Work and Kinetic Energy

Work Comparison
The work done on three blocks of different mass are compared when the same size force acts over the same distance. Displayed is the transformation of work into kinetic energy of the blocks. Work Comparsion QT Movie


Work-Energy Theorem:


Derivation of Work-Energy Theorem using a Constant Net Force

Frame of Reference:
Direction of the Net Force.

Start with Newton 2nd Law for one-dimensional motion:

Next use the Equations for Constant Acceleration that does not involve time:

Calculate the Net Work using the above relationships:

Many problems you encounter related to Work and Energy will have constant forces. While you are trying to learn how to use the concepts of Work and Energy, avoid using Newton's Second Law to solve these problems or else you will have missed the opportunity to learn how to use Work and Energy to solve them. It is reasonable to use the Second Law approach as a way to double-check your Work-Energy solutions.


Derivation of Work-Energy Theorem using a Variable Net Force

Frame of Reference:
Direction of the Net Force.

To reduce the complexity of the derivation, we will assume that the direction of the Net Force is constant while the work is being done. The Work-Energy Theorem is still valid if the net force changes direction as well as magnitude while the work is being done, provided the body is rigid.

The key step is to convert the calculus definition for acceleration into an expression that is a derivative of x.

Plug this and the Second Law into the definition for Work, and integrate.

Here we have used the relation with x = v to do the integration. Also observe that we have assumed that the body's mass does not change while the force is being applied so that we can remove it from under the integral.


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