Problem 14.

A body of mass 5 kg slides a distance of 6 m down a rough Inclined plane 30 degree. Then it moves on frictionless horizontal surface and compresses a spring. The coefficient of kinetic friction is 0.1 and the spring constant is 300 N/m. Find the maximum compression of the spring.

Solution:

In this problem we need to use the work-energy theorem, which determines the relation between the work done by the friction force and the change of the net mechanical energy.

According the work-energy theorem the work done by the friction force (which is always negative) is equal to the difference between the final mechanical energy and the initial mechanical energy.

................................................. (1)

The mechanical energy is the sum of kinetic energy, gravitational potential energy, and elastic energy of the spring.

The initial velocity of the body is zero, the initial compression of the spring is zero, the initial height of the body is . Then the initial mechanical energy of the body is

In the final state we have a maximum compression of the spring. It means that in the final state the velocity of the body is zero. The final height of the body is 0 (ground level). Then the final mechanical energy of the body is

Then equation (1) takes the form

Now we need to find the work done by the friction force. The work done by the friction force (see figure) is

where is the displacement of the body over incline plane. The friction force is determined by the normal force and the coefficient of kinetic friction:

To find the normal force we need to write down the second Newton's law for the motion along the incline. There are three forces acting on the body: gravitational force, normal force, and the friction force. Then the second Newton's law takes the form:

...........................................................(2)

The direction of the acceleration is along the incline. It means that the acceleration has only x component (see figure). The y-component of acceleration is zero. Then to find the normal force we need to write down only the y-component of equation (2):

From this equation we can find the normal force:

Now we can find the friction force

and the work done by the friction force

Then we can find the maximum (final) compression of the spring: