Introduction to Work
Work is a fundamental concept in physics, representing the energy transferred to or from an object via the application of force along a displacement.
Expressions for Work Done
1. Constant Force
When a constant force $\vec{F}$ is applied to an object over a displacement $\vec{d}$, the work $W$ is given by the dot product: $W = \vec{F} \cdot \vec{d} = Fd \cos \theta$ where $\theta$ is the angle between the force and displacement vectors.
2. Variable Force
If the force varies with position, we calculate the work by integrating infinitesimal work increments $dW = \vec{F} \cdot d\vec{r}$ along the path from position $x_1$ to $x_2$: $W = \int_{x_1}^{x_2} F(x) dx$
The Work-Energy Theorem
The Work-Energy Theorem states that the net work done on an object equals its change in kinetic energy: $W_{net} = \Delta K = K_f - K_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$
Proof
- By Newton's Second Law: $F_{net} = ma = m \frac{dv}{dt}$.
- Work done $W = \int F dx = \int (m \frac{dv}{dt}) dx$.
- Rearranging using the chain rule: $\frac{dx}{dt} = v$.
- $W = \int_{v_i}^{v_f} mv dv$.
- Performing the integration: $W = [\frac{1}{2}mv^2]_{v_i}^{v_f} = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$.
- Thus, $W = \Delta K$. Q.E.D.