Simple Harmonic Motion

Oscillatory SystemsPeriodic motionElasticityInertiaInterchange of energiesExamples:

Mass on helical spring Cantilever Simple pendulum Vertical rod floating in liquid

Characteristics of SHMOccurs in many systemsIsochronous oscillationPossesses springiness (elasticity) to store P.E.Possesses inertia to store K.E.Period of vibration depends on elastic and in

ertia factorsConstant total energy

Conditions for performing SHM acceleration is always directed towards a fixed

point acceleration varies directly as its distance from

the fixed point i.e. in linear motion

in angular motion

x x

Figures

Terms of Reference

AmplitudePeriodFrequency

unit: Hertz

Rotating Vector Model

As particle N describes uniform circular motion, its projection point P performs simple harmonic motion

computer simulation

Kinematics of SHM

Displacement: x = a cos t

Velocity:

Acceleration:

22 xax

xx 2

Graphs of SHM

General Relation between x & t

) sin( t a xWhere is the phase angle

Experimental Verification of the relationship x2-x

• Experimental set up:

Procedure:Displacement-time graph

Experimental Verification of the relationship x2-x

x-t and tx graphs

Graphxx

x

Solving Problems on SHM

Assume displacement x from the mean position

Draw a diagram showing all forcesApply Newton’s second law with appropriate

sign conventionShow that The constant of proportionality = 2

Period T = 2/

x x

Mass on a Spiral Spring

Motion on a smooth horizontal surface

xm

kx

k

mT 2

Graph

Motion under gravity

xm

kx

k

mT 2

Floating Tube in a Liquid (1)

Floating Tube in a Liquid (2)

xm

gAx

x

h

g

g

hT 2

Effects of viscosity of liquid:

• causes damping

• takes away K.E. from the oscillating tube

Liquid Oscillating in a U-tube

xh

gx

g

hT 2

Simple Pendulum

sl

gs

g

lT 2

Arrangement of Springs (1)

K

mT 2

K = equivalent spring constant

Springs connected in series:

Arrangement of Springs (2)

nkkkK

1......

111

21

Springs connected in parallel:

Arrangement of Springs (3)

nkkkK .....21

Energy in SHM (1)

22

2

1.. xmEP

)(2

1.. 222 xamEK

Energy in SHM (2)

22

2

1am

energyTotal

constant

Phase Difference

Phase leading Phase lagging In phase Out of phase (Antiphase)

Superposition of Two Harmonic Variations

The amplitude and phase of the resultant is obtained by the parallelogram law

)cos()cos(cos tctbta

Experimental Determination of g (1)

a) A simple pendulum:

gT

22 4

• Plot a graph of T2 vs l

• Slope = g

24

b) A loaded spring:i) The static experiment: Plot the extension-load graph

k

g

m

es 1Slope:

Experimental Determination of g (2)

Experimental Determination of g (3)

ii) The dynamic experiment:

measure the period of oscillation for different loads

ks

2

2

4

22

1

4g

s

s

EXAMPLES