Dynamics Example 4 © 2000 D.P. Maloney

When a force of 784 N pushes a 40 kg block 50 cm up a 30° incline, what is the change in the potential energy and the kinetic energy of the block? (The 50 cm is the distance moved along the incline).

Physical Principles and/or Ideas: Definition of work, conversion of work to kinetic and

potential energy, geometry of movement on an incline, definition (relation) for kinetic and gravitational potential energy.

Solution: In this problem, the force is going to do work on the block and that work is going to be converted to kinetic energy and gravitational potential energy. We have to determine how much the kinetic and potential energies of the block change as a result of the force acting on it. We will do this problem by computing the change in gravitational potential energy due to the block moving up a set distance, then we will determine the force which is accelerating the block, calculate the work that force does, and set it equal to the change in kinetic energy.

There are actually several ways to work this problem. We will follow the plan outlined above first and then do the second part again a different way.

The block is going to move 0.5 m up along the incline, but its change in height (height being the straight line vertical distance above the ground) will be different. The reason for this is shown in the figure below.

Now we use the basic relation for gravitational potential energy to calculate the change. We get:

The force that is pushing the block up the incline actually does two “things”. It moves the block a certain distance up the incline (this produces the change in gravitational potential energy) and it accelerates the block (this produces the change in kinetic energy). Actually, it is the net force that accelerates the block, so we need to determine the net force. The force pushing the block is parallel to the incline surface. The only other force parallel to the incline is the component of the weight. Taking the applied force as positive, the component of the weight will be negative. Summing these, we have:

It is the work done by the net force that produces that change in kinetic energy. In calculating this work, we need to use the distance moved along the incline surface since that is the distance moved in the direction of the force. So the work done by the net force is:

A second way we could have found the change in kinetic energy would be to calculate the total work done by the 784 N force and then subtract the change in potential energy; the difference is the change in kinetic energy. Let’s check this out.

We do get the same answer we found above. The reason this works is that one part of the applied force simply moves the block up without changing its speed, and the other part accelerates the block.

How could we use the change in kinetic energy to find the final speed of the block?