Energy Transformations, Work and Power – Work, Power and Energy (Senior 1 Physics) | Ganzaa

Work, Power and Energy

Lessons

Senior 1 · Physics

Work, Power and Energy

5 lessons

Lesson 5 of 5

Energy Transformations, Work and Power

25 min

The swing that never goes higher

Push a child on a swing once and let go. The swing climbs, stops for an instant, rushes down through the lowest point and climbs the other side, but never higher than where it started. Hold a ball at shoulder height and drop it: it bounces, but each bounce is lower than the one before. In this lesson you will see why: energy keeps changing between potential and kinetic, and the total can never grow.

Definition

Conservation of mechanical energy: when only gravity does work on a body (no friction or air resistance), the total of its potential and kinetic energy stays constant:

E_p + E_k = \text{constant}

So a body that falls from rest through a height $h$ loses potential energy $m g h$ and gains the same amount of kinetic energy:

mgh = \frac{1}{2}mv^2 \quad\Rightarrow\quad v = \sqrt{2gh}

Experiment: the pendulum

You need: a retort stand and clamp, a metre of string, a heavy bob (a stone in a bag or a metal ball), a metre rule and chalk.

Hang the bob so it just clears the bench. Mark its lowest position B.
Pull the bob aside to position A, keeping the string tight, and measure its height above B.
Release it from rest (do not push). Watch where it stops on the other side, C, and measure that height too.
Count how many swings it takes for the height to fall to half.

Figure 5.1: At A and C the bob stops for an instant: all its mechanical energy is potential. At B it is at its lowest and fastest: all its mechanical energy is kinetic. In between, it has some of each.

Observation: C is at almost exactly the same height as A. Over many swings the height slowly falls.

Explanation: the energy changes $E_p \to E_k \to E_p$ every swing and the total stays nearly constant, so the bob rises back to almost the same height. A little energy is lost each swing, as heat and sound, because of air resistance and friction at the clamp, so the swing slowly dies away.

Important

Safety. Clamp the stand firmly or weight its base so it cannot topple, and stand clear of the swinging bob.

Worked example 1: a pendulum bob

A 0.50 kg bob is released from rest 0.20 m above its lowest point. Find (a) its potential energy at the start, (b) its kinetic energy and speed at the lowest point. Ignore air resistance.

\text{(a)}\quad E_p = mgh = 0.50 \times 9.8 \times 0.20 = 0.98\,\mathrm{J}

\text{(b)}\quad E_k = 0.98\,\mathrm{J}, \qquad v = \sqrt{\frac{2E_k}{m}} = \sqrt{\frac{2 \times 0.98}{0.50}} = 1.98\,\mathrm{m\,s^{-1}} \approx 2.0\,\mathrm{m\,s^{-1}}

Worked example 2: a stone dropped from a cliff

A 2.0 kg stone falls from rest from a height of 20 m. Complete the energy table and find the speed at which it hits the ground (ignore air resistance).

Height (m)	$E_{p} = m g h$ (J)	$E_{k}$ (J)	Total (J)
20	392	0	392
15	294	98	392
10	196	196	392
5	98	294	392
0	0	392	392

v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 20} = 19.8\,\mathrm{m\,s^{-1}}

Notice that the mass cancels in $v = \sqrt{2gh}$ : a 20 kg stone would land at the same speed.

Where does the energy go?

Figure 5.2: At each bounce some kinetic energy becomes heat (the ball and floor warm slightly) and sound. The energy is not destroyed; it just stops being mechanical energy.

Work, power and energy: how they are linked

The three quantities in this unit are really one idea seen three ways.

Work is energy transferred. When a force does 500 J of work on a body, 500 J of energy is transferred to it: $W = \Delta E$ . Lifting a body does work that becomes potential energy; pushing a trolley from rest does work that becomes kinetic energy.
Power is the rate of that transfer. $P = \frac{W}{t} = \frac{E}{t}$ , so the energy transferred in a time $t$ is $E = P t$ .

Quantity	Symbol	Formula	SI unit
Work	$W$	$W = F d$	joule (J)
Kinetic energy	$E_{k}$	$E_k = \frac{1}{2}mv^2$	joule (J)
Potential energy	$E_{p}$	$E_{p} = m g h$	joule (J)
Power	$P$	$P = \frac{W}{t} = \frac{E}{t}$	watt (W), $1\,\mathrm{W} = 1\,\mathrm{J\,s^{-1}}$

Worked example 3: a crane on a building site

A crane lifts a 500 kg load of blocks steadily through 12 m in 30 s. Find (a) the work done, (b) the load's gain in potential energy, (c) the crane's power.

\text{(a)}\quad W = Fd = mg \times h = 500 \times 9.8 \times 12 = 58\,800\,\mathrm{J}

(b) All the work goes into lifting the load, so $\Delta E_p = 58\,800\,\mathrm{J}$ .

\text{(c)}\quad P = \frac{W}{t} = \frac{58\,800\,\mathrm{J}}{30\,\mathrm{s}} = 1960\,\mathrm{W} \approx 2.0\,\mathrm{kW}

Worked example 4: power from falling water

At a small hydroelectric plant, 1000 kg of water falls through a height of 40 m every second. Find the potential energy lost by the water each second, and so the greatest power the plant could produce.

E_p = mgh = 1000 \times 9.8 \times 40 = 392\,000\,\mathrm{J} \text{ each second}

P = \frac{E}{t} = \frac{392\,000\,\mathrm{J}}{1\,\mathrm{s}} = 392\,000\,\mathrm{W} \approx 390\,\mathrm{kW}

Friction in the tunnel, the turbines and the generator waste some of this, so the real electrical output is a little smaller. A big plant such as Nyabarongo I, at 28 MW, needs about 70 times as much water each second, or a greater height.

Warning

When does $v = \sqrt{2gh}$ apply? Only when the body starts from rest and no energy is lost to friction or air resistance. For a real falling leaf, a parachutist or a cyclist freewheeling downhill, air resistance takes much of the energy, and the speed is less. Also, $h$ is the vertical drop, not the length of the slope.

Tip

Energy first, forces later. For a question about speeds and heights, write the energy at the start and at the end and set them equal. It is often far quicker than using forces and accelerations.

Try it: in PhET: Energy Skate Park: Basics (version 1.1.36), turn on the bar graph and watch the skater's potential and kinetic energy swap as she rolls, then add friction and watch thermal energy appear.

Work, energy and power (CrashCourse Physics). Watch for the link between work and the change in kinetic energy, and for power as the rate of doing work.

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Practice

State the energy changes in one swing of a pendulum from one end to the other.
Where is a pendulum bob moving fastest, and where does it have the most potential energy?
A 0.30 kg ball is dropped from 45 m. Ignoring air resistance, find its kinetic energy and speed just before it lands.
A skateboarder of mass 50 kg starts from rest at the top of a 5.0 m ramp. What is her greatest possible speed at the bottom?
A 2.0 kW motor runs for 30 s. How much energy does it transfer? If all of it lifted a 60 kg load, how high would the load rise?
A ball dropped from 2.0 m bounces back to 1.5 m. What fraction of its energy is kept after the bounce, and where did the rest go?
Explain why a swing on its own slowly comes to rest, even though energy is conserved.

Show answersHide answers

Potential → kinetic (falling to the lowest point) → potential (rising to the other end), with a little changed to heat and sound.
Fastest at the lowest point; most potential energy at the two ends of the swing.
$E_k = E_p = 0.30 \times 9.8 \times 45 = 132.3\,\mathrm{J}$ ; $v = \sqrt{2 \times 9.8 \times 45} \approx 29.7\,\mathrm{m\,s^{-1}}$
$v = \sqrt{2 \times 9.8 \times 5.0} \approx 9.9\,\mathrm{m\,s^{-1}}$ ; the mass does not matter.
$E = Pt = 2000 \times 30 = 60\,000\,\mathrm{J}$ ; $h = \frac{60\,000}{60 \times 9.8} \approx 102\,\mathrm{m}$
$\frac{1.5}{2.0} = 0.75$ , so 75% is kept. The other 25% became heat and sound at the bounce.
Air resistance and friction at the pivot change some mechanical energy into heat and sound each swing. The total energy is conserved, but the mechanical energy decreases.

Key points

With no friction, $E_{p} + E_{k}$ is constant: potential and kinetic energy swap as a body rises and falls.
Falling from rest through $h$ : $mgh = \frac{1}{2}mv^2$ , so $v = \sqrt{2gh}$ , whatever the mass.
Work done is energy transferred; power is the rate of transfer: $P = \frac{W}{t} = \frac{E}{t}$ .
Real machines always lose some energy as heat and sound, so their useful output is less than their input.

1The swing that never goes higher

2Experiment: the pendulum

3Where does the energy go?

4Work, power and energy: how they are linked

The swing that never goes higher

Experiment: the pendulum

Where does the energy go?

Work, power and energy: how they are linked