ ❖s ❡①❡♠♣❧♦s ❛❜❛✐①♦ ✐❧✉str❛♠✳

❊①❡♠♣❧♦ ✶✳ ❈♦♥s✐❞❡r❡ ❛ r❡❧❛çã♦ y(x) = ax2 + c ❝♦♠ a ❡ c ❝♦♥st❛♥t❡s✳ ❈♦♠♣❛r❡♠♦s ❝♦♠❛ ❡q✉❛çã♦ ❞❛ r❡t❛

y(x) = ax2 + c ✭✹✳✸✳✷✮

y′(x′) = A′x′ +B′ ✭✹✳✸✳✸✮

❱❡♠♦s q✉❡ y′ = y✱ x′ = x2✱ A′ = a ❡ B′ = c✳ ❊♥tã♦ ❡st❛s sã♦ ❛s r❡❧❛çõ❡s ❞❡ tr❛♥s❢♦r♠❛çã♦ q✉❡tr❛♥s❢♦r♠❛♠ ❛ ♣❛rá❜♦❧❛ ✭✹✳✸✳✷✮ ♥❛ r❡t❛ ✭✹✳✸✳✸✮✳

❊①❡♠♣❧♦ ✷✳ ❙❡❥❛ ❛ ❢✉♥çã♦ T = 2π

√

m

kq✉❡ r❡♣r❡s❡♥t❛ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ ♣❡rí♦❞♦ ❞❡ ✉♠

♦s❝✐❧❛❞♦r ❤❛r♠ô♥✐❝♦ ❝♦♠ ❛ ♠❛ss❛ ♣r❡s❛ ❛ ✉♠❛ ♠♦❧❛ ✐❞❡❛❧ ❞❡ ❝♦♥st❛♥t❡ ❡❧ást✐❝❛ k✳ ❙✉♣♦♥❤❛♠♦sq✉❡ ❞✐❢❡r❡♥t❡s ♠❛ss❛s s❡❥❛♠ ❧✐❣❛❞❛s à ♠❡s♠❛ ♠♦❧❛✳ ❆ss✐♠✱ ❡s❝♦❧❤❡♥❞♦✲s❡ ❛❧t❡r❛r ♦ ✈❛❧♦r ❞❛♠❛ss❛ ♣❛r❛ ♠❡❞✐r ♦ ♣❡rí♦❞♦✱ ♦❜t❡♠♦s q✉❡ T = T (m)✱ ♦✉ s❡❥❛✱ ♦ ♣❡rí♦❞♦ ❞❡ ♦s❝✐❧❛çã♦ é ❛✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ q✉❡ ❞❡♣❡♥❞❡ ❞❛ ♠❛ss❛ ✭✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡✮ ❝♦❧♦❝❛❞❛ ❛ ♦s❝✐❧❛r✳ ❆♥t❡s ❞❡❝♦♠♣❛r❛r♠♦s ❞✐r❡t❛♠❡♥t❡ ❛ ❡q✉❛çã♦ ❝♦♠ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛✱ ❞❡✈❡♠♦s ♦❜s❡r✈❛r q✉❡ ❛ ♠❛ss❛ ❡ ❛❝♦♥st❛♥t❡ ❞❛ ♠♦❧❛ ❡stã♦ ❞❡♥tr♦ ❞♦ r❛❞✐❝❛❧✱ s❡♥❞♦ ♥❡❝❡ssár✐♦ ♣r✐♠❡✐r❛♠❡♥t❡ ✏s❡♣❛rá✲❧❛s✑✳ ❆ss✐♠✱

é ❝♦♥✈❡♥✐❡♥t❡ q✉❡ ❡s❝r❡✈❛♠♦s T =2π√k

√m ❞❡ ♠♦❞♦ q✉❡ t♦❞❛s ❛s ❝♦♥st❛♥t❡s ♣❡r♠❛♥❡ç❛♠ ❥✉♥t❛s

♥✉♠❛ ú♥✐❝❛ ❝♦♥st❛♥t❡✳ ❆❣♦r❛ ❝♦♠♣❛r❡♠♦s

T (m) =2π√k

√m ✭✹✳✸✳✹✮

y′(x′) = A′x′ +B′ ✭✹✳✸✳✺✮

✸✻ ❈❆P❮❚❯▲❖ ✹✳ ●❘➪❋■❈❖❙

❱❡♠♦s q✉❡ ❛s r❡❧❛çõ❡s ❞❡ tr❛♥s❢♦r♠❛çã♦ ✭❧✐♥❡❛r✐③❛çã♦✮ sã♦ y′ = T ✱ x′ =√m✱ A′ =

2π√k

❡

B′ = 0✳ ❖❜s❡r✈❡ q✉❡ ♥❡st❡ ❝❛s♦ ♦ ❝♦❡✜❝✐❡♥t❡ ❧✐♥❡❛r ❞❛ r❡t❛ é ③❡r♦✱ ♦ q✉❡ ♣❡r♠✐t❡ ❡st✐♠❛r s❡ ❛r❡t❛ tr❛ç❛❞❛ ❛ ♠ã♦ ❡stá ❜♦❛ ♦✉ ♥ã♦✳

❯♠❛ ♦✉tr❛ ♠❛♥❡✐r❛ ❞❡ ❧✐♥❡❛r✐③❛r ❛ ❡q✉❛çã♦ ✭✹✳✸✳✹✮ é ❡❧❡✈❛♥❞♦✲❛ ❛♦ q✉❛❞r❛❞♦ ❡ ❝♦♠♣❛r❛♥❞♦✲❛♥♦✈❛♠❡♥t❡ ❝♦♠ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛✳

T 2 =4π2

km

y′ = A′x′ +B′

❆❣♦r❛ t❡♠♦s y′ = T 2✱ x′ = m✱ A′ =4π2

k❡ ♥♦✈❛♠❡♥t❡ B′ = 0✳ ❉❡✈❡ s❡r ♦❜s❡r✈❛❞♦✱ ♥❡st❡ ❝❛s♦✱

q✉❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s r❡t❛s sã♦ ❞✐❢❡r❡♥t❡s✱ t♦❞❛✈✐❛✱ ♦ ✈❛❧♦r ❞❛ ❝♦♥st❛♥t❡ ❡❧ást✐❝❛ ❞❛ ♠♦❧❛✱ k✱❞❡✈❡ s❡r ♦ ♠❡s♠♦✳

❊①❡♠♣❧♦ ✸✳ ❈♦♥s✐❞❡r❡ ❛❣♦r❛ ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✱ t❛❧ q✉❛❧ y(x) = αeβx ❡♠ q✉❡ ❛s❝♦♥st❛♥t❡s sã♦ α ❡ β✳ ❙❡ ❛ ❝♦♥st❛♥t❡ β ❢♦r ❝♦♥❤❡❝✐❞❛✱ ♣♦❞❡r❡♠♦s ✉s❛r y′ = y✱ x′ = eβx✱ A′ = α❡ B′ = 0✳ ◆♦ ❡♥t❛♥t♦✱ ♠✉✐t❛s ❞❛s ✈❡③❡s ❛ ❝♦♥st❛♥t❡ β t❛♠❜é♠ ♣r❡❝✐s❛ s❡r ❞❡t❡r♠✐♥❛❞❛✳ ❆ss✐♠✱❛♥t❡s ❞❛ ❝♦♠♣❛r❛çã♦ ❝♦♠ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ ❞❡✈❡♠♦s ❛♣❧✐❝❛r ♦ ❧♦❣❛r✐t♠♦ ♥❛t✉r❛❧ ♥♦s ❞♦✐s ❧❛❞♦s❞❛ r❡❧❛çã♦ ❛✜♠ ❞❡ ✏r❡t✐r❛r♠♦s✑ ❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡ x ❞♦ ❛r❣✉♠❡♥t♦ ❞❛ ❡①♣♦♥❡♥❝✐❛❧✳ ❆ss✐♠♦❜t❡♠♦s

ln(y) = ln(α) + βx ✭✹✳✸✳✻✮

y′ = A′x′ +B′ ✭✹✳✸✳✼✮

❆ss✐♠✱ é ✐♠❡❞✐❛t♦ q✉❡ y′ = ln(y)✱ x′ = x✱ A′ = β ❡ B′ = ln(A) é ❛ ❧✐♥❡❛r✐③❛çã♦ ♣❛r❛ ❡st❡ t✐♣♦❞❡ ❢✉♥çã♦✳ ❈❛s♦ ❛ ❜❛s❡ ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ♥ã♦ s❡❥❛ ♦ ♥ú♠❡r♦ ❞❡ ◆❡♣❡r ✭e✮ ❞❡✈❡✲s❡ ❛♣❧✐❝❛r♦ ❧♦❣❛r✐t♠♦ ❞❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❜❛s❡✱ ❝♦♠♦ log2 ♦✉ log10✳

❊①❡♠♣❧♦ ✹✳ ❆ r❡s✐stê♥❝✐❛ ❡❧étr✐❝❛ ❞❡ ✉♠ ❝♦♥❞✉t♦r é ♦❜t✐❞❛ ❛♣❧✐❝❛♥❞♦✲s❡ ✉♠❛ ❞✐❢❡r❡♥ç❛❞❡ ♣♦t❡♥❝✐❛❧ ❡❧étr✐❝♦ ✭♦✉ t❡♥sã♦✮ ❡♥tr❡ ❛s ❡①tr❡♠✐❞❛❞❡s ❞♦ ❝♦♥❞✉t♦r ❡ ♠❡❞✐❞♦✲s❡ ❛ ✐♥t❡♥s✐❞❛❞❡

❞❛ ❝♦rr❡♥t❡ ❡❧étr✐❝❛ q✉❡ ♦ ❛tr❛✈❡ss❛✳ ❆ r❡❧❛çã♦ R =V

i❞❡✜♥❡✱ ❡♥tã♦✱ ❛ r❡s✐stê♥❝✐❛ ❡❧étr✐❝❛✳

❆❧❣✉♥s ❝♦♥❞✉t♦r❡s✱ ❝❤❛♠❛❞♦s ❞❡ ô❤♠✐❝♦s✱ ♣♦ss✉❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ s✉❛ r❡s✐stê♥❝✐❛ ❡❧étr✐❝❛s❡r ❝♦♥st❛♥t❡ ❡ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ ✈❛❧♦r ❡ ❞❛ ♣♦❧❛r✐❞❛❞❡ ❞❛ t❡♥sã♦ ❛♣❧✐❝❛❞❛✳ ■st♦ ❢❛③ ❝♦♠ q✉❡ ❛❝♦rr❡♥t❡ ❡❧étr✐❝❛ s❡❥❛ ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ à t❡♥sã♦ ❛♣❧✐❝❛❞❛ ❡ ✉♠❛ r❡❧❛çã♦ ❧✐♥❡❛r ❡♥tr❡ V

❡ i é ♦❜t✐❞❛✱ t❛❧ q✉❛❧ i =V

R✳ ◆❡st❡ ❝❛s♦ ❛ ❧✐♥❡❛r✐③❛çã♦ ❢♦r♥❡❝❡ A′ = 1/R✳ ◆♦ ❡♥t❛♥t♦✱ ♠✉✐t♦s

♠❛t❡r✐❛✐s ❡ ❞✐s♣♦s✐t✐✈♦s ❝♦♥❞✉t♦r❡s ❛♣r❡s❡♥t❛♠ ♣r♦♣r✐❡❞❛❞❡s t❛✐s q✉❡ s✉❛ r❡s✐stê♥❝✐❛ ✈❛r✐❛ ❡✱❡♠ ❣❡r❛❧✱ s❡ ♦❜té♠ r❡❧❛çõ❡s ♥ã♦✲❧✐♥❡❛r❡s ❡♥tr❡ ❛ ❝♦rr❡♥t❡ ❡❧étr✐❝❛ ❡ ❛ t❡♥sã♦✳ ❈♦♥s✐❞❡r❡ ✉♠❞✐s♣♦s✐t✐✈♦ q✉❡ t❡♥❤❛ t❛✐s ♣r♦♣r✐❡❞❛❞❡s✳ ❊♥tã♦ ✈❛❧❡ ❛ r❡❧❛çã♦ i = aV b ❝♦♠ a ❡ b ❝♦♥st❛♥t❡s✳❖❜s❡r✈❡ q✉❡ ♦ ❝❛s♦ ❞❡ b ❝♦♥❤❡❝✐❞♦ é s❡♠❡❧❤❛♥t❡ ❛♦ ❝❛s♦ ❞♦ ❊①❡♠♣❧♦ ✷ ♠❛s✱ ❡♠ ❣❡r❛❧✱ b ét❛♠❜é♠ ❞❡s❝♦♥❤❡❝✐❞♦✳ ❖❜s❡r✈❡ ❛✐♥❞❛ ❛ ❞✐❢❡r❡♥ç❛ ❞❡st❡ ❝❛s♦ ❡♠ r❡❧❛çã♦ àq✉❡❧❡ ❞♦ ❊①❡♠♣❧♦ ✸♥♦ q✉❛❧ ❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡ ❡st❛✈❛ ♥♦ ❛r❣✉♠❡♥t♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✳ ❆❣♦r❛ t❡♠♦s✉♠❛ ♣♦tê♥❝✐❛ ❞❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡✳ ❙ã♦ s✐t✉❛çõ❡s ❜❡♠ ❞✐❢❡r❡♥t❡s✦ ❆q✉✐ t❛♠❜é♠ ❞❡✈❡♠♦s❛♣❧✐❝❛r ❛ ❢✉♥çã♦ ❧♦❣❛r✐t♠♦ ❛♥t❡s ❞❡ ❝♦♠♣❛r❛r ❝♦♠ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛✳

log(i) = log(a) + blog(V ) ✭✹✳✸✳✽✮

y′ = A′x′ +B′ ✭✹✳✸✳✾✮

❱❡♠♦s ❢❛❝✐❧♠❡♥t❡ q✉❡ y′ = log(i)✱ x′ = log(V )✱ A′ = b ❡ B′ = log(a)✳

✹✳✸✳ ❈❖◆❙❚❘❯➬➹❖ ❉❊ ●❘➪❋■❈❖❙ ✸✼

✹✳✸✳✺ ❉❡t❡r♠✐♥❛çã♦ ❞❡ ❝♦♥st❛♥t❡s

❯♠❛ ✈❡③ ❢❡✐t❛ ❛ ❧✐♥❡❛r✐③❛çã♦ ♣♦❞❡r❡♠♦s ❝❛❧❝✉❧❛r ❛s ❝♦♥st❛♥t❡s ❞❡s❝♦♥❤❡❝✐❞❛s✱ ❝❛❧❝✉❧❛♥❞♦ ♦s❝♦❡✜❝✐❡♥t❡s ❞❛ r❡t❛ ✭❝♦❡✜❝✐❡♥t❡ ❧✐♥❡❛r ❡ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r✮✳ P❛r❛ ✐st♦✱ ❞❡✈❡♠♦s ❡s❝♦❧❤❡r ❞♦✐s♣♦♥t♦s q✉❛✐sq✉❡r ❞❛ r❡t❛ ❡ ✉t✐❧✐③á✲❧♦s ♥♦ ❝á❧❝✉❧♦ ❞❡ A′ ❡ ❞❡ B′✳ ❏❛♠❛✐s ❞❡✈❡♠♦s ✉s❛r ♦s ♣♦♥t♦s❡①♣❡r✐♠❡♥t❛✐s✳ ❙❡❥❛♠ ♦s ♣♦♥t♦s P1(x

′

1, y′

1) ❡ P2(x

′

2, y′

2) ❞❛ r❡t❛ tr❛ç❛❞❛✳ ❙❡♥❞♦ ♣♦♥t♦s ❞❛ r❡t❛✱

s❛t✐s❢❛③❡♠ ❛s ❡q✉❛çõ❡s

y′1

= A′x′

1+B′

y′2

= A′x′

2+B′

q✉❡ ♣♦❞❡♠ s❡r r❡s♦❧✈✐❞❛s ♣❛r❛ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ r❡t❛✳ ❙✉❜tr❛✐♥❞♦ ❛s ❡q✉❛çõ❡s ♦❜t❡♠♦s

A′ =y′1− y′

2

x′

1− x′

2

. ✭✹✳✸✳✶✵✮

❙✉❜st✐t✉✐♥❞♦ ❡st❡ ✈❛❧♦r ❡♠ q✉❛❧q✉❡r ❞❛s ❡q✉❛çõ❡s ❛❝✐♠❛✱ ❡♥❝♦♥tr❛♠♦s

B′ = y′2x′

1− y′

1x′

2. ✭✹✳✸✳✶✶✮

❖❜s❡r✈❡ q✉❡ ❡♠ ✈ár✐❛s s✐t✉❛çõ❡s é ♣r❡❝✐s♦ ❝♦♥str✉✐r ✉♠❛ ♥♦✈❛ t❛❜❡❧❛ ❝♦♠ ♦s ✈❛❧♦r❡s ❞❡ y′

❡✴♦✉ ❞❡ x′ ♣❛r❛ ❛ ❝♦♥❢❡❝çã♦ ❞♦ ❣rá✜❝♦ ❧✐♥❡❛r✐③❛❞♦ ❡♠ ♣❛♣❡❧ ♠✐❧✐♠❡tr❛❞♦✳ ❈♦♠♦ ✈✐st♦✱ ❛❧❣✉♠❛s❧✐♥❡❛r✐③❛çõ❡s ❡♥✈♦❧✈❡♠ ❛ ❛♣❧✐❝❛çã♦ ❞❡ ❧♦❣❛r✐t♠♦s ❡✱ ♣❛r❛ ❡st❛s s✐t✉❛çõ❡s✱ ❡①✐st❡♠ ♣❛♣é✐s ❡s♣❡❝✐❛✐sq✉❡ ❢❛❝✐❧✐t❛♠ ❛ ❧✐♥❡❛r✐③❛çã♦✳

✹✳✸✳✻ P❛♣é✐s ❡s♣❡❝✐❛✐s

◗✉❛♥❞♦ ❛ ❧✐♥❡❛r✐③❛çã♦ ❡♥✈♦❧✈❡r ❛ ❛♣❧✐❝❛çã♦ ❞❡ ❧♦❣❛r✐t♠♦s✱ ❛♦ ✐♥✈és ❞❡ ❣❡r❛r♠♦s ✉♠❛ ♥♦✈❛t❛❜❡❧❛ ♣❛r❛ ❛ ❝♦♥❢❡❝çã♦ ❞❡ ✉♠ ❣rá✜❝♦ ❧✐♥❡❛r ❡♠ ♣❛♣❡❧ ♠✐❧✐♠❡tr❛❞♦✱ ♣♦❞❡♠♦s ❢❛③❡r ✉s♦ ❞♦ ♣❛♣❡❧♠♦♥♦✲❧♦❣ ❡ ❞♦ ♣❛♣❡❧ ❞✐✲❧♦❣✱ t❛♠❜é♠ ❝❤❛♠❛❞♦ ❞❡ ♣❛♣❡❧ ❧♦❣✲❧♦❣✳ ❙ã♦ ♣❛♣é✐s ❝✉❥❛s ❡s❝❛❧❛s ❢♦r❛♠tr❛♥s❢♦r♠❛❞❛s ❞❡ ♠♦❞♦ ❛ ❛❝♦♠♦❞❛r ♥❛t✉r❛❧♠❡♥t❡ ♦s ♣♦♥t♦s ♦❜t✐❞♦s ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ♥✉♠❛r❡t❛✱ s❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❛ ❝r✐❛çã♦ ❞❡ ✉♠❛ ♥♦✈❛ t❛❜❡❧❛✳

P❛♣❡❧ ♠♦♥♦✲❧♦❣

❊st❡ ♣❛♣❡❧ ❡stá ❛♣r❡s❡♥t❛❞♦ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✳

✸✽ ❈❆P❮❚❯▲❖ ✹✳ ●❘➪❋■❈❖❙

0 1 2 3 4 5 6 7 8 9 101,0×10

-1

1,0×100

1,0×101

1,0×102

❋✐❣✉r❛ ✹✳✸✳✸✿ P❛♣❡❧ ♠♦♥♦✲❧♦❣✳

❈♦♠♦ ♣♦❞❡ s❡r ♦❜s❡r✈❛❞♦ ♣♦r ❡st❛ ✜❣✉r❛✱ ♦ ♣❛♣❡❧ ♠♦♥♦✲❧♦❣ ❛♣r❡s❡♥t❛ ✉♠❛ ❡s❝❛❧❛ ❧✐♥❡❛r✱♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♠✐❧✐♠❡tr❛❞❛✱ ♥♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s ❡ ✉♠❛ ❡s❝❛❧❛ ❧♦❣❛rít♠✐❝❛ ♥♦ ❡✐①♦ ❞❛s♦r❞❡♥❛❞❛s✳ ❊st❛ ❡s❝❛❧❛ é s❡♣❛r❛❞❛ ❡♠ ❞é❝❛❞❛s✱ ♥♦ s❡♥t✐❞♦ q✉❡ ♦s ✈❛❧♦r❡s ❞❡ ✉♠❛ ❞é❝❛❞❛ ❝♦rr❡✲s♣♦♥❞❡♠ ❛ ❞❡③ ✈❡③❡s ♦ ✈❛❧♦r ❞❛ ❞é❝❛❞❛ ❛♥t❡r✐♦r✳ ❈♦♥s✐❞❡r❡ ♦ ✈❛❧♦r 1, 0× 100 ♦✉ s✐♠♣❧❡s♠❡♥t❡1, 0✳ ❖s ♣ró①✐♠♦s ✈❛❧♦r❡s sã♦ 2, 0✱ 3, 0✱ · · · ✱ 10, 0✱ ♦✉ s❡❥❛✱ 1, 0 × 101✳ ❖s ♣ró①✐♠♦s ✈❛❧♦r❡s sã♦20, 0✱ 30, 0✱ · · · ✱ 100, 0 ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ 1, 0 × 102✳ ❆ s❡❣✉✐r✱ sã♦ ♦s ✈❛❧♦r❡s 200, 0✱ 300, 0 ❡❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♦ ♣r✐♠❡✐r♦ ✈❛❧♦r ♥❛ ❡s❝❛❧❛ ❞❛s ♦r❞❡♥❛❞❛s é 1, 0 × 10−1✱♦✉ s❡❥❛✱ 0, 1 s❡❣✉✐❞♦ ♣♦r 0, 2✱ 0, 3✱ · · · ✱ 1, 0✳ ❉❡✈❡ s❡r ♦❜s❡r✈❛❞♦ q✉❡ ú♥✐❝❛ ♠✉❞❛♥ç❛ ❞❡ ❡s❝❛❧❛q✉❡ s❡ ♣♦❞❡ ❢❛③❡r ♥❡st❛ ❡s❝❛❧❛ ❧♦❣❛rít♠✐❝❛ é ❝♦♠ r❡❧❛çã♦ ❛ ♣♦tê♥❝✐❛ ❞❡ ❞❡③✳ P♦❞❡✲s❡ ❝♦♠❡ç❛r ❛❡s❝❛❧❛ ❞❡ 100 ❡ ✐r ❛té 10.000✱ ♦✉ s❡❥❛✱ ❞❡ 1, 0 × 102 ❛té 1, 0 × 104✱ ❞❡♣❡♥❞❡♥❞♦ ❞♦ ✐♥t❡r✈❛❧♦ ❞❡✈❛❧♦r❡s ❛ss✉♠✐❞♦s ♣❡❧❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡✳ ❚❛♠❜é♠ ♣♦❞❡ ♦❝♦rr❡r ❞❡ ♦ ✐♥t❡r✈❛❧♦ s❡r ❞❡ ✈❛❧♦r❡s♠❡♥♦r❡s ❞♦ q✉❡ ❛ ✉♥✐❞❛❞❡✱ ✐♥❞♦ ❞❡ 1, 0 × 10−5 ❛ 1, 0 × 10−2✱ ♣♦r ❡①❡♠♣❧♦✳ ◆❡♥❤✉♠❛ ♦✉tr❛♠✉❞❛♥ç❛ ❞❡ ❡s❝❛❧❛ é ♣❡r♠✐t✐❞❛✱ ❛ ♥ã♦ s❡r✱ ❛ ❞❡ ♣♦tê♥❝✐❛ ❞❡ ❞❡③✳ ❊♠ ✈✐rt✉❞❡ ❞❛ ♠✉❞❛♥ç❛ ♥❛❡s❝❛❧❛ ❞❡ ✉♠ ❞♦s ❡✐①♦s✱ ♣♦❞❡✲s❡ ❝♦❧♦❝❛r ♦s ♣♦♥t♦s ❡①♣❡r✐♠❡♥t❛✐s ❞✐r❡t❛♠❡♥t❡ ♥♦ ❣rá✜❝♦✱ s❡♠❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ ❝♦♥str✉✐r ✉♠❛ ♥♦✈❛ t❛❜❡❧❛✳ ◆♦ ❡♥t❛♥t♦✱ ♦ ❢❛t♦ ❞❡ ❛ ❧✐♥❡❛r✐③❛çã♦ ❧❡✈❛r ❛♦

✹✳✸✳ ❈❖◆❙❚❘❯➬➹❖ ❉❊ ●❘➪❋■❈❖❙ ✸✾

❛♣❛r❡❝✐♠❡♥t♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❡ ❝♦♥s❡q✉❡♥t❡ ✉s♦ ❞♦ ♣❛♣❡❧ ♠♦♥♦✲❧♦❣ ♥ã♦ é ♦ ú♥✐❝♦✐♥❞í❝✐♦ ❞♦ ✉s♦ ❞❡ t❛❧ ♣❛♣❡❧✳ ♥❛ r❡❛❧✐❞❛❞❡✱ é ♣r❡❝✐s♦ t❛♠❜é♠ ♦❜s❡r✈❛r s❡ ♦s ♣♦♥t♦s ❡①♣❡r✐♠❡♥t❛✐s♦❝✉♣❛♠✱ ♣❡❧♦ ♠❡♥♦s✱ ❞✉❛s ❞é❝❛❞❛s ❞♦ ♣❛♣❡❧✱ ❝❛s♦ ❝♦♥trár✐♦✱ s❡ ♦❜t❡rá ✉♠ ❣rá✜❝♦ ❝♦♠ ♦s ♣♦♥t♦s❡①♣❡r✐♠❡♥t❛✐s ❝♦♥❝❡♥tr❛❞♦s ♥✉♠❛ ♣❡q✉❡♥❛ r❡❣✐ã♦ ❞♦ ♣❛♣❡❧✱ ❞✐✜❝✉❧t❛♥❞♦ ❛ ❧❡✐t✉r❛ ❞❡ ♣♦♥t♦s ❡❝á❧❝✉❧♦s ❞❡ ❝♦♥st❛♥t❡s ❞❡ ✐♥t❡r❡ss❡✳ P♦r ❡①❡♠♣❧♦✱ ♣♦❞❡r✐❛✲s❡ t❡r ✉♠❛ s✐t✉❛çã♦ ❞❡ ❞❡s❝❛r❣❛ ❞❡✉♠ ❝❛♣❛❝✐t♦r✱ ❝✉❥❛ ❞❡♣❡♥❞ê♥❝✐❛ t❡♠♣♦r❛❧ é ❞❛❞❛ ♣♦r ✉♠❛ ❢✉♥çã♦ ❞♦ t✐♣♦ AeBt ❝♦♠ A ❡ B ❝♦♥✲st❛♥t❡s ❛ s❡r❡♠ ❞❡t❡r♠✐♥❛❞❛s✱ t❡♥❞♦ s✐❞♦ ♠❡❞✐❞❛ ❛ t❡♥sã♦ ❡❧étr✐❝❛ ❡♥tr❡ ❛s ♣❧❛❝❛s ❞♦ ❝❛♣❛❝✐t♦r❡♠ ❞✐❢❡r❡♥t❡s ✐♥st❛♥t❡s ❞❡ t❡♠♣♦✳ ❉✐❣❛♠♦s q✉❡ ♦s ✈❛❧♦r❡s t❡♥❤❛♠ s✐❞♦

V (V ) ✺✷✱✾ ✺✶✱✵ ✹✾✱✶ ✸✺✱✶ ✸✶✱✸ ✷✻✱✵ ✽✱✹✸ ✶✱✷✾ ✵✱✷✵ ✵✱✵✸t (s) ✵✱✵✺ ✵✱✶✵ ✵✱✶✺ ✵✱✻✵ ✵✱✼✺ ✶✱✵✵ ✷✱✺✵ ✺✱✵✵ ✼✱✺✵ ✶✵✱✵✵

➱ ❜❡♠ ✈✐sí✈❡❧✱ ♣♦r ❡st❛ t❛❜❡❧❛✱ q✉❡ ♦s ♣♦♥t♦s ❡①♣❡r✐♠❡♥t❛✐s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❞❛ t❡♥sã♦✱♦❝✉♣❛♠ ♠❛✐s q✉❡ ✉♠❛ ❞é❝❛❞❛✱ ❥✉st✐✜❝❛♥❞♦ t❛♠❜é♠ ♦ ✉s♦ ❞♦ ♣❛♣❡❧ ♠♦♥♦✲❧♦❣✳ ❈❛s♦ ♦s ♣♦♥t♦s❡①♣❡r✐♠❡♥t❛✐s ♥ã♦ ❛♣r❡s❡♥t❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❛❧♦r❡s q✉❡ ♦❝✉♣❡♠ ♠❛✐s q✉❡ ✉♠❛ ❞é❝❛❞❛ s✉❣❡r❡✲s❡ ❝♦♥str✉✐r ✉♠❛ ♥♦✈❛ t❛❜❡❧❛ ❛✜♠ ❞❡ tr❛ç❛r ✉♠ ❣rá✜❝♦ ❧✐♥❡❛r ❡♠ ♣❛♣❡❧ ♠✐❧✐♠❡tr❛❞♦ ✉s❛♥❞♦✲s❡♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞✐s❝✉t✐❞♦ ❛♥t❡r✐♦r♠❡♥t❡✳

❖✉tr♦ ❝✉✐❞❛❞♦ q✉❡ ❞❡✈❡ s❡r t♦♠❛❞♦ é ♥♦ ❝á❧❝✉❧♦ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❛♥❣✉❧❛r ❡ ❧✐♥❡❛r ❞❛ r❡t❛✳❖❜s❡r✈❡ q✉❡ ♥❛ ❧✐♥❡❛r✐③❛çã♦ ❞❛❞❛ ♣❡❧❛ ❡q✳✭✹✳✸✳✻✮ ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r r❡t❛✱ ♣❡❧❛ ❡q✳✭✹✳✸✳✶✵✮ ✜❝❛

A′ =ln (y2)− ln (y1)

x2 − x1

, ✭✹✳✸✳✶✷✮

✈✐st♦ q✉❡ ❛ ❡s❝❛❧❛ ❞♦ ♣❛♣❡❧ ❢♦✐ ♠♦❞✐✜❝❛❞❛✳ P♦rt❛♥t♦✱ ♥♦ ♣❛♣❡❧ ♠♦♥♦✲❧♦❣ ❢❛③❡♠♦s ❛ ❧❡✐t✉r❛❞❛s ❝♦♦r❞❡♥❛❞❛s ❞♦s ♣♦♥t♦s ❞✐r❡t❛♠❡♥t❡ ♥♦ ❣rá✜❝♦ ❡ ❛♣❧✐❝❛♠♦s ♦ ❧♦❣❛r✐t♠♦ ❛✜♠ ❞❡ ❝❛❧❝✉❧❛r ♦❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛✳

P❛♣❡❧ ❞✐✲❧♦❣

❊st❡ ♣❛♣❡❧ ❡stá ❛♣r❡s❡♥t❛❞♦ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✳

✹✵ ❈❆P❮❚❯▲❖ ✹✳ ●❘➪❋■❈❖❙

1,0×10-1

1,0×100

1,0×101

1,0×102

1,0×10-1

1,0×100

1,0×101

1,0×102

❋✐❣✉r❛ ✹✳✸✳✹✿ P❛♣❡❧ ❞✐✲❧♦❣✳

❉❡ ♠♦❞♦ s❡♠❡❧❤❛♥t❡ ❛♦ ♣❛♣❡❧ ♠♦♥♦✲❧♦❣✱ ❡st❡ ♣❛♣❡❧ ❛♣r❡s❡♥t❛ ❡s❝❛❧❛ ❧♦❣❛rít♠✐❝❛✱ ♣♦ré♠ ♥♦s❞♦✐s ❡✐①♦s✳ ❙✉❛s ♣r♦♣r✐❡❞❛❞❡s sã♦ ❛♥á❧♦❣❛s àq✉❡❧❛s ❝♦❧♦❝❛❞❛s ❛❝✐♠❛ ♣❛r❛ ♦ ♣❛♣❡❧ ♠♦♥♦✲❧♦❣✱ ♦✉s❡❥❛✱ ♦s ❡✐①♦s ❡stã♦ ❞✐✈✐❞✐❞♦s ❡♠ ❞é❝❛❞❛s ❡ ❛ ú♥✐❝❛ ❛❧t❡r❛çã♦ ♣❡r♠✐t✐❞❛ ♥❛s ❡s❝❛❧❛s é ❝♦♠ r❡❧❛çã♦❛ ♣♦tê♥❝✐❛ ❞❡ ❞❡③✳ ❖ ✉s♦ ❞❡st❡ t✐♣♦ ❞❡ ♣❛♣❡❧ é ✐♥❞✐❝❛❞♦ ♥❛q✉❡❧❡s ❝❛s♦s ❡♠ q✉❡ ❛ r❡❧❛çã♦ ❡♥tr❡❛s ✈❛r✐á✈❡✐s é ❞♦ t✐♣♦ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛✱ ❝♦♠ ❛ ♣♦tê♥❝✐❛ ❞❡s❝♦♥❤❡❝✐❞❛✱ ❝♦♠♦ y(x) = AxB ❡♠ q✉❡A ❡ B sã♦ ❝♦♥st❛♥t❡s ❛ s❡r❡♠ ❞❡t❡r♠✐♥❛❞❛s✳ ▼❛s ❡st❡ ♥ã♦ é ♦ ú♥✐❝♦ ❛r❣✉♠❡♥t♦✱ t❛♠❜é♠ ❞❡✈❡s❡r ✈❡r✐✜❝❛❞♦ s❡ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❛s ✈❛r✐á✈❡✐s ♦❝✉♣❛ ♠❛✐s q✉❡ ✉♠❛ ❞é❝❛❞❛✱ ❛✜♠ ❞❡ s❡ ✉t✐❧✐③❛r✉♠❛ r❡❣✐ã♦ ❝♦♥s✐❞❡rá✈❡❧ ❞♦ ♣❛♣❡❧✳ ❈❛s♦ ❝♦♥trár✐♦✱ é ✐♥❞✐❝❛❞❛ ❛ ❝♦♥❢❡❝çã♦ ❞❡ ♥♦✈❛ t❛❜❡❧❛ ❝♦♠♦s ✈❛❧♦r❡s ❞❡ y′ ❡ x′ ♦❜t✐❞♦s ❝♦♠ ❛ ❧✐♥❡❛r✐③❛çã♦ ❞❛❞❛ ♣♦r y′ = log(y)✱ x′ = log(x)✱ B′ = log(A) ❡A′ = B✳ ❊♥tã♦✱ ❞❡ ♣♦ss❡ ❞❡st❛ ♥♦✈❛ t❛❜❡❧❛✱ ❝♦♥stró✐✲s❡ ✉♠ ❣rá✜❝♦ ❧✐♥❡❛r ❡♠ ♣❛♣❡❧ ♠✐❧✐♠❡tr❛❞♦✳

❉❡✈❡✲s❡ ❛t❡♥t❛r ♣❛r❛ ❛ ♠❛♥❡✐r❛ ❝♦rr❡t❛ ❞❡ s❡ ❝❛❧❝✉❧❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❛♥❣✉❧❛r ❡ ❧✐♥❡❛r r❡t❛✱❡♠❜♦r❛ t❛♥t♦ ❡♠ ♣❛♣❡❧ ❞✐✲❧♦❣ q✉❛♥t♦ ❡♠ ♣❛♣❡❧ ♠✐❧✐♠❡tr❛❞♦ ❡❧❡s s❡❥❛♠ ❞❛❞♦s ♣❡❧❛ ❡q✳✭✹✳✸✳✶✵✮ ❡❡q✳✭✹✳✸✳✶✶✮✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖ ❝✉✐❞❛❞♦ ❛ s❡r t♦♠❛❞♦ é q✉❡ ♥♦ ❝❛s♦ ❞♦ ✉s♦ ❞♦ ♣❛♣❡❧ ♠✐❧✐♠❡tr❛❞♦✱♦s ✈❛❧♦r❡s ❞❡ y′ ❡ ❞❡ x′ sã♦ ❧✐❞♦s ♥✉♠❛ ❡s❝❛❧❛ ❧✐♥❡❛r✱ ❡♥q✉❛♥t♦ q✉❡ ♥♦ ♣❛♣❡❧ ❞✐✲❧♦❣✱ ❡❧❡s sã♦ ❧✐❞♦s♥✉♠❛ ❡s❝❛❧❛ ❧♦❣❛rít♠✐❝❛✳ ◆❡st❡ ❝❛s♦✱ ❞❡✈❡♠♦s t♦♠❛r ♦ ❧♦❣❛r✐t♠♦ ♣❛r❛ ♦❜t❡r

A′ =ln (y2)− ln (y1)

ln (x2)− ln (x1)✭✹✳✸✳✶✸✮

✹✳✸✳ ❈❖◆❙❚❘❯➬➹❖ ❉❊ ●❘➪❋■❈❖❙ ✹✶

➱ ó❜✈✐♦ q✉❡ ♥♦ ❝❛s♦ ❞❡ s❡ ❢❛③❡r ✉♠ ❣rá✜❝♦ ❧✐♥❡❛r ❡♠ ♣❛♣❡❧ ♠✐❧✐♠❡tr❛❞♦✱ ❞❡✈❡✲s❡ ❝❛❧❝✉❧❛r ❛❡s❝❛❧❛ ❛❞❡q✉❛❞❛ ♣❛r❛ ❛❝♦♠♦❞❛r ♦s ♣♦♥t♦s ✭❞❛ ♥♦✈❛ t❛❜❡❧❛✮ ♥♦ ♠❛✐♦r ❡s♣❛ç♦ ♣♦ssí✈❡❧ ❞❛ ❢♦❧❤❛✳

✹✷ ❈❆P❮❚❯▲❖ ✹✳ ●❘➪❋■❈❖❙

❈❛♣ít✉❧♦ ✺

■◆❙❚❘❯▼❊◆❚❖❙ ❉❊ ▼❊❉■❉❆❙

❊❞✐♦ ❈✉♥❤❛ ❞❛ ❈♦st❛

✺✳✶ ■♥str✉♠❡♥t♦s ❆♥❛❧ó❣✐❝♦s ❡ ❉✐❣✐t❛✐s

❱✐♠♦s ❛♥t❡r✐♦r♠❡♥t❡ q✉❡ ♦ ♣r♦❝❡ss♦ ❞❡ ♠❡❞✐❞❛ ❝♦♥s✐st❡ ♥❛ ❝♦♠♣❛r❛çã♦ ❞❛ ❣r❛♥❞❡③❛ ❝♦♠ ♦♣❛❞rã♦ ❡ q✉❡✱ ❡♠ ♠✉✐t♦s ❝❛s♦s✱ ❢❛③✲s❡ ♥❡❝❡ssár✐❛ ❛ ❡st✐♠❛t✐✈❛ ❞❡ ✉♠ ❛❧❣❛r✐s♠♦ s✐❣♥✐✜❝❛t✐✈♦s✱❝❤❛♠❛❞♦ ❞❡ ❛❧❣❛r✐s♠♦ ❞✉✈✐❞♦s♦✳ ◆♦ ❡♥t❛♥t♦✱ ❛❧❣✉♥s ✐♥str✉♠❡♥t♦s ❞❡ ♠❡❞✐❞❛ ❞ã♦ t♦❞♦s ♦s❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s✱ s❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ ❢❛③❡r q✉❛❧q✉❡r ❡st✐♠❛t✐✈❛✳ ■st♦ ♥♦s ♣❡r♠✐t❡❝❧❛ss✐✜❝❛r ♦s ✐♥str✉♠❡♥t♦s ❡♠ ❛♥❛❧ó❣✐❝♦s ❡ ♥ã♦✲❛♥❛❧ó❣✐❝♦s✱ t❛♠❜é♠ ❝❤❛♠❛❞♦s ❞❡ ✐♥str✉♠❡♥t♦s❞✐❣✐t❛✐s✳

P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❛ss✐♠ ❞❡✜♥✐r✿ ✐♥str✉♠❡♥t♦ ❛♥❛❧ó❣✐❝♦ é t♦❞♦ ✐♥str✉♠❡♥t♦ ❡♠ q✉❡ é♥❡❝❡ssár✐♦ ❡st✐♠❛r ♦ ú❧t✐♠♦ ❛❧❣❛r✐s♠♦ s✐❣♥✐✜❝❛t✐✈♦✱ s❡♥❞♦ ❡st❡ ♦ ❛❧❣❛r✐s♠♦ ❞✉✈✐❞♦s♦ ❡ ✐♥str✉✲♠❡♥t♦ ❞✐❣✐t❛❧ é t♦❞♦ ✐♥str✉♠❡♥t♦ ❡♠ q✉❡ ♥ã♦ é ♥❡❝❡ssár✐❛ ❛ ❡st✐♠❛t✐✈❛ ❞❡ q✉❛❧q✉❡r ❛❧❣❛r✐s♠♦s✐❣♥✐✜❝❛t✐✈♦✱ ♦✉ s❡❥❛✱ sã♦ ❛q✉❡❧❡s ✐♥str✉♠❡♥t♦s q✉❡ ❞ã♦ ❛ ❧❡✐t✉r❛ ❞❡ t♦❞♦s ♦s ❛❧❣❛r✐s♠♦s s✐❣♥✐✲✜❝❛t✐✈♦s✳

✺✳✷ ■♥str✉♠❡♥t♦s

▲✐st❛♠♦s✱ ❛ s❡❣✉✐r✱ ✈ár✐♦s ✐♥str✉♠❡♥t♦s ❞❡ ♠❡❞✐❞❛s ✉t✐❧✐③❛❞♦s ❡♠ ❧❛❜♦r❛tór✐♦s✱ ❞❡s❞❡ ♦s ♠❛✐ss✐♠♣❧❡s ❞❡ ❝❛rát❡r ❞✐❞át✐❝♦✱ ❛té ♦s ♠❛✐s s♦✜st✐❝❛❞♦s ♥♦s q✉❛✐s ❛s ♣❡sq✉✐s❛s ❛t✉❛✐s sã♦ r❡❛❧✐③❛❞❛s✳➱ ❣r❛♥❞❡ ❛ ❞✐✈❡rs✐❞❛❞❡ ❞❡ ✐♥str✉♠❡♥t♦s ❞❡ ♠❡❞✐❞❛s✱ ♥♦ ❡♥t❛♥t♦✱ ♦♣t❛♠♦s ♣♦r ❛♣r❡s❡♥t❛r ♦s ♠❛✐s✉t✐❧✐③❛❞♦s ♥✉♠ ❧❛❜♦r❛tór✐♦ ❞✐❞át✐❝♦ ❞❡ ❋ís✐❝❛ ❊①♣❡r✐♠❡♥t❛❧✳

❚r❡♥❛

❆ tr❡♥❛✱ ♦✉ ✜t❛ ♠étr✐❝❛✱ é ✉t✐❧✐③❛❞❛ ♣❛r❛ ♠❡❞✐r ❛s ❣r❛♥❞❡③❛s ❢ís✐❝❛s q✉❡ ♣♦ss✉❛♠ ❞✐♠❡♥sã♦❞❡ ❝♦♠♣r✐♠❡♥t♦✱ ❝♦♠♦ ❞✐stâ♥❝✐❛s✱ ❧❛r❣✉r❛s ❡ ❛❧t✉r❛s✱ ❡♥tr❡ ♦✉tr❛s✳ ❊①❡♠♣❧♦s ❝♦♠✉♥s sã♦ ❛s✜t❛s ❞❡ ❝♦st✉r❛✱ ♦ ♠❡tr♦ ❛rt✐❝✉❧❛❞♦ ♦✉ ✉♠❛ ré❣✉❛ ❡s❝♦❧❛r✳ ◆❛ s✉❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛✱ ❛s ❡s❝❛❧❛s✉t✐❧✐③❛❞❛s sã♦ ♦ ❝❡♥tí♠❡tr♦✱ ♦ ♠✐❧í♠❡tr♦ ❡ ❛ ♣♦❧❡❣❛❞❛✳ ❆ ✜❣✉r❛ ❛❜❛✐①♦ ❛♣r❡s❡♥t❛ ✉♠❛ ré❣✉❛✱✉♠❛ tr❡♥❛ ❡ ✉♠ ♠❡tr♦ ❛rt✐❝✉❧❛❞♦✳

✹✸

❈❛♣ít✉❧♦ ✻

❊❳❊❘❈❮❈■❖❙ P❘❖P❖❙❚❖❙

✶✳ ❊❢❡t✉❡ ❛s s❡❣✉✐♥t❡s ♦♣❡r❛çõ❡s ❡①♣r❡ss❛♥❞♦ ♦ r❡s✉❧t❛❞♦ ❡♠ ♥ú♠❡r♦ ❝♦rr❡t♦ ❞❡ ❛❧❣❛r✐s♠♦ss✐❣♥✐✜❝❛t✐✈♦s✿

✭❛✮ 0, 234cm+ 23, 3mm− 1, 4dm

✭❜✮ 23, 3J × 10, 239s =

✭❝✮0, 234N

1, 4m2=

✭❞✮√

2, 33 + 3, 6

7, 891=

✷✳ ❯♠ ❡st✉❞❛♥t❡ ♦❜s❡r✈❛♥❞♦ ❡ ♠❡❞✐❞♦ ❛ ♣♦s✐çã♦ ❞❡ ✉♠ ❞❡s❧✐③❛❞♦r ♥✉♠ tr✐❧❤♦ ❞❡ ❛r ❢❡③ ❛ss❡❣✉✐♥t❡s ♠❡❞✐❞❛s ♠♦st❛❞❛s ♥❛ t❛❜❡❧❛ ❛❜❛✐①♦✿

●r❛♥❞❡③❛ ❱❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧ ■♥t❡r✈❛❧♦ ♣r♦✈á✈❡❧♣♦s✐çã♦✱ x 53, 5 53, 1 ❛ 53, 5 ✭cm✮

✈❡❧♦❝✐❞❛❞❡✱ v −13, 5 −14, 0 ❛ −13, 0 ✭cm/s✮❛❝❡❧❡r❛çã♦✱ a 93 90 ❛ 96 ✭cm/s2✮

❊①♣r❡ss❡ ❛s ♠❡❞✐❞❛s ♥❛ ❢♦r♠❛ ❝♦rr❡t❛✱ ♦✉ s❡❥❛✱ ♥❛ ❢♦r♠❛ 〈x〉 ± δx✳

✸✳ ❊s❝r❡✈❛ ♦s s❡❣✉✐♥t❡s r❡s✉❧t❛❞♦s ❝♦♠ ♦ ♥ú♠❡r♦ ❝♦rr❡t♦ ❞❡ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s✿

✭❛✮ ♠❡❞✐❞❛ ❞❡ ❛❧t✉r❛❂5, 03± 0, 04329m

✭❜✮ ♠❡❞✐❞❛ ❞❡ t❡♠♣♦❂2, 937± 1 s

✭❝✮ ♠❡❞✐❞❛ ❞❡ ❝❛r❣❛ ❡❧étr✐❝❛❂−3, 21× 10−19 ± 2, 67× 10−20 C

✭❞✮ ♠❡❞✐❞❛ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛❂0, 000.000.563± 0, 000.000.07m

✭❡✮ ♠❡❞✐❞❛ ❞❡ ♠♦♠❡♥t♦ ❧✐♥❡❛r❂5, 911× 103 ± 55 g · cm/s

✹✳ ❯♠ ❝❛rr✐♥❤♦ ❞❡s❧✐③❛ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠ ♣❧❛♥♦ ✐♥❝❧✐♥❛❞♦✱ ♣r❛t✐❝❛♠❡♥t❡ s❡♠ ❛tr✐t♦✳ ❆ ❛❝❡❧✲❡r❛çã♦ ❞❡ ❞❡s❝✐❞❛✱ ♣r❡✈✐st❛ t❡♦r✐❝❛♠❡♥t❡✱ é ❞❛❞❛ ♣♦r gsenθ ❡♠ q✉❡ g é ❛ ❛❝❡❧❡r❛çã♦ ❞❛❣r❛✈✐❞❛❞❡ ❡ θ é ♦ â♥❣✉❧♦ ❞❡ ✐♥❝❧✐♥❛çã♦ ❞♦ ♣❧❛♥♦✳ ❯♠ ❡st✉❞❛♥t❡ r❡❛❧✐③♦✉ ❛s ♠❡❞✐❞❛s ❛♣r❡✲s❡♥t❛❞❛s ♥❛ t❛❜❡❧❛ ❛❜❛✐①♦✿▼❡❞✐❞❛ ❛❝❡❧❡r❛çã♦

a (m/s2)❛❝❡❧❡r❛çã♦ ❡s♣❡r❛❞❛

gsenθ (m/s2)✶ 2, 04± 0, 04 2, 36± 0, 1✷ 3, 58± 0, 06 3, 88± 0, 08✸ 4, 32± 0, 08 4, 57± 0, 05✹ 4, 85± 0, 09 5, 05± 0, 04✺ 5, 53± 0, 1 5, 72± 0, 03

✹✾

✺✵ ❈❆P❮❚❯▲❖ ✻✳ ❊❳❊❘❈❮❈■❖❙ P❘❖P❖❙❚❖❙

❆ ♣❛rt✐r ❞❡st❡s ❞❛❞♦s✱ ❝❛❧❝✉❧❡ ❛ ❞✐❢❡r❡♥ç❛ a−gsenθ ❡ s✉❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ✐♥❝❡rt❡③❛✳ P♦❞❡✲s❡ ❝♦♥❝❧✉✐r q✉❡ ♦ ✈❛❧♦r ❞❛ ❛❝❡❧❡r❛çã♦ é ❞❛❞♦ ♣♦r gsenθ❄ ❏✉st✐✜q✉❡✳

✺✳ ❙❛❜❡✲s❡ q✉❡ ❛ ❡♥❡r❣✐❛ ❝✐♥ét✐❝❛ ❞❡ ✉♠ ❝❛rr✐♥❤♦ ❞❡ ♠❛ss❛ m ❡ ✈❡❧♦❝✐❞❛❞❡ v é ❞❛❞❛ ♣♦rE = mv2/2✳ ❯♠ ❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛s r❡❛❧✐③❛❞❛s ♣♦r ✉♠ ❡①♣❡r✐♠❡♥t❛❞♦s ❢♦r♥❡❝❡✉ ♦s✈❛❧♦r❡sm = (402, 3±0, 1) g ❡ v = (1, 23±0, 12) cm/s✳ ❊s❝r❡✈❛ ❛ ❡q✉❛çã♦ ❞♦ ❡rr♦ ♣r♦♣❛❣❛❞♦♣❛r❛ ❛ ❡♥❡r❣✐❛ ❝✐♥ét✐❝❛ ❞♦ ❝❛rr✐♥❤♦ ❡♠ t❡r♠♦s ❞❛s ❣r❛♥❞❡③❛s ❢ís✐❝❛s ❡♥✈♦❧✈✐❞❛s✱ ♦✉ s❡❥❛✱m ❡ v✳ ❈❛❧❝✉❧❡ ♦ ✈❛❧♦r ❞❛ ❡♥❡r❣✐❛ ❝✐♥ét✐❝❛ ❞♦ ❝❛rr✐♥❤♦ ✉s❛♥❞♦ ♦s ✈❛❧♦r❡s ❛♣r❡s❡♥t❛❞♦s ❡❡①♣r❡ss❡ ♦ r❡s✉❧t❛❞♦ ♥❛ ❢♦r♠❛ ❝♦rr❡t❛✱ ❝❛❧❝✉❧❛♥❞♦ t❛♠❜é♠ ♦ ❡rr♦ ♣r♦♣❛❣❛❞♦✳

✻✳ ◆✉♠ ❡①♣❡r✐♠❡♥t♦ ♣❛r❛ ❛♥❛❧✐s❛r ❛ ✈❛❧✐❞❛❞❡ ❞❛ ▲❡✐ ❞❡ ❍♦♦❦❡✱ ✉♠❛ ♠♦❧❛ ❞❡ ❝♦♥st❛♥t❡❡❧ást✐❝❛ k✱ ❞❡s❝♦♥❤❡❝✐❞❛✱ ❢♦✐ ✉t✐❧✐③❛❞❛✳ ❖ ❡①♣❡r✐♠❡♥t♦ ❝♦♥s✐st✐✉ ❡♠ s❡ ♠❡❞✐r ❛ ❞❡❢♦r♠❛çã♦❞❛ ♠♦❧❛ ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡ ♠❛ss❛s ✉t✐❧✐③❛❞❛s ♥✉♠ s✉♣♦rt❡ ❞❡ ♠❛ss❛s✳ ❆ t❛❜❡❧❛❛❜❛✐①♦ ♠♦str❛ ♦s ✈❛❧♦r❡s ♦❜t✐❞♦s✳ ❙❛❜❡♥❞♦ q✉❡ ❛ s✐t✉❛çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❝♦rr❡s♣♦♥❞❡ ❛mg = kx ♦❜t❡♥❤❛✱ ♣❛r❛ ❝❛❞❛ ♠❛ss❛ ❡ ❝♦rr❡s♣♦♥❞❡♥t❡ ❞❡❢♦r♠❛çã♦ ❞❛ ♠♦❧❛✱ ♦ ✈❛❧♦r ❞❛❝♦♥st❛♥t❡ ❡❧ást✐❝❛✳ ❈❛❧❝✉❧❡ ❛✐♥❞❛ ♦ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧ ❞❡ k✱ ♦ ❞❡s✈✐♦ ♠é❞✐♦ ❡ ♦ ❞❡s✈✐♦♣❛❞rã♦✳x (cm) ✷✱✸✹ ✹✱✺✵ ✼✱✷✶ ✾✱✸✸ ✶✶✱✷✺ ✶✹✱✸✻ ✶✻✱✹✵ ✶✽✱✼✺ ✷✷✱✵✺ ✷✸✱✹m (g) ✶✵✱✵ ✷✵✱✵ ✸✵✱✵ ✹✵✱✵ ✺✵✱✵ ✻✵✱✵ ✼✵✱✵ ✽✵✱✵ ✾✵✱✵ ✶✵✵✱✵

✼✳ ❈♦♥s✐❞❡r❡ ♦s ❞❛❞♦s ❞❛ t❛❜❡❧❛ ❞♦ ❡①❡r❝í❝✐♦ ❛♥t❡r✐♦r✳ ❙❛❜❡♥❞♦ q✉❡ ❛ ❞❡❢♦r♠❛çã♦ ❞❛ ♠♦❧❛é ♠❡❞✐❞❛ ❛♣ós ❛ ❛❧❡t❡r❛çã♦ ❞❛ ♠❛ss❛ ♥♦ s✉♣♦rt❡ ❞❡ ♠❛ss❛s✱ ✐♥❞✐q✉❡✱ ❥✉st✐✜❝❛♥❞♦✱ ❛s✈❛r✐á✈❡✐s ❞❡♣❡♥❞❡♥t❡ ❡ ✐♥❞❡♣❡♥❞❡♥t❡✳ ❈♦♥str✉❛ ✉♠ ❣rá✜❝♦ ❧✐♥❡❛r✱ ❡♠ ♣❛♣❡❧ ❛❞❡q✉❛❞♦✱ ❡❝❛❧❝✉❧❡ ❛ ♣❛rt✐r ❞♦ ❣rá✜❝♦ ♦ ✈❛❧♦r ❞❛ ❝♦♥st❛♥t❡ k ❞❛ ♠♦❧❛✳ ❈♦♠♣❛r❡ ♦ ✈❛❧♦r ♦❜t✐❞♦ ♣❡❧♦❣rá✜❝♦ ❝♦♠ ♦ ✈❛❧♦r ♦❜t✐❞♦ ♣❡❧♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞♦ ❡①❡r❝í❝✐♦ ❛♥t❡r✐♦r✱ ❛tr❛✈és ❞♦ ❝á❧❝✉❧♦ ❞♦❡rr♦ ♣❡r❝❡♥t✉❛❧✳ ❚♦♠❡ ♦ ✈❛❧♦r ♦❜t✐❞♦ ♣❡❧♦ ❣rá✜❝♦ ❝♦♠♦ ✈❛❧♦r ❞❡ r❡❢❡rê♥❝✐❛✳

✽✳ ❈♦♥s✐❞❡r❡ ♦s ❞❛❞♦s ❛♣r❡s❡♥t❛❞♦s ♥❛ t❛❜❡❧❛ ✸✳✶ ❞❛ s❡çã♦ ✸✳✶✳ ❈❛❧❝✉❧❡ ♦ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧❞♦ t❡♠♣♦ ❞❡ q✉❡❞❛ ❞❛ ❡s❢❡r❛✳ ❆ ♣❛rt✐r ❞❡st❡ t❡♠♣♦✱ ❝❛❧❝✉❧❡ ♦ ✈❛❧♦r ❞❛ ❛❝❡❧❡r❛çã♦ ❞❛❣r❛✈✐❞❛❞❡✳ ❆❣♦r❛ ❝❛❧❝✉❧❡ ♦ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧ ❞♦ ✈❛❧♦r ❞❡ g ❝❛❧❝✉❧❛♥❞♦ ♦ ✈❛❧♦r ♣❛r❛ ❝❛❞❛❧❛♥ç❛♠❡♥t♦✳ ❈♦♠♣❛r❡ ♦s ❞♦✐s ✈❛❧♦r❡s ♦❜t✐❞♦s✳ ❊r❛ ❞❡ s❡ ❡s♣❡r❛r ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s❄❏✉st✐✜q✉❡✳

✾✳ ❯♠❛ ♠❛♥❡✐r❛ ❞❡ s❡ ❡st✉❞❛r ♦ ❡❢❡✐t♦ ❞❛ r❡s✐stê♥❝✐❛ ❞♦ ❛r ♥❛ q✉❡❞❛ ❞♦s ❝♦r♣♦s é✱ s✐♠♣❧❡s✲♠❡♥t❡✱ ♠❡❞✐r ♦ t❡♠♣♦ ❞❡ q✉❡❞❛ ❞❡ ♠♦❞♦ s❡♠❡❧❤❛♥t❡ àq✉❡❧❡ ❢❡✐t♦ ♥❛ s❡çã♦ ✸✳✶✳ ◆✉♠❞❡t❡r♠✐♥❛❞♦ ❡①♣❡r✐♠❡♥t♦✱ ✉t✐❧✐③❛♥❞♦ ✉♠ ❡q✉✐♣❛♠❡♥t♦ ❛❞❡q✉❛❞♦✱ ❢♦r❛♠ ♠❡❞✐❞♦s ♦ ✐♥t❡r✲✈❛❧♦ ❞❡ t❡♠♣♦ ❞❡ q✉❡❞❛ ❞❡ ✉♠❛ ♣❡q✉❡♥❛ ❡s❢❡r❛ ♠❡tá❧✐❝❛ ♣r❡s❛ ❛ ✉♠ ❞✐s❝♦ ❞❡ ♣❛♣❡❧ã♦ ❝♦♠❞✐â♠❡tr♦ d✱ ♣❛r❛ ❞✐❢❡r❡♥t❡s ❛❧t✉r❛s ❞❡ ❧❛♥ç❛♠❡♥t♦✳ ❆ t❛❜❡❧❛ ❛❜❛✐①♦ ♠♦str❛ ♦s ✈❛❧♦r❡s♦❜t✐❞♦s✳ ❋❛ç❛ ✉♠ ❣rá✜❝♦ ❡♠ ♣❛♣❡❧ ❞✐❧♦❣ ❝♦♠ ♦s ♣♦♥t♦s ❞❡st❛ t❛❜❡❧❛✳ ❊r❛ ❞❡ ❡s♣❡r❛r ❛❝✉r✈❛ ♦❜t✐❞❛❄ ❏✉st✐✜q✉❡✳ ❙✉♣♦♥❤❛ q✉❡ s❡❥❛ ✈á❧✐❞❛ ❛ r❡❧❛çã♦ h = gt2/2✳ ❈♦♠ ♦s ❞❛❞♦s ❞❛t❛❜❡❧❛ ❡ ❞♦ ❣rá✜❝♦✱ ❞❡t❡r♠✐♥❡ ♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ✈❛❧♦r ❞❡ g✳ ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ h = gtn✳❉❡t❡r♠✐♥❡ ♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ✈❛❧♦r❡s ❞❡ g ❡ n✳

✺✶

h (cm) t (ms)✷✱✵ ✻✽✸✱✵ ✽✹✹✱✵ ✾✹✺✱✵ ✶✵✺✻✱✵ ✶✶✺✼✱✵ ✶✷✹✽✱✵ ✶✸✷✾✱✵ ✶✹✶✶✵✱✵ ✶✹✾✶✺✱✵ ✶✽✷✷✺✱✵ ✷✸✻✸✺✱✵ ✷✽✷✹✺✱✵ ✸✷✸✻✵✱✵ ✸✼✺✼✵✱✵ ✹✵✽✽✺✱✵ ✹✺✶✾✵✱✵ ✹✻✻

✺✷ ❈❆P❮❚❯▲❖ ✻✳ ❊❳❊❘❈❮❈■❖❙ P❘❖P❖❙❚❖❙

❆P✃◆❉■❈❊

❖✉tr♦s ❙✐st❡♠❛s ❞❡ ❯♥✐❞❛❞❡s

❙✐st❡♠❛ ❝❣s

❊①✐st❡♠ ♥❛ ❋ís✐❝❛ ♠✉✐t♦s ♦✉tr♦s s✐st❡♠❛s ❞❡ ✉♥✐❞❛❞❡s✱ ❝♦♠♦ ♦ cgs ❞❡ ❝❡♥tí♠❡tr♦✱ ❣r❛♠❛ ❡s❡❣✉♥❞♦✳ ❆ ♣❛rt✐r ❞❡st❛s três ♣♦❞❡✲s❡ ♦❜t❡r ✉♥✐❞❛❞❡s ❞❡r✐✈❛❞❛s✳ P♦r ❡①❡♠♣❧♦✱ ❛ ✉♥✐❞❛❞❡ ❞❡❢♦rç❛ é grama · cm

s2✱ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ ❢♦rç❛ q✉❡✱ ❛♣❧✐❝❛❞❛ ❛ ♠❛ss❛ ❞❡ ✉♠ ❣r❛♠❛ ❢♦r♥❡❝❡ ❛

❛❝❡❧❡r❛çã♦ ❞❡ ✉♠ ❝❡♥tí♠❡tr♦ ♣♦r s❡❣✉♥❞♦ ❛♦ q✉❛❞r❛❞♦✳ ❈♦♠♣❛r❛♥❞♦ ❛♦ ❙■ ♦❜té♠✲s❡

g · cms2

=10−3kg × 10−2m

s2= 10−5 N

❝❤❛♠❛❞❛ ❞❡ ❞✐♥❛ ❡ r❡♣r❡s❡♥t❛❞❛ ♣♦r dyn✳ ❆ss✐♠ 1dyn = 10−5N✳❖✉tr❛ ✉♥✐❞❛❞❡ ❞❡r✐✈❛❞❛ é ♦ erg ❡ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♥✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛✳ ❙❡♥❞♦ J = N ·m =

105dyn × 102cm✱ ♦✉ s❡❥❛✱ 1J = 107dyn · cm ❞❡ ♦♥❞❡ t✐r❛♠♦s ❛ ✉♥✐❞❛❞❡ erg✳ ❆ss✐♠ 1erg ❝♦rr❡✲s♣♦♥❞❡ ❛♦ tr❛❜❛❧❤♦ r❡❛❧✐③❛❞♦ ♣❡❧❛ ❢♦rç❛ ❞❡ 1dyn ♥♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞❡ 1cm✳

❙✐st❡♠❛ ❞❡ ✉♥✐❞❛❞❡s ♥❛t✉r❛✐s

❯♠ s✐st❡♠❛ ❞❡ ✉♥✐❞❛❞❡s ♥❛t✉r❛✐s ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ❡♠ t♦r♥♦ ❞❡ q✉❛✐sq✉❡r ❝♦♥st❛♥t❡s ❢ís✐❝❛s✉♥✐✈❡rs❛✐s ❞❡ ❢♦r♠❛ q✉❡ ❡ss❛s ❝♦♥st❛♥t❡s t❡♥❤❛♠ ✈❛❧♦r 1✳ ❘❡❧❛❝✐♦♥❛♥❞♦ ❡ss❛s ❝♦♥st❛♥t❡s ❞❡❞✐❢❡r❡♥t❡s ♠♦❞♦s é ♣♦ssí✈❡❧ s❡ ❝❤❡❣❛r ❛s ✉♥✐❞❛❞❡s ❞❡ t❡♠♣♦✱ ❞✐stâ♥❝✐❛✱ t❡♠♣❡r❛t✉r❛✱ ♠❛ss❛❡ ❝❛r❣❛ ❡❧étr✐❝❛✳ ❊ss❡s s✐st❡♠❛s sã♦ ❝❤❛♠❛❞♦s ❞❡ ♥❛t✉r❛✐s ♣♦r ❞❡♣❡♥❞❡r❡♠ s♦♠❡♥t❡ ❞❡ ♣r♦✲♣r✐❡❞❛❞❡s ❞❛ ♥❛t✉r❡③❛✱ ❡ ❝♦st✉♠❛♠ s❡r ✉t✐❧✐③❛❞♦s ❝♦♠♦ ❢♦r♠❛ ❞❡ s✐♠♣❧✐✜❝❛r ❧❡✐s ❢ís✐❝❛s✳

❯♠ ❡①❡♠♣❧♦ ❞❡ s✐st❡♠❛ ❞❡ ✉♥✐❞❛❞❡s ♥❛t✉r❛✐s ♠✉✐t♦ ✉s❛❞♦ sã♦ ❛s ❯♥✐❞❛❞❡s ❞❡ P❧❛♥❝❦✱❞❡✜♥✐❞❛ ❡♠ t❡r♠♦s ❞❛s ❝♦♥st❛♥t❡s ✉♥✐✈❡rs❛✐s c ✭✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③✮✱ ~ ✭❝♦♥st❛♥t❡ ❞❡ P❧❛♥❝❦r❡❞✉③✐❞❛✮✱ G ✭❝♦♥st❛♥t❡ ❞❛ ❣r❛✈✐t❛çã♦ ✉♥✐✈❡rs❛❧✮ ❡ kB ✭❝♦♥st❛♥t❡ ❞❡ ❇♦❧t③♠❛♥♥✮✳ P♦r ❡①❡♠✲

♣❧♦✱ ♥❡ss❡ s✐st❡♠❛ ❛ ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ é ❞❡✜♥✐❞❛ ❝♦♠♦√

~Gc3✱ ✈❛❧♦r q✉❡✱ ♥♦ ❙■✱ ✈❛❧❡

1, 61609735× 10−35m✳❆❧é♠ ❞❛s ✉♥✐❞❛❞❡s ❞❡ P❧❛♥❝❦✱ ❡①✐st❡♠ t❛♠❜é♠ ♦✉tr♦s s✐st❡♠❛s ❞❡ ✉♥✐❞❛❞❡s ♥❛t✉r❛✐s✱ ❝♦♠♦

♦s ❞❡ ❙t♦♥❡②✱ ❙❝❤rö❞✐♥❣❡r ❡ ❍❛rtr❡❡✱ ❡♥tr❡ ♦✉tr♦s✳

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✺✹ ❈❆P❮❚❯▲❖ ✻✳ ❊❳❊❘❈❮❈■❖❙ P❘❖P❖❙❚❖❙

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

❬✶❪ ❈❊❘❱❖✱ ❆♠❛❞♦ ▲✉✐③❀ ❇❊❘❱■❆◆✱ P❡❞r♦ ❆❧❝✐♥♦✳▼❡t♦❞♦❧♦❣✐❛ ❈✐❡♥tí✜❝❛✿ ♣❛r❛✉s♦ ❞♦s ❡st✉❞❛♥t❡s ✉♥✐✈❡rs✐t❛r✐♦s✳ ❙ã♦ P❛✉❧♦✿ ▼❛❝●r❛✇✲❍✐❧❧ ❞♦ ❇r❛s✐❧✱ ✶✾✼✺✳

❬✷❪ ❤tt♣✿✴✴✇✇✇✳❜✐♣♠✳♦r❣✴❡♥✴s✐✴s✐❴❜r♦❝❤✉r❡✴

❬✸❪ ❍✳ ▼❛r✐♦♥✱ ❋✳ P❡r❡✐r❛ ❉♦s ❙❛♥t♦s✱ ▼✳ ❆❜❣r❛❧❧✱ ❙✳ ❩❤❛♥❣✱ ❨✳ ❙♦rt❛✐s✱ ❙✳ ❇✐③❡✱ ■✳▼❛❦s✐♠♦✈✐❝✱ ❉✳ ❈❛❧♦♥✐❝♦✯✱ ❏✳ ●rü♥❡rt✱ ❈✳ ▼❛♥❞❛❝❤❡❸✱ P✳ ▲❡♠♦♥❞❡✱ ●✳ ❙❛♥t❛r❡❧❧✐✱P❤✳ ▲❛✉r❡♥t✱ ❛♥❞ ❆✳ ❈❧❛✐r♦♥✳ ❙❡❛r❝❤ ❢♦r ❱❛r✐❛t✐♦♥s ♦❢ ❋✉♥❞❛♠❡♥t❛❧ ❈♦♥✲st❛♥ts ✉s✐♥❣ ❆t♦♠✐❝ ❋♦✉♥t❛✐♥ ❈❧♦❝❦s✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs ✾✵✱ ✶✺✵✽✵✶✭✷✵✵✸✮✳

❬✹❪ ▼❖◆❚❊■❘❖✱ ▲✉✐③ ❍✳ ❆✳✱ ❙✐st❡♠❛s ❉✐♥â♠✐❝♦s✳ ❊❞✐t♦r❛✿ ▲■❱❘❆❘■❆ ❉❆❋■❙■❈❆✱ ✷✵✵✻✳

❬✺❪ ❚❆❨▲❖❘✱ ❏♦♥❤ ❘✳ ❆♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ ❡rr♦r ❛♥❛❧②s✐s✳ ✷➟ ❊❞✳ ❯♥✐✈❡rs✐t②❙❝✐❡♥❝❡ ❇♦♦❦s✱ ✶✾✾✼✳

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