A factory worker accidentally releases a crate of mass m tha

A factory worker accidentally releases a crate of mass m that was being held at rest at the top of a ramp with length d which is inclined at an angle to the horizontal. The coefficient of kinetic friction between the crate and the ramp, and between the crate and the horizontal factory floor, is _k. (a) How fast is the crate moving as it reaches the bottom of the ramp? (b) How far will it subsequently slide across the floor? (Assume that the crate\'s kinetic energy does not change as it moves from the ramp onto the floor.) State your answers in terms of the given variables (use g where applicable).

Solution

Kinetic energy at bottom k = U_top + W_friction

U_top = m g d sin theta

W_friction = -f_k d

f_k = mu_k N = mu_k m g d cos theta

k = m g d (sin theta - mu_k cos theta) = 1/2 m v²

v = sqrt [2 g d (sin theta - mu_k cos theta)]

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N = m g

1/2 m v² = mu_k m g d\'

d\' = v² / 2 mu_k g