A warehouse employee is slowly (at a constant velocity) lowering a crate of mass

Question

A warehouse employee is slowly (at a constant velocity) lowering a crate of mass m down a ramp inclined at an angle ? = 30 degrees relative to the horizontal. The coefficient of kinetic friction is ?k. To do this they are exerting a horizontal force on the crate using a rigid pole.
For the following, choose axes so that the x-axis lies along the ramp and the y-axis is perpendicular to the ramp.

If the crate has a mass of 60 kg and the employee is able to wax the ramp to a value of the coefficient of kinetic friction of their choosing, at what value of the coefficient of kinetic friction does the direction of the force that they exert, change from a pull to a push?

Explanation / Answer

if we assume the x-axis lies along the ramp and the y-axis is perpendicular to the ramp

(Muk)(mgcos30+Fsin30)=Fcos30+mgsin30

Muk=[1.732F+588]/[1018.44+F]

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