Work, energy and power
Introduced| English | Meaning |
|---|---|
| work done/wɜːk dʌn/ | Energy transferred when a force acts through a displacement, calculated using the component of force along that displacement. |
| displacement/dɪˈspleɪsmənt/ | The straight-line change in position, including direction. |
| conservation of energy/ˌkɒnsəˈveɪʃn ɒv ˈenədʒi/ | The principle that energy cannot be created or destroyed, only transferred or transformed. |
| efficiency/ɪˈfɪʃənsi/ | The fraction of total energy or power input converted to useful output. |
| thermal energy/ˈθɜːml ˈenədʒi/ | Energy associated with the random motion and interactions of particles in a substance. |
| power/ˈpaʊə/ | The rate of energy transfer or work done. |
Holding a bag does no work
- Hold a heavy bag still and your arm aches — but in physics you do zero work.
- Work needs the force to move something along its own direction.
- No movement (or a sideways force) means no work done 功.
Work done by a force
- $W = F s \cos\theta$, where $\theta$ is the angle between force and displacement 位移.
- Only the part of the force along the motion does work. Unit: $\text{J} = \text{N}\cdot\text{m}$.
Work, energy & power
PE + KE = constant
Work transfers energy; as it falls, PE becomes KE with the total fixed.
A force of $10\ \text{N}$ pushes a box $3.0\ \text{m}$ in the direction of the force. How much work is done?
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Force is along the motion ($\theta = 0$), so $W = Fs = 10 \times 3.0 = 30\ \text{J}$.
Positive, negative, zero
- Force along the motion → positive work (energy given to the object).
- Force opposite the motion → negative work (e.g. friction takes energy away).
- Force at right angles → zero work.
Only the component F cos(theta) along the displacement does work
A force acting at right angles to the motion does how much work?
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$W = Fs\cos 90^{\circ} = 0$. The normal contact force on a car on a flat road does no work.
Friction on a sliding object does negative work.
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Friction points opposite to the motion ($\theta = 180^{\circ}$), so $W = -Fs$ — it takes energy away.
Conservation of energy 能量守恒
- Energy is never made or destroyed — it only changes form or moves between objects.
- In a closed system the total energy stays constant.
Positive work (force along motion) and negative work (force opposite motion)
Energy cannot be created or destroyed, only ____ from one form to another.
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That is conservation of energy — the total in a closed system stays the same.
Match each term to the definition the examiner marks.
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The work definition's last phrase is what makes a force at right angles to the motion do no work at all.
Efficiency 效率
- $\text{efficiency} = \dfrac{\text{useful output}}{\text{total input}} \times 100\%$.
- It is always below $100\%$ — some energy ends up as "useless" thermal energy 热能.
The same work done in less time means more power
Energy flow & efficiency
The input energy divides into useful work and wasted energy; efficiency = useful ÷ input, and useful + wasted always equals the input.
A machine takes in $200\ \text{J}$ and gives out $60\ \text{J}$ of useful energy. What is its efficiency (in %)?
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$\dfrac{60}{200} \times 100\% = 30\%$. The other $140\ \text{J}$ is wasted, mostly as heat.
Power 功率
- Power is the rate of doing work: $P = \dfrac{W}{t} = \dfrac{\Delta E}{\Delta t}$.
- Unit: $\text{W} = \dfrac{\text{J}}{\text{s}}$.
Power, force and velocity
- For a force $F$ along the motion at speed $v$: $P = Fv$.
- A car at steady speed needs $P = F_{\text{resist}} \times v$; lifting a weight at speed $v$ needs $P = mgv$.
A force $F$ acts on an object moving at speed $v$ in the direction of the force. The power delivered is:
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In time $\Delta t$ the displacement is $v\Delta t$ and the work is $Fv\Delta t$; dividing by $\Delta t$ gives $P = Fv$.
Worked example: a car at steady speed
- A car travels at a steady $25\ \text{m/s}$ against a total resistive force of $600\ \text{N}$. Find the useful output power of its engine.
- At steady speed the resultant force is zero, so the driving force equals the resistive force: $F = 600\ \text{N}$.
- $P = Fv = 600 \times 25 = 1.5\times10^{4}\ \text{W} = 15\ \text{kW}$.
- The step that carries the mark is stating why the driving force equals $600\ \text{N}$: constant velocity means zero resultant force.
- If the car were accelerating, the driving force would be larger than $600\ \text{N}$ and you could not use the resistive force in $P = Fv$ at all.
Marks that slip away
- Efficiency is useful output divided by total input. The input is always the larger number, so an answer above $100\%$ means the two were swapped.
- In $P = Fv$, $F$ is the driving force. It equals the resistive force only at constant speed.
- Use the vertical height in $mg\Delta h$, never the distance along a slope.
- Work done by a force is the force times the distance moved in the direction of the force. A force at right angles to the motion does no work.
- On a force-extension graph the area is the energy stored. The gradient is the stiffness, and reading one for the other is a whole question lost.
A car travels at a steady 25 m/s against a total resistive force of 600 N. What is the useful output power, in kW?
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Steady speed means zero resultant force, so the driving force is 600 N and P = Fv = 15 kW. Saying WHY the driving force equals the resistance is what carries the mark.
In which cases does a force do no work? Select all that apply.
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Braking does plenty of work, just negative work. The orbit is the neat case: gravity is always perpendicular to the motion, so the satellite's speed never changes.
Put an efficiency calculation in order.
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The input is always the larger number. An efficiency above 100% means the two were swapped, and dividing by the wasted energy is the other common wrong route.
You've got it
- work $W = Fs\cos\theta$ — only the part of $F$ along the motion counts
- energy is conserved; efficiency = useful ÷ total (always < 100%)
- power $P = \dfrac{W}{t} = Fv$