A crate of mass m = 91 kg rests on a rough surface inclined at an angle of θ = 31° with the horizontal. A massless rope to which a force can be applied parallel to the surface is attached to the crate and leads to the top of the incline. In its present state, there is no tension in the rope and the crate is just ready to slip and start to move down the plane. The coefficient of kinetic friction is given by μk=fμs where μs refers to the coefficient of static friction and f = 0.75

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A crate of mass m = 91 kg rests on a rough surface inclined at an angle of θ = 31° with the horizontal. A massless rope to which a force can be applied parallel to the surface is attached to the crate and leads to the top of the incline. In its present state, there is no tension in the rope and the crate is just ready to slip and start to move down the plane. The coefficient of kinetic friction is given by μk=fμs where μs refers to the coefficient of static friction and f = 0.75

a) With a slightly greater applied force than what is found to be the maximum force the rope can exert parallel to the surface of the plane before the crate begins to move, the crate will slide up the plane. Write an equation for the acceleration of the crate when it begins to slide upwards. Give your equation in terms of variables from the problem statement together with g for acceleration due to gravity and μs for the coefficient of static friction. 

b) After the crate begins to move, we reduce the force on the rope. Write an equation for the force exerted upwards by the rope at which the moving crate stops accelerating and moves at a constant speed. Give your equation in terms of variables from the problem statement together with g for acceleration due to gravity and μs for the coefficient of static friction. 

c) Consider a case where no force is being exerted by the rope and the crate is at rest. Write an equation for the acceleration of the crate down the plane if it is given a slight nudge to get it started. Give your answer in terms of variables from the problem statement together with g for acceleration due to gravity and μs for the coefficient of static friction. 

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