Lesson 1 of 5Energy and Society

Unit 3 · Lesson 1 of 5

Work, Power and Efficiency

The precise physics meaning of work — including when a force does none at all — and the two quantities that describe how fast and how well energy is transferred.

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Key things to remember

1 $W = F d cos θ$ , measured in joules. Work is a scalar even though force and displacement are vectors.
2A force perpendicular to the displacement does zero work — carrying a box across a level room is the standard case.
3Negative work removes energy: friction, braking and lowering a load all do negative work.
4 $Σ W = W_{n e t} = W_{+} + W_{-}$ .
5 $P = \frac{W}{Δ t}$ , in watts, where $1 W = 1 J/s$ . Not on the formula sheet.
6 $efficiency = \frac{E _{u se f u l}}{E _{t o t a l}} \times 100% .$ Not on the formula sheet.
7The 'lost' energy in an inefficient device is almost always thermal energy dispersed into the surroundings.

In everyday speech, holding a heavy box still is hard work. In physics it is no work at all. Work is done only when a force causes a displacement, and it is the mechanism by which energy is transferred from one thing to another. Getting the definition exactly right is what makes the rest of the unit consistent.

W = F \cdot d = F d cos θ

Work is measured in joules (J). One joule is one newton acting through one metre.

Work is a scalar — it has no direction, even though it is built from two vectors. The angle $θ$ is the angle between the force and the displacement, and it is what decides whether the force helps, hinders, or does nothing.

Fig 1Only the component of force along the displacement does work.

Angle $θ$	$cos θ$	Work	Example
$0^{\circ}$	$1$	$W = F d$ , maximum positive	Pushing a box in the direction it moves
Between $0^{\circ}$ and $9 0^{\circ}$	Between 0 and 1	Positive, reduced	Pulling a sled with a rope at an angle
$9 0^{\circ}$	$0$	Zero	Carrying a box horizontally — the lift force is vertical
Between $9 0^{\circ}$ and $18 0^{\circ}$	Negative	Negative	A rope dragging a sled backwards and upwards
$18 0^{\circ}$	$- 1$	$W = - F d$ , maximum negative	Braking force against the motion

Three ways to do zero work

No displacement — holding a box still, or pushing a wall that does not move. Force perpendicular to motion — carrying a box across a level room, where the upward force and the horizontal displacement are at

9 0^{\circ}

. No force — an object coasting in space. In all three, the energy of the object is unchanged, which is the real test.

Negative work is not a trick of the algebra. A force that opposes the motion removes energy from the object. Friction does negative work on a sliding box, which is precisely why the box slows down. Catching a ball, braking a car and lowering a load all involve negative work.

Σ W = W_{n e t} = W_{+} + W_{-}

Net work is the sum of the positive and negative contributions, just like net force.

Power: how fast the work is done

Not on your formula sheet

P = \frac{W}{Δ t}

is required by this course and is not printed on the SPH3U formula sheet. Memorize it. The unit is the watt (W), where

1 W = 1 J/s

— and note the unfortunate clash:

W

is the symbol for work and the abbreviation for watt.

Two people carry identical loads up identical stairs. They do exactly the same work, because the force and displacement are the same. The one who runs up has the greater power, because they did that work in less time. Power measures rate, not amount.

A $60 W$ lamp converts $60 J$ of electrical energy every second.
A fit person can sustain roughly $100 W$ of mechanical output for a long period, and several hundred watts in a short burst.
One horsepower is about $746 W$ , a unit that survives from steam-engine marketing.
A kilowatt-hour is not a unit of power but of energy: $1 kW$ for $1 hour = 3.6 \times 1 0^{6} J$ . It is what your hydro bill charges for.

Efficiency: how much of it was useful

Also not on your formula sheet

efficiency = \frac{E _{u se f u l}}{E _{t o t a l}} \times 100% .

Memorize this one too. It works identically with power in place of energy, since the time cancels.

Fig 2The width of each arrow is the energy it carries. Nothing vanishes — it just stops being useful.

No real device is 100% efficient. Energy is never destroyed, but some of it always ends up in a form nobody wanted — almost always thermal energy, spread out into the surroundings where it cannot be recovered. Efficiency is a statement about usefulness, not about conservation.

Device	Typical efficiency	Where the rest goes
Electric motor	85 – 95%	Heat in the windings, friction in bearings
LED bulb	40 – 50%	Heat at the junction
Incandescent bulb	About 5%	Heat — it is essentially a small heater that glows
Petrol car engine	20 – 30%	Exhaust heat, coolant, engine friction
Electric vehicle drivetrain	About 80%	Battery and motor losses
Gas furnace	80 – 98%	Hot exhaust up the flue

Why an electric heater is 100% efficient and it does not help

Every joule going into an electric baseboard heater ends up as heat in the room, so by the definition above it is 100% efficient. A heat pump is more than 100% by that arithmetic, because it moves existing heat rather than making it. Efficiency is always relative to what you wanted — the same device can be rated very differently depending on the job.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

A person pulls a sled with a force of

85 N

along a rope at

3 0^{\circ}

above the horizontal, moving it

22 m

along level ground. How much work is done by the rope?

Example 2

55 kg

student runs up a staircase

4.2 m

high in

6.5 s

. Find the work done against gravity and the power developed.

Example 3

A motor is supplied with

2400 J

of electrical energy and does

1860 J

of useful mechanical work. Find its efficiency and the energy wasted.

Example 4

A student carries a

12 kg

box at constant speed across a level room

15 m

wide. How much work do they do on the box?

Where marks get lost

Forgetting $cos θ$ when the force is applied at an angle, which overstates the work every time.
Leaving the calculator in radian mode, so $cos 3 0^{\circ}$ comes out as $0.154$ instead of $0.866$ .
Claiming work is done while holding something still. No displacement means no work, however tiring it is.
Treating work as a vector and giving it a direction. Work is a scalar; only its sign carries meaning.
Dropping the minus sign on the work done by friction, which then appears to speed the object up.
Confusing power with energy. A watt is a joule per second; a kilowatt-hour is an amount of energy, not a rate.
Putting the wrong quantity on top in the efficiency formula. Useful output goes on top, always.

Flashcards 10

Work

When does a force do zero work?

Negative work

Joule

+ 6 more

Practice

A force of

45 N

pushes a box

12 m

in the direction of the force. How much work is done, in joules?

A rope pulls a cart with

120 N

4 0^{\circ}

above the horizontal, moving it

8.0 m

horizontally. How much work does the rope do, in joules?

A waiter carries a tray at constant height across a level room. How much work does he do on the tray?

A machine does

3600 J

of work in

12 s

. What is its power output, in watts?

A device takes in

850 J

and produces

510 J

of useful output. What is its efficiency, as a percentage?

Two students carry identical boxes up the same staircase. One takes 5 s, the other 10 s. Compare their work and their power.

What is the SI unit of power?

Friction does

- 250 J

of work on a sliding crate. What does the negative sign mean?

Match each angle between force and displacement to the work done.

A.Maximum negative work, $W = - F d$
B.Maximum positive work, $W = F d$
C.Positive but reduced, $W = 0.5 F d$
D.Zero work

0^{\circ}

6 0^{\circ}

9 0^{\circ}

18 0^{\circ}

Order these devices from most efficient to least efficient.

1Gas furnace (about 85%)
2Incandescent bulb (about 5%)
3Petrol car engine (about 25%)
4Electric motor (about 90%)

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