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Задать вопрос экспертам

Консультация № 198136

04.04.2020, 01:49

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Образование

Здравствуйте! У меня возникли сложности с данной задачей:
Частица движется в плоскости ХY по криволинейному участку траектории с радиусом кривизны R = 2 м. Скорость
частицы изменяется во времени по закону v=4sin(3t)*i+4cos(3t)*j . Чему равно нормальное ускорение частицы в момент времени t = 1 c?

Обсуждение

давно

Алексеев Владимир Николаевич

Мастер-Эксперт
259041
7459

08.04.2020, 10:33

общий

это ответ

Здравствуйте, woefulking!
Дано : Скорость частицы V(t) = 4·sin(3·t)·i^[$8594$] + 4·cos(3·t)·j^[$8594$] , радиус кривизны R = 2 м.
Вычислить нормальное ускорение a_n в момент времени t₁ = 1 c.

Решение : Проекция скорости на ось X : V_x(t) = 4·sin(3·t)
Проекция скорости на ось Y : V_y(t) = 4·cos(3·t)
Модуль линейной скорости :
V(t) = [$8730$]{[V_x(t)]² + [V_y(t)]²} = [$8730$]{[4·sin(3·t)]² + [4·cos(3·t)]²} = 4·[$8730$]{[sin(3·t)]² + [cos(3·t)]²} = 4·[$8730$](1) = 4 м/сек
потому что [sin([$945$])]² + [cos([$945$])]² = 1

Нормальное ускорение вычисляем по формуле : a_n = V² / R = 4² / 2 = 8 м/сек²
Как видно, в этой задаче частица движется по оружности с постоянной линейной скоростью V(t) = 4 м/сек, и её нормальное (центростремительное) ускорение НЕ зависит от времени.
Ответ : нормальное ускорение частицы в момент времени t₁ = 1 c (и в любой др момент) равно 8 м/сек².

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