# 8.12: Work-Energy Theorem

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## Kinetic Energy and Work-Energy Theorem

The work-energy theorem states that the work done by all forces acting on a particle equals the change in the particle’s kinetic energy.

learning objectives

- Outline the derivation of the work-energy theorem

### The Work-Energy Theorem

The principle of work and kinetic energy (also known as the work-energy theorem) states that the work done by the sum of all forces acting on a particle equals the change in the kinetic energy of the particle. This definition can be extended to rigid bodies by defining the work of the torque and rotational kinetic energy.

**Kinetic Energy**: A force does work on the block. The kinetic energy of the block increases as a result by the amount of work. This relationship is generalized in the work-energy theorem.

The work *W* done by the net force on a particle equals the change in the particle’s kinetic energy K*E*:

\[\mathrm{W=ΔKE=\dfrac{1}{2}mv_f^2−\dfrac{1}{2}mv_i^2}\]

where *v _{i}* and

*v*are the speeds of the particle before and after the application of force, and

_{f}*m*is the particle’s mass.

### Derivation

For the sake of simplicity, we will consider the case in which the resultant force *F* is constant in both magnitude and direction and is parallel to the velocity of the particle. The particle is moving with constant acceleration *a *along a straight line. The relationship between the net force and the acceleration is given by the equation *F* = *ma* (Newton’s second law), and the particle’s displacement *d,* can be determined from the equation:

\[\mathrm{v_f^2=v_i^2+2ad}\]

obtaining,

\[\mathrm{d=\dfrac{v^2_f−v^2_i}{2a}}\]

The work of the net force is calculated as the product of its magnitude (F=ma) and the particle’s displacement. Substituting the above equations yields:

\[\mathrm{W=Fd=ma\dfrac{v^2_f−v^2_i}{2a}=\dfrac{1}{2}mv^2_f−\dfrac{1}{2}mv^2_i=KE_f−KE_i=ΔKE}\]

## Key Points

- The work
*W*done by the net force on a particle equals the change in the particle’s kinetic energy K*E*: \(\mathrm{W=ΔKE=\frac{1}{2}mv_f^2−\frac{1}{2}mv_i^2}\). - The work-energy theorem can be derived from Newton’s second law.
- Work transfers energy from one place to another or one form to another. In more general systems than the particle system mentioned here, work can change the potential energy of a mechanical device, the heat energy in a thermal system, or the electrical energy in an electrical device.

## Key Terms

**torque**: A rotational or twisting effect of a force; (SI unit newton-meter or Nm; imperial unit foot-pound or ft-lb)

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- Work energy theorem.
**Provided by**: Wikipedia.**Located at**:.**en.Wikipedia.org/wiki/Work_energy_theorem%23Work-energy_principle****License**:**CC BY-SA: Attribution-ShareAlike** - torque.
**Provided by**: Wiktionary.**Located at**:.**en.wiktionary.org/wiki/torque****License**:**CC BY-SA: Attribution-ShareAlike** - Sunil Kumar Singh, Work - Kinetic Energy Theorem. February 2, 2013.
**Provided by**: OpenStax CNX.**Located at**:.**http://cnx.org/content/m14095/latest/****License**:**CC BY: Attribution**