13.3 Content
13.3.1 Work
Work is the energy transferred by a force acting through a displacement:
\[W = Fd\cos\theta\]
where: - \(F\) = magnitude of force (N) - \(d\) = displacement (m) - \(\theta\) = angle between force and displacement
SI unit: Joule (J), where 1 J = 1 N·m
- Force parallel to motion (\(\theta = 0°\)): \(W = Fd\) (positive work)
- Force opposite to motion (\(\theta = 180°\)): \(W = -Fd\) (negative work)
- Force perpendicular to motion (\(\theta = 90°\)): \(W = 0\) (no work done)
13.3.2 Kinetic Energy
Kinetic energy is the energy of motion:
\[KE = \frac{1}{2}mv^2\]
- Always positive (mass and \(v^2\) are positive)
- Doubles when speed doubles? No! Quadruples (because \(v^2\))
13.3.3 Interactive: Energy Bar Chart - Motion
Visualise kinetic energy as an object speeds up:
13.3.4 Gravitational Potential Energy
Gravitational potential energy (GPE) is the energy stored due to height:
\[GPE = mgh\]
where: - \(m\) = mass (kg) - \(g\) = gravitational field strength (9.8 m/s² on Earth) - \(h\) = height above reference point (m)
GPE depends on the chosen reference height. Usually, we set GPE = 0 at the lowest point in a problem.
13.3.5 Work-Energy Theorem
The net work done on an object equals its change in kinetic energy:
\[W_{net} = \Delta KE = KE_f - KE_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2\]
This powerful theorem connects force, displacement, and speed change.
13.3.6 Interactive: Energy Conservation - Falling Object
As an object falls, GPE converts to KE:
Key observation: Total mechanical energy (KE + GPE) remains constant (in the absence of friction).
13.3.7 Conservation of Mechanical Energy
In the absence of friction and air resistance:
\[KE_i + GPE_i = KE_f + GPE_f\]
Or equivalently: \[\frac{1}{2}mv_i^2 + mgh_i = \frac{1}{2}mv_f^2 + mgh_f\]
This is one of the most useful equations in physics!
13.3.8 Interactive: Pendulum Energy
A pendulum swings back and forth, continuously converting between KE and GPE:
At the highest points, all energy is GPE (momentarily stationary). At the lowest point, all energy is KE (maximum speed).
13.3.9 Power
Power is the rate of doing work or transferring energy:
\[P = \frac{W}{t} = \frac{E}{t}\]
For constant force and velocity: \[P = Fv\]
SI unit: Watt (W), where 1 W = 1 J/s
13.3.10 Efficiency
Efficiency measures how much input energy becomes useful output:
\[\eta = \frac{E_{useful}}{E_{input}} \times 100\% = \frac{P_{output}}{P_{input}} \times 100\%\]
No machine is 100% efficient—some energy is always lost to friction, heat, sound, etc.
13.3.11 Interactive: Energy with Friction
When friction is present, some mechanical energy is lost:
Friction does negative work, removing mechanical energy and converting it to thermal energy.