Work and Energy

Overview

Energy is a fundamental concept in physics that describes the capacity to do work. The work-energy theorem connects force and motion through the concept of energy.

Work

Definition

Work is done when a force causes displacement.

$W = \vec{F} \cdot \vec{d} = Fd\cos\theta$

Where $\theta$ is the angle between $\vec{F}$ and $\vec{d}$

Units

• SI unit: Joule (J) = N·m = kg·m²/s²

Special Cases

Condition	Work
$\theta = 0°$ ($F$ parallel to $d$)	$W = Fd$
$\theta = 90°$ ($F$ perpendicular to $d$)	$W = 0$
$\theta = 180°$ ($F$ opposite to $d$)	$W = -Fd$

Work Done by Variable Force

$W = \int \vec{F} \cdot d\vec{r}$

For 1D:

$W = \int_{x_1}^{x_2} F(x) \, dx$

Kinetic Energy

Energy associated with motion:

$KE = \frac{1}{2}mv^2$

• Always positive or zero
• Scalar quantity
• Depends on speed squared

Work-Energy Theorem

The net work done on an object equals its change in kinetic energy:

$W_{\text{net}} = \Delta KE = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2$

Potential Energy

Energy associated with position or configuration.

Gravitational Potential Energy

$PE_g = mgh$

(relative to a chosen reference level)

Elastic Potential Energy (Spring)

$PE_s = \frac{1}{2}kx^2$

Where $k$ is spring constant and $x$ is displacement from equilibrium

Conservation of Energy

Conservative Forces

• Work is path-independent
• Work around a closed loop is zero
• Associated with potential energy
• Examples: gravity, spring force, electric force

Non-Conservative Forces

• Work is path-dependent
• Dissipate mechanical energy
• Examples: friction, air resistance

Conservation of Mechanical Energy

When only conservative forces do work:

$E_{\text{total}} = KE + PE = \text{constant}$

$\frac{1}{2}mv_1^2 + mgh_1 = \frac{1}{2}mv_2^2 + mgh_2$

With Non-Conservative Forces

$KE_1 + PE_1 + W_{nc} = KE_2 + PE_2$

Where $W_{nc}$ is work done by non-conservative forces

Work Done by Common Forces

Gravity

$W_g = -mg\Delta h = mg(h_1 - h_2)$

Positive when moving down, negative when moving up

Spring Force

$W_s = \frac{1}{2}kx_1^2 - \frac{1}{2}kx_2^2$

Friction

$W_f = -f_k d = -\mu_k Nd$

Always negative (removes mechanical energy)

Examples

Example 1: Falling Object

A 2 kg ball falls from 10 m. Find velocity just before hitting ground.

Using energy conservation:

$mgh = \frac{1}{2}mv^2$

$v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 10} = 14 \text{ m/s}$

Example 2: Roller Coaster

A 500 kg car starts from rest at $h = 30$ m. Find speed at $h = 10$ m.

$mgh_1 = \frac{1}{2}mv^2 + mgh_2$

$gh_1 = \frac{1}{2}v^2 + gh_2$

$v = \sqrt{2g(h_1 - h_2)} = \sqrt{2 \times 9.8 \times 20} = 19.8 \text{ m/s}$

Example 3: Spring Launch

A spring ($k = 200$ N/m) compressed 0.3 m launches a 0.5 kg ball vertically. Find max height.

$\frac{1}{2}kx^2 = mgh$

$h = \frac{kx^2}{2mg} = \frac{200 \times 0.09}{2 \times 0.5 \times 9.8} = 1.84 \text{ m}$

Example 4: Friction on Incline

A 5 kg block slides down a 3 m, 30° incline ($\mu_k = 0.2$). Find final speed.

Height: $h = 3\sin(30°) = 1.5$ m

Normal force: $N = mg\cos(30°)$

Work by friction: $W_f = -\mu_k N d = -0.2 \times 5 \times 9.8 \times 0.866 \times 3 = -25.5$ J

$mgh + W_f = \frac{1}{2}mv^2$

$5 \times 9.8 \times 1.5 - 25.5 = \frac{1}{2} \times 5 \times v^2$

$v = \sqrt{\frac{2 \times 47.5}{5}} = 4.36 \text{ m/s}$

Power

Rate at which work is done or energy is transferred:

$P = \frac{W}{t} = \frac{dW}{dt}$

$P = \vec{F} \cdot \vec{v} = Fv\cos\theta$

Units

• SI unit: Watt (W) = J/s
• 1 horsepower (hp) = 746 W