Work

In physics, work is done on an object when a force causes a displacement. Work is defined as:

$$W = F \cdot d \cdot \cos\theta$$

where $F$ is the applied force (N), $d$ is the displacement (m), and $\theta$ is the angle between the force vector and the direction of displacement. The SI unit of work is the joule (J), where $1\ \text{J} = 1\ \text{N} \cdot \text{m}$.

Example: A person pulls a suitcase with a force of 50 N at an angle of 30° to the horizontal, moving it 10 m along the floor.<br>$$W = 50 \times 10 \times \cos 30° = 500 \times 0.866 \approx 433\ \text{J}$$

If the force is parallel to the displacement ($\theta = 0°$), then $\cos 0° = 1$ and $W = Fd$. If force is perpendicular to displacement ($\theta = 90°$), no work is done — this is why carrying a bag horizontally does no work against gravity.

Kinetic Energy and Gravitational Potential Energy

Kinetic energy (KE) is the energy an object possesses due to its motion:<br>$$KE = \frac{1}{2}mv^2$$<br>where $m$ is mass (kg) and $v$ is speed (m/s). A 2 kg ball moving at $3\ \text{m/s}$ has $KE = \frac{1}{2} \times 2 \times 3^2 = 9\ \text{J}$.

Gravitational potential energy (GPE) is stored energy due to an object's height above a reference point:<br>$$GPE = mgh$$<br>where $g = 9.81\ \text{m/s}^2$ and $h$ is the height (m). A 3 kg book resting on a shelf 2 m above the floor has $GPE = 3 \times 9.81 \times 2 \approx 58.9\ \text{J}$.

Power

Power is the rate at which work is done, or energy is transferred:

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

where $P$ is power, $W$ is work (J), and $t$ is time (s). The SI unit of power is the watt (W), where $1\ \text{W} = 1\ \text{J/s}$.

Example: An electric motor does 6,000 J of work in 30 s. Its power is $P = 6000 / 30 = 200\ \text{W}$.

An equivalent expression for power when force and velocity are known:<br>$$P = Fv$$<br>This follows because $W = Fd$ and $v = d/t$, so $P = W/t = Fd/t = Fv$.