Kinetics of Particles: Work and Energy

Kinetics of Particles: Work and Energy

The method of work and energy is a powerful tool for solving problems involving force, displacement, and velocity, without explicitly determining acceleration.

Work of a Force

The work done by a force $\mathbf{F}$ moving a particle through a displacement $d\mathbf{r}$ is: $U = \int \mathbf{F} \cdot d\mathbf{r}$

Energy

• Kinetic Energy ($T$): Energy due to motion. $T = \frac{1}{2} m v^2$
• Potential Energy ($V$): Energy due to position.
  • Gravitational: $V_g = mgh$
  • Elastic: $V_e = \frac{1}{2} k x^2$

Principle of Work and Energy

$T_1 + \sum U_{1-2} = T_2$

The initial kinetic energy plus the total work done by all forces equals the final kinetic energy.

Alternatively, using Conservation of Energy (for conservative forces): $T_1 + V_1 = T_2 + V_2$

Example: Spring-Loaded Launcher