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ALL INVOLVE

SIMPLE HARMONIC MOTION

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A body will undergo SIMPLE HARMONIC MOTION when the force that tries to restore the object to its REST POSITION is PROPORTIONAL TO the DISPLACEMENT of the object.

A pendulum and a mass on a spring both undergo this type of motion which can be described by a SINE WAVE or a COSINE

WAVE depending upon the start position.

Displacement x

Time t

+ A

- A

ftAx 2cos

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SHM is a particle motion with an acceleration (a) that is directly proportional to the particle’s displacement (x) from a fixed point (rest point), and this acceleration always points towards the fixed point.

Rest point

x

AAx

xa or xa 2

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Displacement x

time

Amplitude ( A ): The maximum distance that an object moves from its rest position. x = A and x = - A .

+ A

- A

Period ( T ): The time that it takes to execute one complete cycle of its motion.Units seconds,

T

Frequency ( f ): The number or oscillations the object completes per unit time.Units Hz = s-1 .

Tf 1

Angular Frequency ( ω ): The frequency in radians per second, 2π per cycle.

Tf 22

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θr

Arc length s

IN RADIANSrs

FOR A FULL CIRCLE RADIANS 22

rr

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EQUATION OF SHM xa 2

x

a

Acceleration – Displacement graph xmy

Gradient = - ω2

+ A

- A

MAXIMUM ACCELERATION = ± ω2 A = ( 2πf )2 A

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EQUATION FOR VARIATION OF VELOCITY WITH DISPLACEMENT

+x-xx

v

222 xAfv

Maximum velocity, v = ± 2 π f A

Maximum Kinetic Energy, EK = ½ mv2 = ½ m ( 2 π f A )2

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Displacement x

Velocity v

Acceleration a

ftAx 2cos

t

t

t

txv

Velocity = gradient of displacement- time graph

Maximum velocity in the middle of the motion

ZERO velocity at the end of the motion

tva

Acceleration = gradient of velocity - time graph

Maximum acceleration at the end of the motion – where the restoring force is greatest!ZERO acceleration in the middle of the motion!

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THE PENDULUMThe period, T, is the time for one complete cycle.

glT 2

l

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MASS ON A SPRING

M

F = Mg = ke

e

Stretch & Release

A

kmT 2

geT 2k = the spring constant in N m-1

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http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=44

The link below enables you to look at the factors that influence

the period of a pendulum and the period of a mass on a spring

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ENERGY IN SHM

potential

EP

Kinetic

EK

Potential

EP

PENDULUM SPRING

MM

M

potential

kineticpotential

If damping is negligible, the total energy will be constant

ETOTAL = Ep + EK

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Maximum velocity, v = ± 2 π f A

222 xAfvvelocity Energy in SHM

Maximum Kinetic Energy, EK = ½ m ( 2 π f A )2 = 2π2 m f2 A2

Hence TOTAL ENERGY = 2π2 m f2 A2

m

x = 0

F

For a spring,

m

x = A

energy stored = ½ Fx = ½ kx2, [as F=kx]

= MAXIMUM POTENTIAL ENERGY!

MAXIMUM POTENTIAL ENERGY = TOTAL ENERGY = ½ kA2

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Energy in SHM

Energy Change with POSITION

= kinetic = potential = TOTAL ENERGY, E

Energy Change with TIME

x-A +A0

energy E

energy

timeTT/2

N.B. Both the kinetic and the potential energies reach a maximum TWICE in on cycle.

E

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time

DAMPING

DISPLACEMENT

INITIAL AMPLITUDE

THE AMPLITUDE DECAYS EXPONENTIALLY WITH TIME