Table of Contents
Oscillations, Simple Harmonic Motion and Resonance
1. Introduction
Oscillations are repeated movements of an object about an equilibrium position. They occur in mechanical systems, electrical circuits, sound waves, musical instruments, clocks and many natural phenomena.
Simple harmonic motion (SHM) is a special type of periodic oscillation in which the restoring force or acceleration is directly proportional to displacement and directed towards the equilibrium position.
Resonance occurs when a system is driven by an external periodic force at or near one of its natural frequencies, potentially producing a large amplitude of oscillation.
2. Oscillatory Motion
2.1 Definition
Oscillatory motion is the repeated movement of an object to and fro about a fixed or equilibrium position.
Examples include:
- A swinging pendulum
- A vibrating tuning fork
- A mass attached to a spring
- The vibrating strings of musical instruments
- Vibrations in mechanical machinery
2.2 Equilibrium Position
The equilibrium position is the position at which the net force acting on the system is zero.
During oscillation, the object moves on either side of this position.
2.3 Periodic Motion
Periodic motion is motion that repeats itself at equal intervals of time.
All ideal simple harmonic motion is periodic, but not every periodic motion is simple harmonic.
Examples of periodic motion include the revolution of the Earth around the Sun and the oscillation of a pendulum under suitable conditions.
2.4 Amplitude
Amplitude is the maximum displacement of an oscillating object from its equilibrium position.
It is commonly represented by (A).
Amplitude is measured in metres for mechanical displacement.
2.5 Time Period
The time period is the time required to complete one full oscillation.
It is represented by (T) and measured in seconds.
2.6 Frequency
Frequency is the number of complete oscillations per second.
[
f=\frac{1}{T}
]
The SI unit of frequency is hertz (Hz).
One hertz represents one complete cycle per second.
2.7 Angular Frequency
Angular frequency is the rate of change of phase in an oscillatory motion.
[
\omega=2\pi f=\frac{2\pi}{T}
]
Its SI unit is radians per second (rad/s).
3. Simple Harmonic Motion
3.1 Definition
Simple harmonic motion is a periodic motion in which acceleration is directly proportional to displacement from the equilibrium position and is always directed towards that position.
[
a=-\omega^2x
]
Where:
- (a) = acceleration
- (\omega) = angular frequency
- (x) = displacement from equilibrium
The negative sign indicates that acceleration acts opposite to displacement.
3.2 Restoring Force
In SHM, the restoring force is directly proportional to displacement and acts towards the equilibrium position.
[
F=-kx
]
Where:
- (F) = restoring force
- (k) = force constant
- (x) = displacement
For a mass-spring system, the force constant is the spring constant.
3.3 Characteristics of SHM
- Motion repeats periodically.
- The object oscillates about an equilibrium position.
- Acceleration is proportional to displacement.
- Acceleration is directed towards equilibrium.
- Velocity is maximum at the equilibrium position.
- Speed is zero at the extreme positions.
- Acceleration is zero at equilibrium.
- Acceleration has maximum magnitude at the extreme positions.
3.4 Displacement Equation
The displacement of an ideal oscillator can be expressed as:
[
x=A\cos(\omega t+\phi)
]
Where:
- (A) = amplitude
- (\omega) = angular frequency
- (t) = time
- (\phi) = initial phase constant
A sine function may also be used depending on the initial conditions.
3.5 Velocity in SHM
Velocity is the rate of change of displacement.
For SHM:
[
v=-A\omega\sin(\omega t+\phi)
]
The maximum speed is:
[
v_{max}=A\omega
]
Velocity is maximum in magnitude at the equilibrium position and zero at the extreme positions.
3.6 Acceleration in SHM
Acceleration is the rate of change of velocity.
[
a=-\omega^2x
]
Its maximum magnitude is:
[
a_{max}=\omega^2A
]
Acceleration is zero at equilibrium and maximum in magnitude at the extreme positions.
4. Energy in Simple Harmonic Motion
The total mechanical energy of an ideal SHM system remains constant when no energy is lost through friction or other dissipative effects.
4.1 Kinetic Energy
Kinetic energy is associated with the motion of the oscillating object.
[
KE=\frac{1}{2}mv^2
]
It is maximum at the equilibrium position because speed is maximum there.
It is zero at the extreme positions because instantaneous velocity is zero.
4.2 Potential Energy
For an ideal mass-spring system:
[
PE=\frac{1}{2}kx^2
]
Potential energy is maximum at the extreme positions and zero at equilibrium when the equilibrium position is taken as the reference.
4.3 Total Energy
The total energy of an ideal spring-mass oscillator is:
[
E=\frac{1}{2}kA^2
]
It remains constant throughout the oscillation.
At equilibrium, energy is entirely kinetic for the ideal spring-mass model.
At the extreme positions, energy is entirely elastic potential energy.
5. Mass-Spring System
A mass attached to an ideal spring oscillates when displaced from its equilibrium position and released.
The restoring force is:
[
F=-kx
]
Using Newton’s second law:
[
ma=-kx
]
[
a=-\frac{k}{m}x
]
Comparing with the SHM equation:
[
a=-\omega^2x
]
Therefore:
[
\omega=\sqrt{\frac{k}{m}}
]
The time period is:
[
T=2\pi\sqrt{\frac{m}{k}}
]
The frequency is:
[
f=\frac{1}{2\pi}\sqrt{\frac{k}{m}}
]
Important conclusions:
- Increasing mass increases the time period.
- Increasing spring stiffness decreases the time period.
- For an ideal linear spring, the period does not depend on amplitude.
6. Simple Pendulum
6.1 Definition
A simple pendulum consists of a small mass, called a bob, suspended from a fixed support by a light, inextensible string.
When displaced slightly and released, it oscillates about its equilibrium position.
6.2 Time Period of a Simple Pendulum
For small angular oscillations:
[
T=2\pi\sqrt{\frac{L}{g}}
]
Where:
- (T) = time period
- (L) = length of the pendulum
- (g) = acceleration due to gravity
The formula assumes a small angular displacement and negligible air resistance and friction.
6.3 Factors Affecting the Time Period
Length
The time period increases with the square root of the length.
A longer pendulum takes more time to complete one oscillation.
Gravity
The time period decreases as gravitational acceleration increases.
A pendulum generally has a longer period on the Moon than on Earth.
Mass
For an ideal simple pendulum under the small-angle approximation, the period is independent of the mass of the bob.
Amplitude
For small oscillations, the period is approximately independent of amplitude. At larger amplitudes, the period increases slightly.
6.4 Applications of Pendulums
- Traditional pendulum clocks
- Demonstration of periodic motion
- Measurement of gravitational acceleration
- Seismological instruments
- Experimental physics
7. Damped Oscillations
Damped oscillations are oscillations whose amplitude gradually decreases because energy is dissipated through resistance, friction or other mechanisms.
Examples include:
- A pendulum gradually coming to rest.
- A vibrating spring losing energy through air resistance.
- A vehicle suspension system reducing vibrations.
7.1 Types of Damping
Underdamping
The system continues to oscillate, but its amplitude gradually decreases.
Critical Damping
The system returns to equilibrium in the shortest time without oscillating.
Overdamping
The system returns to equilibrium without oscillating, but more slowly than in critical damping.
Damping is important in vehicle shock absorbers, building vibration control and mechanical instruments.
8. Forced Oscillations
Forced oscillations occur when an external periodic force acts on an oscillating system.
The system oscillates at the frequency of the applied driving force in its steady-state response.
The amplitude depends on the driving frequency, damping and strength of the applied force.
Examples include:
- A child being pushed periodically on a swing.
- A machine subjected to periodic vibration.
- A speaker cone driven by an alternating electrical signal.
9. Resonance
9.1 Definition
Resonance is the phenomenon in which the amplitude of forced oscillation becomes large when the driving frequency approaches a natural frequency of the system.
The natural frequency is the frequency at which a system oscillates freely after being disturbed.
The response is influenced by damping. Greater damping generally reduces the maximum resonant amplitude.
9.2 Examples of Resonance
- A swing moves with increasing amplitude when pushed at suitable intervals.
- Musical instruments use resonance to strengthen sound.
- An appropriately tuned radio circuit responds strongly to a selected frequency.
- Buildings and bridges may experience dangerous vibrations under periodic forces.
- Mechanical machines may develop excessive vibration at resonant operating speeds.
9.3 Beneficial Applications of Resonance
- Musical instruments
- Radio and communication circuits
- Resonant sensors
- Medical imaging systems using magnetic resonance
- Frequency-selective electronic circuits
9.4 Harmful Effects of Resonance
- Excessive vibration in machinery
- Structural damage to bridges and buildings
- Fatigue failure of components
- Uncomfortable vibration in vehicles
- Damage to equipment and instruments
Engineers reduce harmful resonance through damping, changes in stiffness or mass, structural redesign and avoiding operating frequencies near dangerous resonances.
10. Natural Frequency
Natural frequency is the frequency at which a system oscillates freely when displaced from equilibrium and then released.
For an ideal mass-spring system:
[
f_n=\frac{1}{2\pi}\sqrt{\frac{k}{m}}
]
For an ideal simple pendulum under the small-angle approximation:
[
f_n=\frac{1}{2\pi}\sqrt{\frac{g}{L}}
]
Natural frequency depends on the physical properties and configuration of the system.
11. Worked Examples
Example 1: Frequency and Time Period
An oscillator completes 20 oscillations in 10 seconds. Calculate its frequency and time period.
[
f=\frac{20}{10}=2,Hz
]
[
T=\frac{1}{f}=\frac{1}{2}=0.5,s
]
Answer: Frequency = 2 Hz; time period = 0.5 s.
Example 2: Angular Frequency
An oscillator has a frequency of 5 Hz. Calculate its angular frequency.
[
\omega=2\pi f
]
[
\omega=2\pi\times5=10\pi
]
[
\omega\approx31.42,rad/s
]
Answer: Approximately 31.42 rad/s.
Example 3: Time Period of a Spring-Mass System
A 1 kg mass is attached to a spring with a spring constant of 100 N/m. Calculate its time period.
[
T=2\pi\sqrt{\frac{m}{k}}
]
[
T=2\pi\sqrt{\frac{1}{100}}
]
[
T=2\pi(0.1)
]
[
T\approx0.628,s
]
Answer: Approximately 0.628 seconds.
Example 4: Time Period of a Simple Pendulum
Calculate the time period of a simple pendulum of length 1 m. Take (g=9.8,m/s^2).
[
T=2\pi\sqrt{\frac{L}{g}}
]
[
T=2\pi\sqrt{\frac{1}{9.8}}
]
[
T\approx2.01,s
]
Answer: Approximately 2.01 seconds.
Example 5: Maximum Speed in SHM
An oscillator has an amplitude of 0.1 m and an angular frequency of 10 rad/s. Calculate its maximum speed.
[
v_{max}=A\omega
]
[
v_{max}=0.1\times10=1,m/s
]
Answer: 1 m/s.
Example 6: Maximum Acceleration
An oscillator has an amplitude of 0.05 m and an angular frequency of 20 rad/s. Calculate its maximum acceleration.
[
a_{max}=\omega^2A
]
[
a_{max}=20^2\times0.05
]
[
a_{max}=20,m/s^2
]
Answer: 20 m/s².
12. Important Examination Distinctions
| Concept | Meaning |
|---|---|
| Oscillation | Repeated to-and-fro motion about equilibrium |
| Periodic motion | Motion that repeats at equal time intervals |
| Amplitude | Maximum displacement from equilibrium |
| Time period | Time for one complete cycle |
| Frequency | Number of cycles per second |
| Angular frequency | Rate of change of phase |
| SHM | Oscillation with restoring acceleration proportional to negative displacement |
| Natural frequency | Frequency of free oscillation |
| Damped oscillation | Oscillation with decreasing amplitude |
| Forced oscillation | Oscillation maintained by an external periodic force |
| Resonance | Large response near a natural frequency |
| Critical damping | Fastest return to equilibrium without oscillation |
13. 36 Key Takeaways for Exams
- Oscillatory motion is repeated to-and-fro motion about an equilibrium position.
- Periodic motion repeats at equal intervals of time.
- Simple harmonic motion is a special type of periodic motion.
- The equilibrium position is where the net force is zero.
- Amplitude is the maximum displacement from equilibrium.
- Time period is the time required for one complete oscillation.
- Frequency is the number of oscillations per second.
- The SI unit of frequency is hertz.
- Frequency is the reciprocal of time period.
- Angular frequency is given by (\omega=2\pi f).
- In SHM, acceleration is proportional to displacement and directed towards equilibrium.
- The SHM acceleration equation is (a=-\omega^2x).
- The restoring force in an ideal spring is (F=-kx).
- The negative sign represents the restoring direction.
- Displacement in SHM can be expressed using a sine or cosine function.
- Velocity is maximum at the equilibrium position.
- Velocity is zero at the extreme positions.
- Acceleration is zero at equilibrium.
- Acceleration has maximum magnitude at the extreme positions.
- Kinetic energy is maximum at equilibrium in ideal SHM.
- Potential energy is maximum at the extreme positions.
- Total mechanical energy remains constant in ideal undamped SHM.
- The time period of a mass-spring system is (2\pi\sqrt{m/k}).
- Increasing the mass of a spring-mass system increases its time period.
- Increasing the spring constant decreases its time period.
- The time period of a simple pendulum is (2\pi\sqrt{L/g}) for small oscillations.
- The simple pendulum’s period is independent of bob mass under ideal small-angle conditions.
- A longer simple pendulum has a longer time period.
- A pendulum’s time period increases when gravitational acceleration decreases.
- Damped oscillations have gradually decreasing amplitude.
- Underdamped systems continue to oscillate while losing amplitude.
- Critically damped systems return to equilibrium rapidly without oscillating.
- Forced oscillations are produced by an external periodic force.
- Resonance occurs when driving frequency approaches a natural frequency.
- Damping reduces the amplitude of resonant oscillations.
- Resonance is useful in musical and electronic systems but can be dangerous in structures and machinery.