Physics A Level
Chapter 5: Work, energy and power 5.7 Power
Physics A Level
Chapter 5: Work, energy and power 5.7 Power
The word power has several different meanings – such as political power, powers of ten or electrical power from power stations. In physics, it has a specific meaning related to these other meanings. Figure 5.17 illustrates what we mean by power in physics.
does many thousands of joules of work each second
The lift shown in Figure 5.17 can lift a heavy load of people. The motor at the top of the building provides a force to raise the lift car, and this force does work against the force of gravity. The motor transfers energy to the lift car. The power P of the motor is the rate at which it does work over a unit of time.
Power is defined as the rate of work done per unit of time. As a word equation, power is given by:
$\begin{array}{l}
Power\, = \,\frac{{Work\,done}}{{time\,taken}}\\
P = \frac{W}{t}
\end{array}$
where W is the work done in a time t.
Units of power: the watt
Power is measured in watts, named after James Watt, the Scottish engineer famous for his development of the steam engine in the second half of the $18th$ century. The watt is defined as a rate of working of 1 joule per second. Hence:
$1\,watt\, = \,1\,joule\,per\,\sec ond$
or
$1\,W\, = \,1\,J\,{{\mathop{\rm s}\nolimits} ^{ - 1}}$
In practice, we also use kilowatts (kW) and megawatts (MW).
$\begin{array}{l}
1000\,watts\, = \,1\,kilowatt\,(1\,kw)\\
1000000\,watts\, = \,1\,megawatt\,(1\,MW)
\end{array}$
The labels on light bulbs display their power in watts; for example, $60 W$ or $10 W$. The values of power on the labels tell you about the energy transferred by an electrical current, rather than by a force doing work.
Questions
15) Calculate how much work is done by a $50 kW$ car engine in a time of 1.0 minute.
16) A car engine does $4200 kJ$ of work in one minute. Calculate its output power, in kilowatts.
17) A particular car engine provides a force of $700 N$ when the car is moving at its top speed of $40\,m\,{s^{ - 1}}$.
a: Calculate how much work is done by the car’s engine in one second.
b: State the output power of the engine.
Moving power
An aircraft is kept moving forwards by the force of its engines pushing air backwards. The greater the force and the faster the aircraft is moving, the greater the power supplied by its engines.
Suppose that an aircraft is moving with velocity v. Its engines provide the force F needed to overcome the drag of the air. In time t, the aircraft moves a distance s equal to $v \times t$.
So, the work done by the engines is:
$\begin{array}{l}
work\,done\, = \,force\, \times \,dis\tan ce\\
W = F\, \times \,v\, \times t
\end{array}$
We know that:
$\begin{array}{l}
power\, = \,\frac{{work\,done}}{{time\,taken}}\\
P = \,\frac{w}{t}
\end{array}$
Substituting W for gives:
$P = \frac{{F \times v \times t}}{t}$
Which can be simplified to:
$\begin{array}{l}
P\, = \,F\, \times \,v\\
power\, = \,force\, \times \,velocity
\end{array}$
It may help to think of this equation in terms of units. The right-hand side is in $N \times m\,{s^{ - 1}}$, and $N m$ is the same as J. So the right-hand side has units of $J\,{s^{ - 1}}$, or W, the unit of power. If you look back to Question 17, you will see that, to find the power of the car engine, rather than considering the work done in $1 s$, we could simply have multiplied the engine’s force by the car’s speed.
Human power
Our energy supply comes from our food. A typical diet supplies $2000–3000 kcal$ (kilocalories) per day. This is equivalent (in SI units) to about $10 MJ$ of energy. We need this energy for our daily requirements – keeping warm, moving about, brainwork and so on. We can determine the average power of all the activities of our body:
$\begin{array}{l}
average\,power\, = \,10\,Mg\,per\,day\\
= \,10\, \times \,\frac{{{{10}^6}}}{{86400}}\\
= \,116\,W
\end{array}$
So we dissipate energy at the rate of about $100 W$. We supply roughly as much energy to our surroundings as a $100 W$ light bulb. Twenty people will keep a room as warm as a $2 kW$ electric heater.
Note that this is our average power. If you are doing some demanding physical task, your power will be greater. This is illustrated in Worked example 7.
Note also that the human body is not a perfectly efficient system; a lot of energy is wasted when, for example, we lift a heavy load. We might increase an object’s g.p.e. by $1000 J$ when we lift it, but this might require five or ten times this amount of energy to be expended by our bodies.
WORKED EXAMPLE
7) A person who weighs $500 N$ runs up a flight of stairs in $5.0 s$ (Figure 5.18). Their gain in height is $3.0 m$. Calculate the rate at which work is done against the force of gravity.
Step 1: Calculate the work done against gravity:
$\begin{array}{l}
work\,done\,W\, = \,F\, \times \,s\\
= \,500\, \times \,3.0\\
= 1500\,J
\end{array}$
Step 2: Now calculate the power:
$\begin{array}{l}
power\,P\, = \,\frac{w}{t}\\
= \,\frac{{1500}}{{5.0}}\\
= \,300\,W
\end{array}$
So, while the person is running up the stairs, they are doing work against gravity at a greater rate than their average power – perhaps three times as great. And, since our muscles are not very efficient, they need to be supplied with energy even faster, perhaps at a rate of $1 kW$. This is why we cannot run up stairs all day long without greatly increasing the amount we eat. The inefficiency of our muscles also explains why we get hot when we exert ourselves.
Question
18) In an experiment to measure a student’s power, she times herself running up a flight of steps. Use the data to work out her useful power.
number of steps $= 28$
height of each step $= 20 cm$
acceleration of free fall $ = 9.8\,m\,{s^{ - 2}}$
mass of student $= 55 kg$
time taken $= 5.4 s$
EXAM-STYLE QUESTIONS
1) How is the joule related to the base units of m, kg and s? [1]
A: $kg\,{m^{ - 1}}\,{s^2}$
B: $kg\,{m^2}\,{s^{ - 2}}$
C: $kg\,{m^2}\,{s^{ - 1}}$
D: $kg\,{s^{ - 2}}$
2) An object falls at terminal velocity in air. What overall conversion of energy is occurring? [1]
A: gravitational potential energy to kinetic energy
B: gravitational potential energy to thermal energy
C: kinetic energy to gravitational potential energy
D: kinetic energy to thermal energy
3) In each case a–c, describe the energy changes taking place:
a: An apple falling towards the ground [1]
b: A car decelerating when the brakes are applied [1]
c: A space probe falling towards the surface of a planet. [1]
[Total: 3]
4) A $120 kg$ crate is dragged along the horizontal ground by a $200 N$ force acting at an angle of ${30^ \circ }$ to the horizontal, as shown.
The crate moves along the surface with a constant velocity of $0.5\,m\,{s^{ - 1}}$. The $200 N$ force is applied for a time of $16 s$.
a: Calculate the work done on the crate by:
i- the $200 N$ force [3]
ii- the weight of the crate [2]
iii- the normal contact force N. [2]
b: Calculate the rate of work done against the frictional force F. [1]
[Total: 8]
5) Explain which of the following has greater kinetic energy?
- A 20-tonne truck travelling at a speed of $30\,m\,{s^{ - 1}}$
- A $1.2 g$ dust particle travelling at $150\,km\,{s^{ - 1}}$ through space. [3]
6) A $950 kg$ sack of cement is lifted to the top of a building $50 m$ high by an electric motor.
a: Calculate the increase in the gravitational potential energy of the sack of cement. [2]
b: The output power of the motor is $4.0 kW$. Calculate how long it took to raise the sack to the top of the building. [2]
c: The electrical power transferred by the motor is $6.9 kW$. In raising the sack to the top of the building, how much energy is wasted in the motor as heat? [3]
[Total: 7]
7) a: Define power and state its unit. [2]
b: Write a word equation for the kinetic energy of a moving object. [1]
c: A car of mass $1100 kg$ starting from rest reaches a speed of $18\,m\,{s^{ - 1}}$ in $25s$. Calculate the average power developed by the engine of the car. [2]
[Total: 5]
8) A cyclist pedals a long slope which is at ${5.0^ \circ }$ to the horizontal, as shown.
The cyclist starts from rest at the top of the slope and reaches a speed of $12\,m\,{s^{ - 1}}$ after a time of $67 s$, having travelled $40 m$ down the slope. The total mass of the cyclist and bicycle is $90 kg$.
a: Calculate:
i- the loss in gravitational potential energy as he travels down the slope [3]
ii- the increase in kinetic energy as he travels down the slope. [2]
b: i- your answers to a to determine the useful power output of the cyclist. [3]
ii- Suggest one reason why the actual power output of the cyclist is larger
than your value in i. [2]
[Total: 10]
9) a: Explain what is meant by work. [2]
b: i- Explain how the principle of conservation of energy applies to a man sliding from rest down a vertical pole, if there is a constant force of friction acting on him. [2]
ii- The man slides down the pole and reaches the ground after falling a distance $h = 15 m$. His potential energy at the top of the pole is $1000 J$.
Sketch a graph to show how his gravitational potential energy ${E_p}$ varies with h. Add to your graph a line to show the variation of his kinetic energy ${E_k}$ with h. [3]
[Total: 7]
10) a: Use the equations of motion to show that the kinetic energy of an object of mass m moving with velocity v is $\frac{1}{2}m{v^2}$ . [2]
b: A car of mass 800 kg accelerates from rest to a speed of $20\,m\,{s^{ - 1}}$ in a time of $6.0 s$.
i- Calculate the average power used to accelerate the car in the first 6.0. [2]
ii- The power passed by the engine of the car to the wheels is constant.
Explain why the acceleration of the car decreases as the car accelerates. [2]
[Total: 6]
11) a: i- Define potential energy. [1]
ii- Identify differences between gravitational potential energy and elastic potential energy.
[2]
b: Seawater is trapped behind a dam at high tide and then released through turbines. The level of the water trapped by the dam falls $10.0 m$ until it is all at the same height as the sea.
i- Calculate the mass of seawater covering an area of $20\,m\,{s^{ - 1}}$ and with a depth of $10.0 m$. (Density of seawater $ = 1030\,kg\,{m^{ - 3}}$. [1]
ii- Calculate the maximum loss of potential energy of the seawater in i when passed through the turbines. [2]
iii- The potential energy of the seawater, calculated in ii, is lost over a period of 6.0 hours. Estimate the average power output of the power station over this time period, given that the efficiency of the power station is $50\% $.
SELF-EVALUATION CHECKLIST
After studying the chapter, complete a table like this:
| I can | See topic… | Needs more work | Almost there | Ready to move on |
| understand the concept of work, and recall and use work done = force $ \times $ displacement in the direction of the force | 5.1 | |||
| recall and apply the principle of conservation of energy | 5.6 | |||
| recall and understand that the efficiency of a system is the ratio of useful energy output from the system to the total energy input | 5.6 | |||
| use the concept of efficiency to solve problems | 5.6 | |||
| define power as work done per unit time and solve problems using | 5.7 | |||
| derive $P = Fv$ and use it to solve problems | 5.7 | |||
| derive, using $W = Fs$, the formula $\Delta {E_p} = mg\Delta h$ | 5.2 | |||
| recall and use the formula $\Delta {E_p} = mg\Delta h$ | 5.5 | |||
| derive, using the equations of motion, the formula ${E_k} = \frac{1}{2}m{v^2}$ and recall and use the formula. | 5.3 |