A 2.0-kg piece of wood slides on a curved surface (Fig. P7.43). The sides of the surface are perfectly smooth, but the rough horizontal bottom is 30 m long and has a kinetic friction coefficient of 0.20 with the wood. The piece of wood starts from rest 4.0 m above the rough bottom. (a) Where will this wood eventually come to rest? (b) For the motion from the initial release until the piece of wood comes to rest, what is the total amount of work done by friction?
Solution 47P Introduction We will first calculate the kinetic energy of the wooden block at the start of rough surface. Since the friction opposes the motion, the block will come to stop when the work done will be equal to the kinetic energy of the block. From this we can calculate both the distance when the block will stop and the work done by friction. Step 1 The initial potential energy of the wood is The frictional force of the rough surface will be Now since the wood is sliding down in smooth surface, it will not lost any energy, hence the kinetic energy of the wood, when it will touch the rough surface, will be equal to the initial potential energy, that is Suppose the block will come to stop after distance x. hence the work done by the friction in this distance should be equal to the kinetic energy of the wood when it came contact with the rough surface. Now the work done by friction on the wood is Equating with the initial kinetic energy we have Hence the wood will come to rest at 20 m from the starting of the rough surface.