5. Work, energy and power

Written by: Adhulan Rajkamal
Formatted by: Adhulan Rajkamal

Index

Work is done when a force moves the point at which it acts in the direction of the force: $$ \text{Work Done} = \text{Force} \times \text{Displacement in the direction of the force} $$ $$ W = Fs $$
An object that can do work must have energy.
Principle of Conservation of Energy → Energy cannot be created or destroyed; it can only be converted from one form to another.
- Example: If a ball falls without air resistance, the increase in its kinetic energy equals the decrease in its gravitational potential energy, keeping total energy constant.
Equation to calculate the efficiency of a system: $$ \text{Efficiency} = \frac{\text{Useful Energy Output}}{\text{Total Energy Input}} $$
Power → Work done per unit time: $$ P = \frac{W}{t} $$
Power can also be calculated using force and velocity in the direction of the force: $$ P = Fv $$
Derivation of $ P = Fv $:
- Work done is given by:
  $ W = Fs $
- Dividing both sides by time $ t $:
  $ \frac{W}{t} = \frac{F s}{t} $
- Since $ \frac{s}{t} $ represents velocity $ v $, we get:
  $ P = Fv $

$$E_p = mgh$$

Derivation of $ \Delta E_p = mgh $:
- $ W = Fs $ (Work done formula)
- $ \Delta E_p$ represents work done against gravity
- $ F = mg $ (Force due to gravity)
- Substituting $ F $ into work formula:
  \[ W = mgh \]
  where $ h $ is the change in height (displacement in the direction of gravity).
Formula for calculating kinetic energy (energy due to motion):

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

Derivation of $E_k = \frac{1}{2} mv^2$:
- Equation of motion:
  $v^2 = u^2 + 2as $
- $ F = ma \Rightarrow a = \frac{F}{m} $ (From Newton’s Second Law)
- Substituting $ a $ into the equation of motion:
  $ v^2 = u^2 + 2 \left(\frac{F}{m}\right) s $
- Multiplying by $ m $ on both sides:
  $ m v^2 = m u^2 + 2Fs $
- Rearrange:
  $ 2Fs = m v^2 – m u^2 $
  $ Fs = \frac{1}{2} m v^2 – \frac{1}{2} m u^2 $
  This equation represents the change in kinetic energy.
- Removing initial kinetic energy term ($\frac{1}{2} m u^2$):
  $ Fs = \frac{1}{2} m v^2 $
  This equation represents the kinetic energy at a given instant.
- Since $ Fs $ is work done (a form of energy), we get:
  $$ E_k = \frac{1}{2} m v^2 $$