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## Worked example 5.5: Sliding down a plane

Question: A block of mass starts at rest at a height of on a plane that has an angle of inclination of with respect to the horizontal. The block slides down the plane, and, upon reaching the bottom, then slides along a horizontal surface. The coefficient of kinetic friction of the block on both surfaces is . How far does the block slide along the horizontal surface before coming to rest?

Answer: The normal reaction of the plane to the block's weight is

Hence, the frictional force acting on the block when it is sliding down the plane is

The change in gravitational potential energy of the block as it slides down the plane is

The work done on the block by the frictional force during this process is

where is the distance the block slides. The minus sign indicates that acts in the opposite direction to the displacement of the block. Hence,

Now, by energy conservation, the kinetic energy of the block at the bottom of the plane equals the decrease in the block's potential energy plus the amount of work done on the block:

The frictional force acting on the block when it slides over the horizontal surface is

The work done on the block as it slides a distance over this surface is

By energy conservation, the block comes to rest when the action of the frictional force has drained all of the kinetic energy from the block: i.e., when . It follows that

Next: Worked example 5.6: Driving Up: Conservation of energy Previous: Worked example 5.4: Roller
Richard Fitzpatrick 2006-02-02