Work Done and Power

When you push a shopping trolley, lift a box onto a shelf or drag a sledge across the snow, you are doing work. In everyday speech "work" means effort, but in physics it has an exact meaning: work is done whenever a force makes something move, and the amount of work done is exactly the amount of energy transferred. Closely linked is the idea of power — not how much energy is transferred, but how quickly it is transferred. A powerful engine and a weak one might transfer the same total energy, but the powerful one does it far faster. This lesson, part of Topic P7 (Energy) of OCR Gateway Science A, defines work done and power, introduces their equations, works through calculations in both directions, and explains why work done against friction heats things up.

By the end of this lesson you should be able to define work done and power, use and rearrange $W = Fd$ and $P = \dfrac{W}{t}$ , carry out calculations with the correct units, and explain why doing work against friction raises the temperature.

What Is Work Done?

Work is done on an object whenever a force causes it to move in the direction of the force. Work done is a way of measuring energy transferred: in fact,

work done = energy transferred.

The two are the same thing measured in the same unit, the joule (J). If you do $50 \text{ J}$ of work on a box, you have transferred $50 \text{ J}$ of energy to it. This is why $W = Fd$ is an energy equation — it tells you how much energy a force has moved from one store to another.

A key point: for work to be done, the object must move, and it must move in the direction of the force. If you push as hard as you like on a wall that does not budge, you do no work on the wall (in the physics sense), because it does not move — even though you feel tired.

Exam Tip: "Work done" and "energy transferred" mean exactly the same thing and share the same unit, the joule. If a question gives you the work done, you also know the energy transferred — and vice versa.

The Work Done Equation

The work done by a force is calculated from:

$W = Fd$

where $W$ is the work done (in joules, $\text{J}$ ), $F$ is the force (in newtons, $\text{N}$ ), and $d$ is the distance moved in the direction of the force (in metres, $\text{m}$ ).

The equation rearranges to make force or distance the subject:

$W = Fd \qquad F = \dfrac{W}{d} \qquad d = \dfrac{W}{F}$

One joule is defined as the work done when a force of one newton moves an object one metre in the direction of the force: $1 \text{ J} = 1 \text{ N} \times 1 \text{ m}$ , so $1 \text{ J} = 1 \text{ N m}$ .

Worked Example 1 — Work done pushing a trolley

A shopper pushes a trolley with a force of $40 \text{ N}$ for a distance of $15 \text{ m}$ . How much work is done?

Step 1 — write the equation: $W = Fd$ .

Step 2 — substitute: $W = 40 \times 15$ .

Step 3 — calculate: $W = 600 \text{ J}$ .

Answer: $600 \text{ J}$ of work is done (and $600 \text{ J}$ of energy is transferred).

Worked Example 2 — Rearranging for force

A crane does $24\,000 \text{ J}$ of work lifting a load through a height of $12 \text{ m}$ . What is the weight of the load (the force the crane exerts)?

Step 1 — rearrange for force: $F = \dfrac{W}{d}$ .

Step 2 — substitute: $F = \dfrac{24\,000}{12}$ .

Step 3 — calculate: $F = 2000 \text{ N}$ .

Answer: the load has a weight of $2000 \text{ N}$ .

Worked Example 3 — Rearranging for distance

A motor does $900 \text{ J}$ of work pulling a sledge with a force of $60 \text{ N}$ . How far does the sledge move?

Step 1 — rearrange for distance: $d = \dfrac{W}{F}$ .

Step 2 — substitute: $d = \dfrac{900}{60}$ .

Step 3 — calculate: $d = 15 \text{ m}$ .

Answer: the sledge moves $15 \text{ m}$ .

Exam Tip: In $W = Fd$ , the distance $d$ must be measured in the direction of the force. For something lifted straight up, $d$ is the height gained; for something pushed along the floor, $d$ is the horizontal distance moved. Always show equation, substitution, answer with unit.

What Is Power?

Power is the rate of transferring energy — how much energy is transferred (or work done) per second. Two machines can transfer the same total energy, but the one that does it in less time has the greater power. Power is measured in watts (W), where one watt is one joule per second: $1 \text{ W} = 1 \text{ J/s}$ .

Because work done equals energy transferred, power can be written in two equivalent ways:

$P = \dfrac{W}{t} \qquad \text{and} \qquad P = \dfrac{E}{t}$

where $P$ is the power (in watts, $\text{W}$ ), $W$ is the work done and $E$ is the energy transferred (both in joules, $\text{J}$ ), and $t$ is the time taken (in seconds, $\text{s}$ ). The two forms are the same equation, because $W$ and $E$ are the same quantity.

These rearrange to:

$P = \dfrac{E}{t} \qquad E = Pt \qquad t = \dfrac{E}{P}$

Worked Example 4 — Power of a motor

A motor transfers $6000 \text{ J}$ of energy in $30 \text{ s}$ . Calculate its power.

Step 1 — write the equation: $P = \dfrac{E}{t}$ .

Step 2 — substitute: $P = \dfrac{6000}{30}$ .

Step 3 — calculate: $P = 200 \text{ W}$ .

Answer: the motor has a power of $200 \text{ W}$ .

Worked Example 5 — Combining work and power

A weightlifter lifts a $1500 \text{ N}$ barbell through a height of $2 \text{ m}$ in $4 \text{ s}$ . Calculate (a) the work done and (b) the power developed.

Step 1 — work done: $W = Fd = 1500 \times 2 = 3000 \text{ J}$ .

Step 2 — power: $P = \dfrac{W}{t} = \dfrac{3000}{4}$ .

Step 3 — calculate: $P = 750 \text{ W}$ .

Answer: (a) $3000 \text{ J}$ of work is done; (b) the power developed is $750 \text{ W}$ .

Worked Example 6 — Comparing two cranes

Crane A and crane B both lift a $5000 \text{ J}$ load. Crane A takes $10 \text{ s}$ and crane B takes $25 \text{ s}$ . Which is more powerful, and by how much?

Step 1 — crane A: $P = \dfrac{5000}{10} = 500 \text{ W}$ .

Step 2 — crane B: $P = \dfrac{5000}{25} = 200 \text{ W}$ .

Step 3 — compare: crane A is more powerful by $500 - 200 = 300 \text{ W}$ .

Answer: crane A is more powerful, by $300 \text{ W}$ , because it transfers the same energy in less time.

Exam Tip: Power is energy per second, so the time must be in seconds. If a question gives a time in minutes, convert first (multiply by 60). A bigger power means the same energy transferred faster, or more energy in the same time.

Work Done and Power

Work Done and Power

What Is Work Done?

The Work Done Equation

Worked Example 1 — Work done pushing a trolley

Worked Example 2 — Rearranging for force

Worked Example 3 — Rearranging for distance

What Is Power?

Worked Example 4 — Power of a motor

Worked Example 5 — Combining work and power

Worked Example 6 — Comparing two cranes

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