CH.19.

OSCILLATIONS
18.1 FREE AND FORCED OSCILLATION

Free Oscillation :

❑ Constant amplitude and period

❑ without any external force
❑ Ideally, free oscillation does not undergo damping
18.1 FREE AND FORCED OSCILLATION

Damped Oscillation
❑ Due to damping, the amplitude of oscillation reduces with time.
❑ Damping/ external forces like friction, air resistance and other resistive forces.
18.1 FREE AND FORCED OSCILLATION

Forced Oscillation

❑ influenced by an external periodic force, it is called forced oscillation.

❑ Oscillation experiences damping
❑ But Here, the amplitude of oscillation remain constant,
18.2 OBSERVING OSCILLATION

❑ pendulum

❑ Spring mass system ❑ A ruler vibrating freely

Oscillations :
a repetitive back-and-forth or up and down motion
The restoring force is the force that brings the object back to its equilibrium position
Towards EP

(+ ) max
displacement

(- ) max
displacement
18.3 DESCRIBING THE OSCILLATION

❑ Object accelerates as it moves toward equilibrium position

❑ Moving faster at the centre
❑ Object decelerates as it moves towards the centre of the oscillation
18.3 DESCRIBING THE OSCILLATION

❑ Object accelerates as it moves toward equilibrium position

❑ Moving faster at the centre
❑ Object decelerates as it moves towards the centre of the oscillation
18.3 DESCRIBING THE OSCILLATION

❑ Maximum displacement
❑ Maximum acceleration
❑ Minimum speed (reverses its direction)

❑ Minimum displacement
❑ Minimum acceleration
❑ Maximum speed (reverses its direction)
18.3 DESCRIBING THE OSCILLATION

❑ Amplitude ❑ period ❑ Frequency

Maximum displacement of a particle Time taken to make one complete Number of oscillation per unit time
from its equilibrium position oscillation
18.3 DESCRIBING THE OSCILLATION

❑ phase ❑ Phase difference

The point that an oscillating particle The difference in the phases of two oscillating particles
has reached within the complete measured in degrees or radians
cycle of an oscillation
18.4 SIMPLE HARMONIC MOTION

Motion of

❑ Constant amplitude
❑ Acceleration is proportional and oppositely directed to the
displacement of the body from a position of equilibrium

a ∝ -x
2. SIMPLE HARMONIC MOTION

The restoring force is the force that brings the object back to its equilibrium position
❑ When displacement maximum ❑ When displacement positive (upwards)
➢ Max acceleration ➢ Acceleration and force will be negative (downwards)
➢ Max force
 𝑦 𝑥
3. EQUATIONS OF SHM sin 𝜃 = cos 𝜃 =
𝑅 𝑅
a. Equations of displacement (X)

y = A sinωt
Maximum displacement = A

Y = Displacement at time t (m) A = Amplitude (m) ω = angular speed (rad/s) 𝜔 = 2𝜋𝑓

 𝑦 𝑥
3. EQUATIONS OF SHM sin 𝜃 = cos 𝜃 =
𝑅 𝑅
a. Equations of displacement (X)

y = A sinωt
Maximum displacement = A

Y = Displacement at time t (m) A = Amplitude (m) ω = angular speed (rad/s) 𝜔 = 2𝜋𝑓

3. EQUATIONS OF SHM
b. Equations of velocity (v)

𝑣 = ±𝜔 𝐴2 − 𝑦 2 vmax = ωA
V = velocity at time t (m)
v = ωA cosωt
𝑣 = ±𝜔 𝐴2 − 𝑦 2
3. EQUATIONS OF SHM
c. Equations of acceleration (v)

Graph of acceleration a against

displacement x

Grgradient of the grapf = -ω2

a = − ω2A sinωt a = acceleration at time t (m)

amax = -ω2A
4. ENERGY CHANGE IN S.H.M

MECHANICAL ENERGY IN S.H.M

• ME = PE + KE • ME = KEMAX = PEMAX

1 2 1
KEMAX = PE + KE KEMAX = 𝑚𝑣𝑚𝑎𝑥 = 𝑚𝜔2 𝐴2
2 2

1 1
mω2 A2 = PE + mω2 (A2 −y 2 )
2 2
1
PE = mω2 y 2
2
5. PENDULUM AND S.H.M
Period of Oscillation

Restoring Force: − m. g. sin θ

m. a = − m. g. sin θ
𝑦
− mg sin θ = −mω2 y ; sin θ ≅
𝐿
𝑦
g = ω2 y
𝐿
𝑔
= ω2
𝐿
L
T = 2π
g
Joan has two pendula, one has a length of 1 meter, and the other one is longer. She sets them both
swinging at the same time. After the 1 meter pendulum has completed 12 oscillations, the longer one
has only completed 11. How long is the longer pendulum?
6. SPRING AND S.H.M
Period of Oscillation
You find a spring in the laboratory.
Restoring Force: −𝑘𝑦 When you hang 100 grams at the end of the spring it
m. a = −ky stretches 10 cm. You pull the 100 gram mass 6 cm from
its equilibrium position and let it go at t = 0. Find an
−m. ω2 y = −ky equation for the position of the mass as a function of
−m. ω2 y = −ky time t.

k
ω = m

m
𝑇 = 2π k
The length of nylon rope from which a mountain climber is suspended has a
force constant of 1.40 × 104N/m

(a) What is the frequency at which he bounces, given his mass and the mass of
his equipment are 90.0 kg?
7. DAMPED OSCILLATIONS
Amplitude of the oscillation decrease as the friction transfer energy away

Damping Force :
FD = −bv b = damping coefficient

෍ 𝐹 = m. a
−kx − bv = m. a
m. a + bv + kx = 0
𝑑2 𝑥 𝑑𝑥
m. 𝑑𝑡 2 + b 𝑑𝑡 + kx = 0
7. DAMPED OSCILLATIONS
Solution for the equation : 𝑥 = 𝑒 𝝀𝑡
𝑑𝑥
= λ𝑒 λ𝑡
𝑑𝑡

𝑑2 𝑥
= λ2 𝑒 λ𝑡
𝑑𝑡 2

mλ2 𝑒 λ𝑡 + bλ𝑒 λ𝑡 + k𝑒 λ𝑡 = 0
𝑏 𝑘
λ2 𝑒 λ𝑡 + 𝑚 λ𝑒 λ𝑡 + 𝑚 𝑒 λ𝑡 = 0
𝑏 𝑘
λ2 + 𝑚 λ + 𝑚 = 0

𝑑2 𝑥 𝑑𝑥 𝑏 𝑏2 𝑘 −𝑏 ± 𝑏2 − 4𝑘𝑚
m 𝑑𝑡 2 + b 𝑑𝑡 + kx = 0 −𝑚 ± − 4 =
𝑚2 𝑚 2𝑚
λ=
2
 Critically damping
Heavy Damping
−𝑏
𝑥= 𝐴𝑒 2𝑚𝑡
Critically damping
−𝑏
𝑥= 𝐴𝑒 2𝑚𝑡 cos 𝜔𝑡

Light Damping

Critical damping is minimum amount of damping required to return the oscillator to its equilibrium
position without oscillating
Light Damping : oscillate with gradually decreasing amplitude
Critical Damping : return to rest at its equilibrium position in the shortest possible time without oscillating
Heavy Damping : take a long time to return to its equilibrium position without oscillating
8. RESONANCE

Frequency of the driving force is equal to the

natural frequency of oscillating system.

The system absorbs the maximum energy from the

driver and has a maximum amplitude
Resonance can be a problem, but it can also be very useful

Resonance be a problem Resonance be very useful

8. RESONANCE
8. RESONANCE
8. RESONANCE
Tacoma Bridge Millennium Bridge
8. RESONANCE

This magnetic resonance imaging (MRI)

picture resonate. shows a man, a woman
and a nine-year-old child
In resonance, energy is transferred from the driver to the resonating
system more efficiently than when resonance does not occur.

The following statements apply to any system in

resonance:
■■ Its natural frequency is equal to the frequency of the driver.
■■ Its amplitude is maximum.
■■ It absorbs the greatest possible energy from the driver
8. RESONANCE
8. RESONANCE
upthrust and weight

Upthrust is greater than the weight

A, ρ, g and M are constant

either acceleration ∝ – displacement or acceleration ∝

displacement and (– sign indicates) a and x in opposite
directions
ω = 2 π / 1.3 = 4.8 rad s–1