Energy, work & power
Work done
Work done
Work is done when a force causes an object to move in the direction of the force.
\[ W = Fd \]
1 joule = 1 newton × 1 metre = 1 N·m. Work done equals the energy transferred.
| Symbol | Quantity | Unit |
|---|---|---|
| \( W \) | Work done (= energy transferred) | joule, J |
| \( F \) | Force (in the direction of motion) | newton, N |
| \( d \) | Distance moved in the direction of the force | metre, m |
Conditions for work to be done:
- A force must act on the object.
- The object must move — if \( d = 0 \), then \( W = 0 \) (no matter how large the force).
- The force (or the component of the force) must have a component in the direction of motion.
A person holding a heavy bag stationary does no work on the bag — there is a force but no displacement. A satellite in a circular orbit does no work against gravity — the gravitational force is perpendicular to the motion.
Worked examples
Example 1 — calculate work done
A person pushes a trolley with a force of 40 N for a distance of 15 m. Calculate the work done.
\( W = Fd = 40 \times 15 = 600\,\text{J} \)
Example 2 — find force from work done
A builder does 3600 J of work pushing a wheelbarrow 12 m. Find the force applied.
\( F = \dfrac{W}{d} = \dfrac{3600}{12} = 300\,\text{N} \)
Example 3 — work against gravity (lifting)
A crane lifts a 500 kg steel beam a height of 8.0 m. Calculate the work done against gravity. (g = 10 N/kg)
Force needed = weight = \( mg = 500 \times 10 = 5000\,\text{N} \)
\( W = Fd = 5000 \times 8.0 = 40\,000\,\text{J} = 40\,\text{kJ} \)
This equals the gain in gravitational potential energy of the beam.
Force–distance graphs
The area under a force–distance graph equals the work done (energy transferred).
\[ W = \text{area under } F\text{–}d \text{ graph} \]
For a constant force, the graph is a horizontal line — the area is a rectangle (\( F \times d \)). For a varying force (like a spring), split the area into rectangles and triangles.
Work done and energy transferred are equivalent. When a force acts through a distance, energy moves from one store to another. Always check units: force in N, distance in m → work in J. Convert cm → m or kN → N before substituting.