A wooden crate is released from rest at the top of a ramp inclined at 37^ to the horizontal. When the ramp is perfectly smooth, the crate reaches the bottom with a speed v . When the ramp is rough, the crate reaches the bottom with a speed of v 3 . What is the coefficient of kinetic friction between the crate and the rough ramp? (Take 37^ = 3 4 )

Step-by-step solution

Let the length of the ramp be L and the angle of inclination be = 37^ . For the smooth ramp, the acceleration of the crate is a₁ = g . Using the equation of motion, v^2 = 2a₁L = 2gL . For the rough ramp, the acceleration of the crate is a₂ = g - _k g . The velocity at the bottom is v 3 , so ( v 3 )^2 = 2a₂L = 2gL( - _k ) . Dividing the second equation by the first, we get: 1 9 = - _k 1 9 = 1 - _k _k = 1 - 1 9 = 8 9 _k = 8 9 Given 37^ = 3 4 , substituting this value: _k = 8 9 3 4 = 2 3 Answer: 2 3