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Simple Harmonic Motion (SHM) Notes | Concepts, Derivations, Numericals & Applications

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is one of the most important topics in mechanics and wave motion. It helps us understand how objects vibrate, oscillate, and move back and forth about a fixed position. Many physical systems around us such as a pendulum, vibrating strings, atoms in solids, and even electrical circuits show simple harmonic motion.

In this chapter, we study the concept of SHM in a clear and logical way, starting from basic ideas of oscillatory motion and gradually moving towards equations, physical interpretation, and applications.

Oscillatory and Periodic Motion

An object is said to perform oscillatory motion when it moves to and fro about a fixed point repeatedly. This fixed point is called the mean position or equilibrium position.

If this oscillatory motion repeats itself after equal intervals of time, then the motion is called periodic motion.

Examples:

  • A swinging pendulum of a clock
  • A mass attached to a spring
  • Vibrations of a tuning fork
  • Motion of atoms in a solid lattice

Not all oscillatory motions are simple harmonic motions. SHM is a special type of oscillatory motion with specific conditions.

Definition of Simple Harmonic Motion

A particle is said to perform Simple Harmonic Motion if the restoring force acting on it is:

  • Always directed towards the mean position
  • Directly proportional to the displacement from the mean position

Mathematically, restoring force is given by:

F ∝ −x

The negative sign indicates that the force acts opposite to the displacement.

Since force is proportional to acceleration (Newton’s second law), we can also write:

a ∝ −x

This is the most important condition for SHM.

Restoring Force and Its Physical Meaning

The restoring force is the force that tries to bring the particle back to its mean position whenever it is displaced.

If the particle is displaced to the right of the mean position, the restoring force acts to the left. If displaced to the left, the force acts to the right.

This force is responsible for the continuous oscillatory motion.

Example:

Consider a mass attached to a spring. When the mass is pulled downward, the spring stretches and produces a force upward, trying to restore the mass to its original position.

Mathematical Equation of SHM

Let the displacement of a particle from mean position be x. If the restoring force is proportional to displacement:

F = −kx

where k is a constant called the force constant or spring constant.

Using Newton’s second law:

m d²x/dt² = −kx

Rewriting:

d²x/dt² + (k/m)x = 0

This is the standard differential equation of simple harmonic motion.

Displacement in SHM

The displacement of a particle executing SHM varies with time according to a sinusoidal function.

The general expression for displacement is:

x = A sin(ωt + φ)

or

x = A cos(ωt + φ)

where:

  • A = Amplitude (maximum displacement)
  • ω = Angular frequency
  • t = Time
  • φ = Phase constant

The sine or cosine form depends on the initial conditions of motion.

Amplitude of SHM

Amplitude is the maximum displacement of the particle from its mean position.

It determines how far the particle moves on either side of the equilibrium position.

Larger amplitude means greater energy of oscillation, but it does not affect the time period of SHM.

Time Period and Frequency

Time Period (T) is the time taken by the particle to complete one full oscillation.

Frequency (f) is the number of oscillations completed per second.

They are related by:

f = 1/T

Angular frequency is related to time period as:

ω = 2π/T

Velocity in Simple Harmonic Motion

Velocity of a particle in SHM is not constant. It depends on its position.

Velocity is maximum at the mean position and zero at the extreme positions.

The expression for velocity is:

v = ω √(A² − x²)

This shows that velocity decreases as the particle moves away from the mean position.

Acceleration in SHM

Acceleration in SHM is always directed towards the mean position.

It is maximum at the extreme positions and zero at the mean position.

Acceleration is given by:

a = −ω²x

This equation confirms the defining condition of SHM.

Energy in Simple Harmonic Motion

In SHM, energy continuously changes between kinetic energy and potential energy.

  • At mean position: kinetic energy is maximum, potential energy is minimum
  • At extreme positions: potential energy is maximum, kinetic energy is zero

Total mechanical energy remains constant (if no damping).

Examples of Simple Harmonic Motion

  • Mass-spring system
  • Simple pendulum (for small angles)
  • Vibrations of molecules
  • AC current in electrical circuits

Practice Questions

  1. Define simple harmonic motion.
  2. What is the restoring force in SHM?
  3. Why is acceleration maximum at extreme positions?
  4. State the condition for SHM.
  5. What is amplitude?
  6. What is angular frequency?
  7. Write the equation of SHM.
  8. How does velocity vary with displacement?
  9. What happens to energy during SHM?
  10. Is circular motion an SHM?
  11. What is phase?
  12. Does amplitude affect time period?
  13. Why is SHM called periodic motion?
  14. Give two real-life examples of SHM.
  15. What is the relation between frequency and time period?

Answers (Selected)

  1. SHM is a motion in which restoring force is proportional to displacement and directed towards mean position.
  2. Restoring force brings the particle back to equilibrium.
  3. Because displacement is maximum at extreme positions.
  4. a ∝ −x
  5. Amplitude is maximum displacement from mean position.
  6. Angular frequency is the rate of change of phase.
  7. x = A sin(ωt + φ)
  8. Velocity decreases as displacement increases.
  9. Total energy remains constant.
  10. No, but projection of circular motion is SHM.
  11. Phase defines the state of oscillation.
  12. No, time period is independent of amplitude.
  13. Because motion repeats after equal time intervals.
  14. Spring-mass system, pendulum.
  15. f = 1/T

Phase, Phase Difference and Phase Constant

In simple harmonic motion, the concept of phase is used to describe the state of motion of a particle at any given instant of time.

The quantity (ωt + φ) in the displacement equation x = A sin(ωt + φ) is called the phase of the motion.

Phase tells us:

  • Where the particle is
  • In which direction it is moving
  • What its velocity and acceleration are at that instant

The constant φ is called the phase constant. It depends on the initial conditions of motion, such as initial displacement and initial velocity.

If two particles executing SHM have a difference in phase, they do not reach the same state of motion at the same time. This difference is called phase difference.

Graphical Representation of SHM

Graphs play a very important role in understanding SHM clearly.

Displacement–Time Graph

The displacement–time graph of SHM is a sine or cosine curve.

  • Maximum displacement occurs at extreme positions
  • Zero displacement occurs at mean position
  • The motion is smooth and continuous

Velocity–Time Graph

Velocity is maximum at the mean position and zero at extreme positions. The velocity–time graph is also sinusoidal but shifted in phase.

Acceleration–Time Graph

Acceleration is maximum at extreme positions and zero at mean position. It is always opposite in direction to displacement.

These graphs clearly show that displacement, velocity, and acceleration are all periodic but not in the same phase.

SHM as Projection of Uniform Circular Motion

One of the best ways to understand SHM is to consider it as the projection of uniform circular motion on a diameter of the circle.

Consider a particle moving in a circle with constant angular speed. The projection of this particle on any diameter performs simple harmonic motion.

This explains why sine and cosine functions appear naturally in SHM equations.

This concept helps in visualizing:

  • Phase
  • Angular frequency
  • Velocity variation

Simple Harmonic Motion of a Spring–Mass System

Consider a mass m attached to a spring of spring constant k. When the mass is displaced and released, it oscillates about its equilibrium position.

According to Hooke’s law:

F = −kx

Using Newton’s second law:

m d²x/dt² = −kx

Comparing with SHM equation, angular frequency is:

ω = √(k/m)

Hence, time period is:

T = 2π√(m/k)

Important observations:

  • Time period depends on mass and spring constant
  • It is independent of amplitude
  • Heavier mass oscillates slowly

Simple Pendulum and SHM

A simple pendulum consists of a small heavy bob suspended by a light, inextensible string. When displaced slightly and released, it performs oscillatory motion.

For small angular displacements, the restoring force is proportional to displacement, making the motion approximately simple harmonic.

The time period of a simple pendulum is:

T = 2π√(l/g)

where:

  • l = length of pendulum
  • g = acceleration due to gravity

Time period does not depend on:

  • Mass of the bob
  • Amplitude (for small angles)

Energy Distribution in SHM (Detailed)

The total energy of a particle executing SHM is constant if no energy is lost.

Potential Energy

Potential energy at displacement x is:

PE = ½ kx²

Kinetic Energy

Kinetic energy is:

KE = ½ mω² (A² − x²)

Total Energy

Total energy remains constant:

E = ½ kA²

This continuous exchange between kinetic and potential energy explains oscillatory motion.

Effect of Damping on SHM (Introduction)

In real systems, energy is lost due to friction and air resistance. This causes damping.

Damped oscillations gradually lose amplitude with time.

Types of damping:

  • Light damping
  • Critical damping
  • Heavy damping

Though damped motion is not pure SHM, its study is important in real-life applications.

Solved Numerical Examples

Example 1: A particle performs SHM with amplitude 0.1 m and angular frequency 10 rad/s. Find maximum velocity.

Solution:

vmax = ωA = 10 × 0.1 = 1 m/s

Example 2: Find time period of a spring-mass system with m = 1 kg and k = 400 N/m.

Solution:

T = 2π√(m/k) = 2π√(1/400) = 0.314 s

Practice Questions

  1. Explain the concept of phase in SHM.
  2. What is phase difference?
  3. Draw and explain displacement-time graph.
  4. Why velocity is maximum at mean position?
  5. Explain SHM as projection of circular motion.
  6. Derive time period of spring-mass system.
  7. What assumptions are made in simple pendulum?
  8. Why pendulum motion is SHM only for small angles?
  9. Explain energy exchange in SHM.
  10. What is damping?
  11. Does damping change frequency?
  12. Write expression for total energy.
  13. What factors affect time period of spring?
  14. What happens to amplitude in damped motion?
  15. Is real SHM possible?

Answers (Selected)

  1. Phase defines the state of motion of a particle at any instant.
  2. Difference in phase between two oscillations.
  3. It is a sine or cosine curve showing periodic motion.
  4. Because restoring force is zero there.
  5. Projection of circular motion on diameter gives SHM.
  6. T = 2π√(m/k).
  7. Small angle, light string, point mass.
  8. Because sinθ ≈ θ only for small θ.
  9. KE converts to PE and vice versa.
  10. Loss of energy due to resistance.
  11. Yes, slightly.
  12. E = ½kA².
  13. Mass and spring constant.
  14. Amplitude decreases with time.
  15. No, ideal SHM is theoretical.

Resonance in Simple Harmonic Motion

Resonance is a very important phenomenon related to simple harmonic motion. It occurs when an external periodic force acts on a system and the frequency of the external force becomes equal to the natural frequency of the system.

When resonance occurs:

  • The amplitude of oscillation becomes maximum
  • The system absorbs maximum energy from the external force
  • Oscillations become very large

Resonance is useful in many applications, but it can also be dangerous if not controlled properly.

Examples of Resonance:

  • Tuning of radio and television receivers
  • Musical instruments like guitar and violin
  • Breaking of glass by sound waves of same frequency
  • Collapse of bridges due to rhythmic forces

Forced Oscillations

When a body is made to oscillate by applying an external periodic force, the motion is called forced oscillation.

In forced oscillations:

  • The frequency of motion depends on the external force
  • Amplitude depends on damping and driving frequency

If the frequency of the applied force matches the natural frequency, resonance takes place and oscillations grow in amplitude.

Natural Frequency of a System

The natural frequency of a system is the frequency with which it oscillates when disturbed slightly and then left free.

For different systems:

  • Spring–mass system: depends on mass and spring constant
  • Simple pendulum: depends on length and gravity

Natural frequency plays a key role in resonance and stability of structures.

Comparison Between SHM and Other Motions

SHM vs Uniform Circular Motion

  • SHM is linear motion; circular motion is rotational
  • SHM is projection of circular motion on diameter
  • Both have same angular frequency

SHM vs Periodic Motion

  • All SHM is periodic
  • All periodic motion is not SHM
  • SHM requires restoring force proportional to displacement

Real-Life Applications of Simple Harmonic Motion

Simple harmonic motion is not just a theoretical concept. It has many practical applications in daily life and technology.

  • Clocks and watches use pendulum or balance wheel oscillations
  • Seismographs use oscillatory systems to detect earthquakes
  • Vehicle suspension systems work on oscillatory principles
  • Atoms in solids vibrate in SHM about equilibrium positions
  • AC generators and electrical oscillators are based on SHM concepts

Limitations of Simple Harmonic Motion

Though SHM is very useful, it is an idealized concept and has limitations:

  • Perfect SHM does not exist in nature
  • Energy losses due to friction are unavoidable
  • Large amplitude motion often deviates from SHM

Despite these limitations, SHM provides an excellent approximation for many physical systems.

Important Points for Examinations

  • Always mention restoring force proportional to displacement
  • Remember velocity and acceleration variations
  • Time period is independent of amplitude
  • Energy remains constant in ideal SHM
  • Use small-angle approximation for pendulum problems

Numerical Problems (Practice)

Problem 1: A particle executes SHM with time period 2 s. Find angular frequency.

Answer: ω = 2π/T = π rad/s

Problem 2: Find total energy of a spring system with k = 100 N/m and A = 0.2 m.

Answer: E = ½kA² = 2 J

Practice Questions

  1. What is resonance?
  2. Define forced oscillations.
  3. What is natural frequency?
  4. Why resonance is dangerous?
  5. Give two examples of resonance.
  6. How does damping affect oscillations?
  7. Compare SHM and periodic motion.
  8. Why SHM is idealized?
  9. What happens at resonance?
  10. Define forced frequency.
  11. Explain SHM in atoms.
  12. What is small-angle approximation?
  13. Why bridges collapse due to resonance?
  14. Write two applications of SHM.
  15. Is resonance always useful?
  16. Does frequency change in forced oscillation?
  17. What limits amplitude growth?
  18. Is damping always harmful?
  19. Define equilibrium position.
  20. What controls natural frequency?
  21. Give two limitations of SHM.
  22. What is driving force?
  23. Why oscillations die out?
  24. What is steady-state oscillation?
  25. Can SHM exist without restoring force?

Answers (Selected)

  1. Resonance occurs when driving frequency equals natural frequency.
  2. Oscillations produced by an external periodic force.
  3. Frequency of free oscillation of a system.
  4. Because amplitude becomes very large.
  5. Radio tuning, musical instruments.
  6. Damping reduces amplitude.
  7. All SHM is periodic but not vice versa.
  8. Because friction cannot be eliminated.
  9. Maximum energy transfer occurs.
  10. Frequency of applied force.
  11. Atoms vibrate about mean position.
  12. sinθ ≈ θ for small θ.
  13. External force matches natural frequency.
  14. Clocks, seismographs.
  15. No, it can be harmful.
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