Simple Harmonic MotionLevel 1 Physics

Simple Harmonic MotionWhen a force causes back and forth motion that is directly proportionalto the displacement, it is said to be Simple Harmonic Motion

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Fs α x

Hooke’s LawEasiest example of SHM would be Hooke’s Law

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Fs α x

k = Constant of Proportionality Spring Constant Units: N/m

F = - kx Negative sign indicatesthat the force is in the opposite direction ofthe displacement

“F” is a Restoring Force – object comesback towards equilibrium

Hooke’s LawIn Hooke’s Law, you have to determine what force is stretching/compressingthe spring. Examples are;

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Fs = kx

Fs = Fg → mg = kx

Fs = Fnet → ma = kx

Example

A load of 50 N attached to a spring hanging vertically stretches the spring 5.0 cm. The spring is now placed horizontally on a table and stretched 11.0 cm. What force is required to stretch the spring this amount?

k

k

kxFs

)05.0(501000 N/m

s

s

s

F

F

kxF

)11.0)(1000(

110 N

Hooke’s Law from a Graphical Point of View

x(m) Force(N)

0 0

0.1 12

0.2 24

0.3 36

0.4 48

0.5 60

0.6 72

graph x vs.F a of Slope

kx

Fk

kxF

s

sSuppose we had the following data:

Force vs. Displacement y = 120x + 1E-14

R2 = 1

0

10

20

30

40

50

60

70

80

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Displacement(Meters)

Fo

rce(

New

ton

s)

k =120 N/m

F vs. x again

Force vs. Displacement y = 120x + 1E-14

R2 = 1

0

10

20

30

40

50

60

70

80

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Displacement(Meters)

Fo

rce(

New

ton

s)

Area = Elastic Potential Energy

When you compress or stretcha spring, you must get something in return. In this case, the area under the curve is the energy.

Fx!

Energy is stored when a spring is stretched/compressed. This type ofenergy is classified as Elastic Potential Energy (EPE).

Elastic Potential EnergyThe graph of F vs. x for a spring will always producea linear line that has a positiveslope (IDEAL spring!). Thereforethe area under the curve will be a triangle.

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At =1

2bh

At =1

2Fx

At =1

2kx( )x[ ]

At =1

2kx 2 ⇒ EPE =

1

2kx 2

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At =1

2kx 2

Conservation of Energy in Springs

ExampleA slingshot consists of a light leather cup, containing a

stone, that is pulled back against 2 rubber bands. It takes a force of 30 N to stretch the bands 1.0 cm (a) What is the potential energy stored in the bands when a 50.0 g stone is placed in the cup and pulled back 0.20 m from the equilibrium position? (b) With what speed does it leave the slingshot?

v

vmvU

KUEEc

kkxUb

kkkxFa

s

sAB

s

s

22

2

)050.0(21

21

)

)20)(.(5.021)

)01.0(30) 3000 N/m

300 J

109.54 m/s

Springs are like Waves and Circles

The amplitude, A, of a wave is the same as the displacement ,x, of a spring. Both are in meters.

CREST

Trough

Equilibrium Line

Period, T, is the time for one revolution or in the case of springs the time for ONE COMPLETE oscillation (One crest and trough). Oscillations could also be called vibrations and cycles. In the wave above we have 1.75 cycles or waves or vibrations or oscillations.

Ts=sec/cycle. Let’s assume that the wave crosses the equilibrium line in one second intervals. T =3.5 seconds/1.75 cycles. T = 2 sec.

FrequencyThe FREQUENCY of a wave is the inverse of the

PERIOD. That means that the frequency is the #cycles per sec. The commonly used unit is HERTZ(HZ).

Tf

fT

Hzsccyc

fFrequency

scyc

sTPeriod

11

5.05.0sec5.3

75.1

seconds

cycles

275.1

5.3

cycles

seconds

PendulumsPendulums, like springs,

oscillate back and forth exhibiting simple harmonic behavior.

A shadow projector would show a pendulum moving in synchronization with a circle. Here, the angular amplitude is equal to the radius of a circle.

The Period of a Pendulum

g

lTpendulum 2

alityProportion ofConstant 4

4

2

2

22

g

lT

g

lT

ExampleA visitor to a lighthouse wishes to determine

the height of the tower. She ties a spool of thread to a small rock to make a simple pendulum, which she hangs down the center of a spiral staircase of the tower. The period of oscillation is 9.40 s. What is the height of the tower?

2

2

2

222

)141592.3(4

)8.9(4.9

4

4

2

gTl

g

lT

heightlg

lT

PP

P

L = Height = 21.93 m

Physics AppletTake a look at the following simulation:

UC Irvine Physics of Music Simple Harmonic Motion Applet Demonstrations