Work
Work is energy transferred by a force acting through a displacement:
W = Fd cos θ
- F = force (N)
- d = displacement (m)
- θ = angle between the force and displacement
| Angle | cos θ | Work |
|---|---|---|
| 0° (force along motion) | 1 | Positive: W = Fd |
| 90° (force perpendicular) | 0 | Zero |
| 180° (force opposite motion) | −1 | Negative (e.g., friction) |
Example: A sled is pulled 10 m with a 50 N force at 30° above horizontal.
W = 50 × 10 × cos 30° = 50 × 10 × 0.866 = 433 J
Positive work adds energy to an object; negative work removes it.
Kinetic energy
KE = ½mv²
Example: A 0.145 kg baseball thrown at 40 m/s has KE = ½ × 0.145 × 40² = 116 J.
The work–energy theorem
The work–energy theorem says the net work done on an object equals its change in kinetic energy:
W_net = ΔKE = ½mv_f² − ½mv_i²
Example: What net work is needed to accelerate a 1000 kg car from 10 m/s to 20 m/s?
W = ½(1000)(20²) − ½(1000)(10²) = 200 000 − 50 000 = 150 000 J
This explains stopping distance: braking friction does negative work equal to the car’s KE. Since KE depends on v², doubling speed quadruples stopping distance.
Potential energy
Gravitational potential energy: PE_g = mgh, where h is height above a chosen reference level.
Elastic potential energy (in a spring): PE_e = ½kx², where k is the spring constant (N/m) and x is the stretch or compression.
Conservation of energy
The law of conservation of energy: energy cannot be created or destroyed, only transformed or transferred. The total energy of an isolated system stays constant.
Mechanical energy
Mechanical energy = KE + PE. If no friction or air resistance acts, mechanical energy is conserved:
KE_i + PE_i = KE_f + PE_f
Example: A 50 kg skateboarder starts from rest at the top of a 5.0 m ramp. What is her speed at the bottom (ignoring friction)?
- PE_top = mgh = 50 × 9.8 × 5.0 = 2450 J
- KE_bottom = 2450 J
- ½(50)v² = 2450 → v² = 98 → v ≈ 9.9 m/s
Notice the mass cancels: any object would reach the same speed.
When friction acts
Friction converts mechanical energy into thermal energy:
E_initial = E_final + E_lost to friction
Total energy is still conserved; it just changes into less useful forms.
Pendulum
A pendulum trades PE and KE back and forth:
| Position | PE | KE | Speed |
|---|---|---|---|
| Highest point | Maximum | Zero | Zero |
| Lowest point | Minimum | Maximum | Maximum |
Over time, air resistance and friction at the pivot convert mechanical energy to heat, and the swings get smaller.
Power and efficiency
Power = work ÷ time = energy ÷ time (W, watts)
Power can also be written P = Fv for an object moving at constant velocity.
Efficiency = useful energy output ÷ total energy input × 100%
| Energy conversion | Typical efficiency |
|---|---|
| Electric motor | 85–95% |
| Gasoline engine | 20–35% |
| Solar panel | about 20–23% |
| Human muscle | about 20–25% |
Energy in society
Electricity generation involves energy chains:
- Hydroelectric: gravitational PE of water → KE → electrical
- Wind: KE of air → electrical
- Fossil fuel: chemical → thermal → KE of steam → electrical
Hydroelectricity supplies about 60% of Canada’s electricity.
Common mistakes
- Ignoring the angle in W = Fd cos θ.
- Thinking energy is "lost." It’s transformed, often into heat.
- Forgetting to square v in KE.
Try it yourself
- A 2 kg ball is dropped from 10 m. Find its speed just before it hits the ground.
- A 60 kg cyclist slows from 12 m/s to 4 m/s. How much work did friction and braking do?
- A spring (k = 200 N/m) is compressed 0.10 m. How much energy does it store?
- A roller coaster car starts at 30 m high at rest. What is its speed at 10 m high (no friction)?
(Answers: 14 m/s; −3840 J; 1.0 J; v² = 2 × 9.8 × 20 = 392, v ≈ 19.8 m/s)
Quick recap
- W = Fd cos θ; work transfers energy.
- KE = ½mv²; PE_g = mgh; PE_e = ½kx².
- Net work equals the change in kinetic energy.
- Mechanical energy is conserved without friction; total energy is always conserved.
- Efficiency compares useful output to input.