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Консультация № 172488

22.09.2009, 17:37

0.00 руб.

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Образование

Здравствуйте, товарищи!) Помогите пожалуйста вычислить длину одной арки циклоиды x=3(t-sint) y=3(1-cost) (0<t<2П).
Заранее благодарен!

Обсуждение

Неизвестный

22.09.2009, 23:34

общий

это ответ

Здравствуйте, Попов Антон Андреевич.

График циклоиды представлен тут (изображено две арки)

Длина кривой, заданной в параметрическом виде, вычисляется по формуле:

L = [$8747$]^t2_t1 [$8730$][ (dx/dt)² + (dy/dt)² ]*dt

Вычисляем производные:

dx/dt = (3*(t - sint))' = 3*(1 - cost)

dy/dt = (3*(1 - cost))' = 3*sint

Тогда:

(dx/dt)² + (dy/dt)² = [3*(1 - cost)]² + [3*sint]² = 9*(1 - 2*cost + cos²t + sin²t) = 9*(1 - 2*cost + 1) =

= 18*(1 - cost) = 18*2*sin²(t/2) = 36*sin²(t/2)

Тогда:

L = [$8747$]^t2_t1 [$8730$][ (dx/dt)² + (dy/dt)² ]*dt = [$8747$]^2*pi₀ [$8730$][ 36*sin²(t/2) ]*dt =

= [$8747$]^2*pi₀ 6*sin(t/2)*dt = - 6*2*cos(t/2) |^2*pi₀ = - 12*[ cos(pi) - cos(0) ] = - 12*[ - 1 - 1 ] = 24

Итак, длина одной арки циклоиды равна 24 (единиц длины)

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