Work-Energy Theorem

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

Key Takeaways

Key Points

  • The work W done by the net force on a particle equals the change in the particle’s kinetic energy KE: [latex]\text{W}=\Delta \text{KE}=\frac{1}{2} \text{mv}_\text{f}^2-\frac{1}{2} \text{mv}_\text{i}^2[/latex].
  • 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)

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.

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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 KE:

[latex]\text{W}=\Delta \text{KE}=\frac{1}{2} \text{mv}_\text{f}^2-\frac{1}{2} \text{mv}_\text{i}^2[/latex]

where vi and vf are the speeds of the particle before and after the application of force, and 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:

[latex]\text{v}_\text{f}^2 = \text{v}_\text{i}^2 + 2\text{ad}[/latex]

obtaining,

[latex]\text{d}=\frac{\text{v}_\text{f}^2-\text{v}_\text{i}^2}{2\text{a}}[/latex]

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:

[latex]\text{W}=\text{Fd}=\text{ma}\frac{\text{v}_\text{f}^2-\text{v}_\text{i}^2}{2\text{a}}=\frac{1}{2} \text{mv}_\text{f}^2-\frac{1}{2} \text{mv}_\text{i}^2=\text{KE}_\text{f}-\text{KE}_\text{i}=\Delta \text{KE}[/latex]