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S P E C I A L F E A T U R E : A L L T H E F U N O F T H E FA I R

www.iop.org/journals/physed

The physics of having a swing

Yanqing Ju

College of Equipment and Materials, Liaodong University, Dandong 118003,

Peoples Republic of China

E-mail: [email protected]

AbstractTo swing higher and higher, a person on a swing stands up and squats downtwice for each swing back and forth. These body movements can increasethe mechanical energy of the system. The person on the swing stands on theseat and is propelled by an initial force so that he or she can rise higher and

higher by body movement alone. How is this done? If we watch people onswings we see that they usually squat down while the swing goes up to itstwo highest points, and then stand up quickly when the swing descends to itslowest point. What is the physics behind this?

Many children (and some adults!) enjoy using a

swing in a playground (figure 1). Children soon

learn that they can make the swing go faster by

standing up at the right moment. Simple analysis

can reveal the best moment in the period of the

swing to do this, and can explain the changes inkinetic and potential energy that take place.

Asshown in figure 2, the personand the swing

formasystemwhosemomentofinertiaforrotation

abouttheaxleis I1 aftersquattingdownandI2 after

standing up. Since the centre of mass is closer

to the axle while standing rather than squatting,

I1 > I2; and because it takes very little time for

a person to stand up and squat down (t 0),

we can assume that the angular momentum of the

system is not changed:

I11 = I22 (1)

where 1 and 2 are the angular velocities of the

swing system before and after a person stands up

or squats down.

Since I1 > I2, 2 > 1. This means that

during the movement of the swing, standing up

suddenly increases the angular velocity of the

swing, but squatting down decreases it.

We cannow analyse thechangein mechanical

energy of the system.

Figure 1. Having a swing.

Standing up

The change in kinetic energy

If the change in kinetic energy caused by standing

up suddenly is Ek, and Ek1 and Ek2 are the

rotational kinetic energies at the moments just

before and after standing up, then

Ek= Ek2 Ek1 =12

I222

12

I121. (2)

534 P H Y S I C S E D U C A T I O N 40 (6) 0031-9120/05/060534+03$30.00 2005 IOP Publishing Ltd


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The physics of having a swing

Figure 2. Changes in moment of inertia. (a) Squattingdown. (b) standing up.

(a) (b)

I11

I22

Using (1),

Ek=12

I2

I11

I2

2

12

I121

=

I21 I1I2

2I221. (3)

Since I1 > I2 > 0, I21 I1I2 > 0, then Ek 0.

This means that if a person stands up during the

movement of the swing, the kinetic energy of the

system increases; and the greater 1 is (I1, I2 do

not change), the greater is Ek. If the person

stands up rapidly at the moment when the angular

velocity (1) is greatest, the kinetic energy of the

system (Ek) will increase to its highest level.

Hence the reason for standing up rapidly at the

lowest point of the swing.

The change in gravitational potential energy

The massof the person is m. As shown in figure 3,

his/her centre of mass is lifted from A to B after

standing up, and AB = h (h is a constant for a

particular person). The angular displacement is

, so the increase in gravitational potential energy(GPE) Ep due to the rise in the centre of mass

will be

Ep = mgh cos . (4)

We can see from this equation that when = 0

(at the lowest point), the increase in GPE will be

greatest. Therefore, the person should also stand

up at the lowest point from the consideration that

the GPE will increase by the greatest amount.

Figure 3. The change in gravitational potential energy.

B

h

Ahcos

O

The change in mechanical energy

It canbe seen from theaboveanalysis that standingup can increase both the kinetic energy and

potential energy, so the mechanical energy of the

system increases. Because the greater the increase

in mechanical energy, the higher the swing will

rise, the person on the swing should stand up

quickly at the lowest point.

Squatting down

The change in kinetic energy

The change in kinetic energy is Ek, then

Ek=12

I121

12

I222

=12

I1

I22

I1

2

12

I222

=

I22 I1I2

2I122. (5)

Since I1 > I2 > 0, I22 I1I2 < 0, then E

k 0,

so this means that squatting down can make the

kinetic energy of the system either decrease or

remain constant, but can never make it increase!

The smaller 2 is, the smaller is E

k. Squatting

down rapidly at the moment when the angularvelocity (2) is at its lowest minimizes the loss in

kinetic energy (Ek), and this is the reason why

people squat down rapidly at the highest point of

the swing.

The change in gravitational potential energy

The GPE will decrease while squatting down

because the centre of mass is lowered. As shown

November 2005 P H Y S I C S E D U C A T I O N 535


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Yanqing Ju

Figure 4. The swing will go higher as a result of the body movement of the person on the swing.

M

Q

N

P

Table 1. Some energy changes in the system caused by standing up and squatting down.

Change in Change in The increase in mechanical

Optimum kinetic Change in mechanical energy after one standing upposition energy GPE energy and one squatting down

The person The lowest +Ek + mgh +Ek + mghstands up point

E = +Ek + mgh(1 cos )The person The highest 0 mgh cos mgh cos squats down point

in figure 3, the loss of GPE is

Ep = mgh cos . (6)

whichmeans thegreater is, the smaller the lossof

GPE. Therefore, to minimize this loss the person

should also squat down at the highest point.

The change in mechanical energy

It can be seen from this analysis that squatting

down causes a loss in mechanical energy. If we

want to minimize this loss, the person on the swing

needs to squat down rapidly at the highest point.

Table 1 shows some energy changes in the system

caused by standing up and squatting down.

From table 1 the overall increase after

standing up and squatting down is E > 0(since cos 1), and as the person on the

swing repeatedly stands up at the lowest point and

squats down at the highest point, the mechanical

energy of the system will continue to increase.

As shown in figure 4, M is the left-hand highest

position of the swing. By standing still on the

seat of the swing, the swing sways from M, using

energy conservation, and can only sway to the

corresponding point N. On the other hand, by

squatting down at M, and standing up rapidly at

the lowest point Q, the mechanical energy of the

system will increase and the swing will rise to a

higherpointP. If this is repeatedon returningfrom

point P (see figure 4(a)), the swing will reach ahigher point than M, and so on. As the mechanical

energy of the system increases, so the swing will

rise higher and higher (figure 4(b)). It is obvious

that the person on the swing has done work in

this cycle that results in the increase in mechanical

energy.

Received 30 August 2005, in final form 27 September 2005

doi:10.1088/0031-9120/40/6/001

Further readingBurns J A 1970 More on pumping a swing Am. J.

Phys. 38 9202Kleppner D and Kolenkow R J 1973 An Introduction to

Mechanics (New York: McGraw-Hill) pp 30810

Yanqing Ju is an assistant professor atLiaodong University, where he has beensince 1986, having obtained his MS atLiaoning Normal University. Hisresearch mainly involves physicsexperiments and mechanics.

536 P H Y S I C S E D U C A T I O N November 2005

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