Oscillations summary

8. OSCILLATIONS 8.1 Periodic Motion

Definitions ➊ Displacement x is the

distance of the oscillating object from its equilibrium position at any instant

➋ Amplitude x0 of the oscillation is the maximum displacement from the equilibrium position.

➌ Period T of an oscillating object is the time it takes to complete one cycle of oscillation.

➍ Frequency f of the oscillations is the number of complete cycles per second made by the oscillating object.

f =number of complete cycles

time taken

➎ Angular frequency ω is

defined as ω =2πT

= 2πf

8.2 Kinematics of Simple Harmonic Motion

Oscillating spring-mass system that begins from the equilibrium position

Oscillating spring-mass system that begins from the maximum displacement

Kinematics graphs (x-t, v-t, a-t) of system in SHM

maximum speed vmax =ωx0

maximum acceleration | amax | =ω 2x0

8.3 Dynamics of Simple Harmonic Motion

Simple harmonic motion is defined as the motion of a body whose acceleration is directly proportional to its displacement from a fixed point (equilibrium position) and is always directed towards that fixed point. (a = −ω 2x )

The negative sign indicates that its acceleration a is always in the opposite direction to its displacement x.

• From Newton’s 2nd law, the resultant force exerted on the body in SHM is Fresultant = −mω 2x

o The resultant force on the body is always in the opposite direction to its displacement.

8.4 Simple Harmonic Motion and Circular Motion

Consider the shadow at position P, x = r sinωt , i.e., motion of the shadow is simple harmonic

x = x0 sinωt

v = v0 cosωt

a = −ω 2x0 sinω t

v = ±ω x02 − x2

a = −ω 2x

8.5 Energy of Body in Simple Harmonic Motion

E

K=

12

mv 2 =12

mω 2x0

2 cos2 ω t E

P=

12

kx2 =12

mω 2x02 sin2 ω t

E

T=

12

mω 2x02

Variation of energies with displacement

The variations of the energies of the mass with displacement are

EK =12mv 2 =

12mω 2 x0

2 − x2( )EP =

12kx2 =

12mω 2x2

ET =12mωx0

2

8.6 Damping Energy is lost continuously due to resistive forces. The total energy and the amplitude decrease.

Exponential decrease of amplitude with light damping

Displacement with time for different degree of damping

• The job of a car suspension is to

o maximise the contact between the tyres and the road surface (as a result leading to better handling in terms of steering stability),

o ensure the comfort of the passengers.

• Roads have subtle imperfections that can interact with the wheels of a car.

o The wheel experiences a vertical force as it passes over a bump.

o Without an intervening structure, the entire car moves in the same direction and the wheels lose contact with the road, causing problem in handling.

o Under gravity, the wheels will slam back into the road surface.

• To minimise such effects, a system that will absorb the energy of the vertically accelerated wheel will allow the car frame to be undisturbed.

• A car suspension system includes springs and shock absorbers.

o When the wheel hits a bump, the vertical force on the wheel compresses the spring, keeping the wheel to remain in contact with the road.

o When the spring expands, the viscous oil in the shock absorber slows its motion, enabling the spring to smoothly return to its equilibrium length without oscillating.

o A good suspension system is one in which the damping is slightly under critical damping as this results in a comfortable ride for the passengers along a bumpy road.

8.7 Interaction of oscillating system with external periodic force

• The periodic external force provides a means of supplying energy to the system.

• The system will response to driving force as follow:

o Initially its oscillation is complicated that includes components of its natural frequency and the driving force frequency.

o Given sufficient time, steady state is reached and the system oscillates in the same frequency as the driving force.

When the frequency of the external periodic force is equal to the natural frequency of the oscillating system, the amplitude of the oscillation is large.

System response to periodic driving force

• As the degree of damping increases,

o amplitude of oscillation at all frequencies is reduced

o frequency at maximum amplitude shifts gradually towards lower frequencies

o peak becomes flatter.