Work energy theorem

LEARNING OBJECTIVE 3.2.A - Describe the work done on an object or system by a given force or collection of forces.
AP MECHANICS -C LEARNING OBJECTIVE INT 4.C -Calculate the net work done on an object that
undergoes a specified change in speed or change in kinetic energy.
Calculate changes in an object’s kinetic energy or changes in speed that result from the application of specified forces.

In the last section, we looked at how to calculate the kinetic energy of an object. Let’s now look at a slightly complex case. Let’s say an object is moving with a velocity $V_i$ and then we do some work W on it and as a result, it is now moving with a velocity $V_f$.

Applying Newton’s law to the object we can write:

$F=ma$

Whereas Work will be:

$W = Fs = m \times ( as )$

Using the basic kinematical equation for motion in a straight line we know that $v^2= u^2 + 2as$, given the ball was initially at rest u = $V_i$,

$v_f^2 = v_i^2 + 2 as \mbox{ hence, } as = \frac{v_f^2 – v_i^2}{2}$

substituting it in the last equation will give:

$W = ( \frac{mv_f^2}{2} ) – ( \frac{mv_i^2}{2} )$

$mv_f^2/2$ is the final KE and $mv_i^2/2$ is the initial KE.

Hence the change $\Delta KE$ can be written as:

$W = \Delta KE$

The equation $W = \Delta KE$ represents the equivalence of Work and Kinetic energy and is called the Work-Energy theorem or simply Kinetic energy law.

Work-Energy Theorem

The meaning of Work- Energy theorem

• Work done on an object leads to a change in the kinetic energy of the object on which the work in done.
• Positive work increases the kinetic energy.
• Negative work decreases the kinetic energy.
• No change in kinetic energy means no work.
• If several external forces act on an object, they must be added together vectorially to give the net force. The work done by the net force can then be related to the change in the object’s kinetic energy by using the work–energy theorem. The work-energy theorem does not apply to the work done by an individual force.
• Alternatively, you can calculate the work done by individual forces and add them arithmetically to arrive at the total work done – This is also called the superposition principle of work. Then use the work-energy theorem. The work-energy theorem does not apply to the work done by an individual force but to the total work done by all forces.