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2016-04-04 3:31 pm
A disk of mass m and radius R is released from rest at the top of an inclined plane and rolls without slipping on the plane. If the angle of inclination is theta and the height is h, find:

(a) The velocity of the disk when it reaches the lowest point of the inclined plane.
(b) The angular acceleration a and the linear acceleration a along the inclined plane.
(c) The time it takes the disk to roll down.

回答 (1)

2016-04-04 4:34 pm
✔ 最佳答案
A disk of mass m and radius R is released from rest at the top of an inclined plane and rolls without slipping on the plane. If the angle of inclination is theta and the height is h, find:

(a) The velocity of the disk when it reaches the lowest point of the inclined plane.
potential gravitational energy Epg = m*g*h
kinetic energy Ek = 1/2mv^2 (translating)+1/4mV^2 (rolling) = 3/4*m*V^2
energies equate...and m cross
V = √4*g*h/3 = 2√g*h/3

(b) The angular acceleration a and the linear acceleration a along the inclined plane.
ℓ = h/sin Θ
time t = 2ℓ/V = (h/sin Θ)/(2√g*h/3)
ω = V/R = (2√g*h/3)/R
dω/dt = ((2√g*h/3)/R)/(h/sin Θ)/(2√g*h/3) = 4*g*h*sin Θ/(3h)
a = dω/dt*R = 4*g*h*sin Θ*R/(3h)

(c) The time it takes the disk to roll down.
time t = 2ℓ/V = (h/sin Θ)/(2√g*h/3)


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