Work, Energy and Power

1. Work

Work is done when a force applied to a body produces a displacement. If a constant force $\vec{F}$ moves a body through a displacement $\vec{S}$, and $\theta$ is the angle between them, then work is the scalar (dot) product:

$$ \begin{aligned} W &= \vec{F}\cdot\vec{S} \ &= FS\cos\theta \end{aligned} $$

Work is a scalar quantity. Its SI unit is the joule (J) and its CGS unit is the erg ($1\ \text{J} = 10^7\ \text{erg}$).

Special cases:

• Force along displacement ($\theta = 0^{\circ}$): $W = FS\cos 0^{\circ} = FS$, the maximum work done.
• Force perpendicular to displacement ($\theta = 90^{\circ}$): $W = FS\cos 90^{\circ} = 0$, no work done.
• Force opposing displacement ($\theta = 180^{\circ}$): $W = -FS$, negative work.

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2. Energy

Energy is the capacity of a body to do work. Like work, it is a scalar measured in joules (erg in CGS). Mechanical energy is of two kinds: kinetic and potential.

Kinetic energy

Kinetic energy is the energy possessed by a body due to its motion. Consider a body of mass $m$ initially at rest, acted on by a constant force $F$ over a distance $s$, reaching final velocity $v$. The work done is $W = Fs = mas$. From the equation of motion $v^2 = u^2 + 2as$ with $u = 0$:

$$as = \frac{v^2}{2}$$

Therefore:

$$ \begin{aligned} W &= m\left(\frac{v^2}{2}\right) \ &= \frac{1}{2}mv^2 \end{aligned} $$

This work is stored in the body as kinetic energy:

$$\text{K.E.} = \frac{1}{2}mv^2$$

Potential energy

Potential energy is the energy possessed by a body due to its position or configuration. The gravitational potential energy of a body of mass $m$ at height $h$ above the ground is:

$$\text{P.E.} = mgh$$

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