Mechanics Problem - Work, Energy and Power?
A roofer is replacing the slates on a roof inclined at 40° to the horizontal. An old broken slate of mass 0.4 kg, which may be treated as a particle, is placed on the roof and slides from rest. It slides 5 m down the roof and then falls a further vertical distance of 6 m to the ground, as shown in the diagram. While the slate is sliding down the roof the resistance to its motion is a constant 2 N; when the slate reached the edge of the roof it falls freely and air resistance may be neglected.
(i) Calculate the gravitational potential energy lost by the slate in moving from the
point of release to the ground. Calculate also the speed with which the slate hits the ground. 
A crate of new slates of total mass 12 kg is pulled up from the ground by the roofer using a light rope. This crate is lifted 6 m vertically from the ground and then slides 5 m up the sloping roof before coming to rest. When the crate is being raised vertically there is negligible resistance to motion. When the crate is sliding up the roof the coefficient of friction between the crate and the roof is 0.6 and the rope is parallel to the roof.
(ii) It takes the roofer 25 seconds to pull up the crate of slates. Calculate the average power he must develop to achieve this. 
(iii) Show that, if the crate is not secured when the rope is removed, it will slide back down the roof. What would be the least value of the coefficient of friction between the crate and the roof for this not to happen? 
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