📖 Introduction

In nature, we observe many types of motion that repeat themselves after regular intervals of time. The swinging of a pendulum, the vibration of a guitar string, the rotation of the Earth, the oscillation of a mass attached to a spring—all these are examples of periodic motion.

🎯 Definition of Periodic Motion

Periodic Motion is defined as motion that repeats itself at regular intervals of time. The time taken for one complete cycle of motion is called the period, and the number of cycles completed per unit time is called the frequency.

Among all periodic motions, Simple Harmonic Motion (SHM) is the most fundamental and important. It serves as the basis for understanding more complex oscillatory systems in physics, engineering, and even biological systems (like heartbeats and breathing cycles).

2.1 Definition of Simple Harmonic Motion

🔹 What is Simple Harmonic Motion?

Simple Harmonic Motion (SHM) is a special type of periodic motion in which the restoring force acting on a particle is directly proportional to its displacement from the mean position and is always directed towards the mean position.

Figure 2.1: Mass-Spring System showing restoring force direction (always toward equilibrium)
Labels: \( x = -A \) (Compressed) | \( x = 0 \) (Equilibrium) | \( x = +A \) (Extended)

🔹 Mathematical Definition of SHM

A particle is said to execute simple harmonic motion if the restoring force \( F \) acting on it is given by:

Where:

• \( F \) = restoring force
• \( k \) = force constant (spring constant)
• \( x \) = displacement from mean position
• Negative sign indicates force is opposite to displacement

🔹 Characteristics of SHM

🔹 Equation of Simple Harmonic Motion

Let's derive the equation of motion for a particle executing SHM.

From Newton's second law: \( F = ma \)
From the definition of SHM: \( F = -kx \)
Equating these: \( ma = -kx \)

We know acceleration \( a = \frac{d^2x}{dt^2} \):

Rearranging:

Let \( \omega^2 = \frac{k}{m} \) (where \( \omega \) is angular frequency)

🔹 Solution of the SHM Equation

The general solution of the differential equation \( \frac{d^2x}{dt^2} + \omega^2x = 0 \) is:

Where:

• \( A \) = Amplitude (maximum displacement)
• \( \omega \) = Angular frequency (rad/s)
• \( t \) = Time
• \( \phi \) = Initial phase (phase constant)
• \( (\omega t + \phi) \) = Phase at time \( t \)

🔹 Related Quantities

Velocity in SHM:

Acceleration in SHM:

🔹 Important Relationships

2.2 Energy in Simple Harmonic Motion

A particle executing SHM possesses both kinetic energy (due to its motion) and potential energy (due to its displacement). Let's derive expressions for these energies.

🔹 Derivation of Kinetic Energy

Consider: A particle of mass \( m \) executing SHM with displacement \( x = A \sin(\omega t) \)

Velocity of the particle is: \( v = \frac{dx}{dt} = \frac{d}{dt}[A \sin(\omega t)] = A\omega \cos(\omega t) \)

Kinetic energy is: \( KE = \frac{1}{2}mv^2 = \frac{1}{2}m(A\omega \cos(\omega t))^2 \)

Using the identity \( \cos^2(\omega t) = 1 - \sin^2(\omega t) \):

Since \( x = A \sin(\omega t) \), we have \( \sin^2(\omega t) = x^2/A^2 \)

🔹 Derivation of Potential Energy

The potential energy is the work done by the restoring force in displacing the particle from mean position to position \( x \).

Restoring force: \( F = -kx = -m\omega^2x \)

Work done by external force (against restoring force) in moving from 0 to \( x \):

Integrating:

This work is stored as potential energy.

🔹 Total Energy in SHM

Total Energy \( E = KE + PE \)

Figure 2.2: Energy Distribution in SHM - KE, PE, and Total Energy vs Displacement
Graph Axes: Displacement (x) vs Energy
Curves: Total Energy (constant), PE = ½kx², KE = ½k(A²-x²)

🔹 Energy at Different Positions

📌 Important Conclusions

• Total energy remains constant throughout the motion (conservation of energy)
• At mean position: All energy is kinetic (\( KE = \text{maximum}, PE = 0 \))
• At extreme positions: All energy is potential (\( PE = \text{maximum}, KE = 0 \))
• Energy continuously converts between KE and PE
• Total energy is proportional to the square of amplitude (\( E \propto A^2 \))
• If amplitude doubles, total energy becomes four times

2.3 Period of Vertical Oscillation - Mass-Spring System

When a mass is suspended from a vertical spring, it undergoes simple harmonic motion. Let's derive the expression for its period.

Figure 2.3: Vertical Mass-Spring System showing equilibrium and displaced positions
Labels: Natural Length (\( L_0 \)) | Equilibrium Position (\( mx_0, mg, kx_0 \)) | Displaced Position (\( x, F = -kx \))

🔹 Derivation of Period Formula

Consider: A mass \( m \) suspended from a spring with spring constant \( k \).

Analysis:

At Equilibrium Position:
When the mass is at rest, the spring extends by \( x_0 \). At equilibrium:
Spring force (upward) = Weight (downward)
\[ kx_0 = mg \quad \text{... (i)} \]

When Displaced by Distance \( x \) Below Equilibrium:
Total extension of spring = \( x_0 + x \)
Upward spring force = \( k(x_0 + x) \)
Downward weight = \( mg \)

Net Force on Mass:
Net upward force \( F = k(x_0 + x) - mg \)
\( F = kx_0 + kx - mg \)
Using equation (i): \( kx_0 = mg \)
\( F = mg + kx - mg \)
\[ F = kx \]

Direction:
Since this force is upward (opposite to displacement), we write:
\[ F = -kx \]

This is the condition for SHM!
Comparing with \( F = ma \):
\( ma = -kx \)
\( a = -(k/m)x \)
\( a = -\omega^2x \), where \( \omega^2 = k/m \)

Angular frequency: \( \omega = \sqrt{k/m} \)

Period of oscillation: \( T = 2\pi/\omega = 2\pi/\sqrt{k/m} \)

Where:

• \( m \) = mass of the object (kg)
• \( k \) = spring constant (N/m)

Frequency: \[ f = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{k}{m}} \]

🔹 Alternative Form Using Extension

From equilibrium condition: \( kx_0 = mg \), we get \( k = mg/x_0 \)

Substituting in period formula:

This shows that the period can also be found from the equilibrium extension \( x_0 \).

📌 Important Points

• Period is independent of amplitude - whether you displace the mass by 1 cm or 5 cm, the period remains the same (for small oscillations)
• Period is independent of gravitational acceleration - The effect of gravity cancels out in the derivation
• Stiffer spring (larger \( k \)) → shorter period (faster oscillations)
• Heavier mass → longer period (slower oscillations)
• The equilibrium position becomes the mean position for oscillations

2.4 Angular Simple Harmonic Motion

Just as a mass can oscillate linearly in SHM, a body can also oscillate rotationally about a fixed axis. This is called Angular Simple Harmonic Motion or Torsional Oscillation.

🔹 Definition

A body is said to execute angular simple harmonic motion when the restoring torque acting on it is directly proportional to the angular displacement and is directed opposite to it.

Where:

• \( \tau \) = restoring torque
• \( C \) = torsional constant (restoring torque per unit angular displacement)
• \( \theta \) = angular displacement from equilibrium

Figure 2.4: Torsional Pendulum - Angular SHM showing restoring torque opposing angular displacement
Labels: Equilibrium (\( \theta = 0 \)) | Twisted CCW (\( \theta, \tau \)) | Twisted CW (\( \theta, \tau \))

🔹 Common Example: Torsional Pendulum

A disc or cylinder suspended by a wire from a fixed support. When the disc is rotated and released, the wire develops a restoring torque that makes the disc oscillate.

🔹 Derivation of Period

From the rotational analog of Newton's second law: \( \tau = I\alpha \)
where \( I \) = moment of inertia, \( \alpha \) = angular acceleration

From the definition of angular SHM: \( \tau = -C\theta \)

Equating these: \( I\alpha = -C\theta \)

Since \( \alpha = \frac{d^2\theta}{dt^2} \): \( I\frac{d^2\theta}{dt^2} = -C\theta \)

Rearranging: \( \frac{d^2\theta}{dt^2} = -\frac{C}{I}\theta \)

Comparing with standard SHM equation \( \frac{d^2\theta}{dt^2} = -\omega^2\theta \):
\( \omega^2 = \frac{C}{I} \rightarrow \omega = \sqrt{\frac{C}{I}} \)

Period of angular SHM: \( T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{I}{C}} \)

Where:

• \( I \) = moment of inertia of the body (kg m²)
• \( C \) = torsional constant of the wire (N m/rad)

🔹 Comparison: Linear vs Angular SHM

2.5 Period of Simple Pendulum

A simple pendulum is one of the most common examples of simple harmonic motion. It consists of a point mass (bob) suspended by a light, inextensible string from a fixed support.

Figure 2.5: Simple Pendulum - The tangential component \( mg \sin \theta \) provides the restoring force
Labels: m (Equilibrium) | m (Displaced) | L | θ | s = Lθ | mg | mg sin θ (Restoring) | mg cos θ | T (Tension)

🔹 Derivation of Period for Simple Pendulum

Consider: A simple pendulum with bob of mass \( m \), suspended by string of length \( L \), displaced by small angle \( \theta \) from vertical.

Forces Acting on the Bob:

• Weight \( mg \) acts vertically downward
• This can be resolved into:
  • Radial component: \( mg \cos \theta \) (balanced by tension)
  • Tangential component: \( mg \sin \theta \) (provides restoring force)

Restoring Force:
\( F = -mg \sin \theta \)
(Negative sign because force is opposite to displacement)

For Small Angles (\( \theta < 10^\circ \) or \( \theta < 0.17 \) rad):
\( \sin \theta \approx \theta \) (in radians)
Therefore: \( F = -mg\theta \)

Relating \( \theta \) to Arc Length:
Arc length \( s = L\theta \)
Therefore: \( \theta = s/L \)

Substituting in Force Equation:
\( F = -mg(s/L) = -(mg/L)s \)

Comparing with SHM Force \( F = -kx \):
We can identify: \( k = mg/L \)
This shows the motion is SHM with effective spring constant \( k = mg/L \)

Using Newton's Second Law:
\( F = ma \)
\( -(mg/L)s = m(\frac{d^2s}{dt^2}) \)
\( \frac{d^2s}{dt^2} = -(g/L)s \)

Angular Frequency:
Comparing with \( \frac{d^2s}{dt^2} = -\omega^2s \):
\( \omega^2 = g/L \rightarrow \omega = \sqrt{g/L} \)

Period:
\( T = 2\pi/\omega = 2\pi/\sqrt{g/L} \)

Where:

• \( L \) = length of the pendulum (m)
• \( g \) = acceleration due to gravity (m/s²)

Frequency: \[ f = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{g}{L}} \]

🔹 Important Characteristics

🔹 Effect of Various Factors

🔹 Applications

2.6 Damped Oscillation

In real-world oscillations, we always observe that the amplitude gradually decreases over time, and eventually the oscillation stops. This is called damped oscillation or damped motion.

🔹 What Causes Damping?

Damping is caused by dissipative forces such as:

• Air resistance - Opposes motion of oscillating body
• Friction - At contact surfaces or pivot points
• Internal friction - Within the material of springs or strings
• Viscous forces - In fluids (like oil dampers)

These forces convert mechanical energy into heat, causing the amplitude to decay.

🔹 Mathematical Description

For a damped harmonic oscillator, the damping force is usually proportional to velocity:

where \( b \) is the damping constant.

The equation of motion becomes:

Solution (for underdamped case):

Where:

• \( A_0 \) = initial amplitude
• \( \gamma = b/(2m) \) = damping coefficient
• \( \omega' = \sqrt{\omega_0^2 - \gamma^2} \) = damped angular frequency
• \( \omega_0 = \sqrt{k/m} \) = natural angular frequency (without damping)
• \( e^{-\gamma t} \) = exponential decay factor

Figure 2.6: Damped Oscillation - Amplitude decreases exponentially while frequency remains nearly constant
Graph Axes: Time (t) vs Displacement (x)
Envelope: \( A_0e^{-\gamma t} \) and \( -A_0e^{-\gamma t} \)

🔹 Types of Damping

Based on the amount of damping, we classify oscillations into three types:

1. Underdamped (Light Damping)
   Condition: \( \gamma < \omega_0 \) or \( b < 2\sqrt{km} \)
   System oscillates with gradually decreasing amplitude
   Most common in practical systems
   Examples: Pendulum in air, guitar string, car suspension
2. Critically Damped
   Condition: \( \gamma = \omega_0 \) or \( b = 2\sqrt{km} \)
   System returns to equilibrium as quickly as possible without oscillating
   No overshooting past equilibrium
   Desired in many engineering applications
   Examples: Door closers, shock absorbers, galvanometer
3. Overdamped (Heavy Damping)
   Condition: \( \gamma > \omega_0 \) or \( b > 2\sqrt{km} \)
   System returns to equilibrium very slowly without oscillating
   Takes longer than critical damping
   Example: Moving through thick oil or honey

Figure 2.7: Comparison of Underdamped, Critically Damped, and Overdamped Oscillations

🔹 Quality Factor (Q-factor)

The quality factor measures how "good" an oscillator is, i.e., how lightly damped it is.

Physical Meaning:

• High Q (\( Q \gg 1 \)): Lightly damped, oscillations persist for many cycles
• Low Q (\( Q < 1 \)): Heavily damped, oscillations die out quickly

🔹 Practical Applications

2.7 Forced Oscillation and Resonance

🔹 Forced Oscillation

When an external periodic force is continuously applied to an oscillating system, the system performs forced oscillation.

Characteristics of Forced Oscillations:

• The system oscillates at the frequency of the driving force (not its natural frequency)
• Initial transients die out, then steady-state oscillation is established
• Amplitude depends on:
  • Amplitude of driving force
  • Frequency of driving force
  • Natural frequency of the system
  • Damping in the system
• Energy is continuously supplied by the external force to compensate for damping losses

🔹 Equation of Forced Oscillation

Consider an external periodic force: \( F(t) = F_0 \cos(\omega t) \)

The equation of motion becomes:

Where:

• \( \omega \) = angular frequency of driving force
• \( F_0 \) = amplitude of driving force

Steady-state solution:

Amplitude:

🔹 Resonance

Resonance is a phenomenon in which the amplitude of forced oscillation becomes maximum when the frequency of the driving force matches (or is very close to) the natural frequency of the system.

At resonance, even a small periodic force can produce oscillations of very large amplitude because the driving force is always in phase with the velocity, continuously adding energy to the system.

Figure 2.8: Resonance Curves - Amplitude vs Driving Frequency for different damping levels
Graph Axes: Driving Frequency (ω) vs Amplitude
Curves: Small damping (high Q), Medium damping, Large damping (low Q)
Peak: Maximum at \( \omega = \omega_0 \)

🔹 Examples of Resonance

Beneficial Resonance:

• Musical Instruments:
  • Guitar: Sound box resonates with string vibrations, amplifying sound
  • Violin: Body resonates to produce rich tones
  • Organ pipes: Air column resonates at specific frequencies
• Radio Tuning: LC circuit in radio resonates at frequency of desired station. Filters out other frequencies. Enables selective reception.
• MRI Machines: Nuclear magnetic resonance. Atoms resonate at specific frequencies in magnetic field. Used for medical imaging.
• Microwave Ovens: Microwaves at resonance frequency of water molecules. Causes water molecules to vibrate and heat up.

Destructive Resonance:

• Tacoma Narrows Bridge (1940): Wind caused forced oscillations at bridge's natural frequency. Amplitude grew until bridge collapsed. Famous example of destructive resonance.
• Earthquake Damage: If earthquake frequency matches building's natural frequency. Resonance amplifies oscillations. Can cause building collapse. Modern buildings designed to avoid this.
• Soldiers Breaking Step on Bridges: Marching in step creates periodic force. If frequency matches bridge natural frequency, resonance occurs. That's why soldiers are ordered to break step when crossing bridges.
• Opera Singer Breaking Glass: Singing at glass's natural frequency. Resonance causes large amplitude vibrations. Glass shatters when stress exceeds strength.

🔹 Key Differences: Free vs Forced vs Damped Oscillations

2.8 Numerical Problems and Conceptual Questions

📝 Solved Numerical Problems

Given:

Solution:

Step 1: Find angular frequency
\[ \omega = 2\pi f = 2\pi \times 2 = 4\pi \text{ rad/s} \approx 12.57 \text{ rad/s} \]

(a) Maximum velocity:
\[ v_{max} = A\omega = 0.1 \times 4\pi = 0.4\pi \text{ m/s} \approx \boxed{1.26 \text{ m/s}} \]

(b) Maximum acceleration:
\[ a_{max} = A\omega^2 = 0.1 \times (4\pi)^2 = 1.6\pi^2 \text{ m/s}^2 \approx \boxed{15.79 \text{ m/s}^2} \]

(c) At \( x = 6 \) cm = 0.06 m:
Velocity: \[ v = \pm\omega\sqrt{A^2 - x^2} = \pm 4\pi\sqrt{0.01 - 0.0036} = \pm 0.32\pi \text{ m/s} \approx \boxed{\pm 1.01 \text{ m/s}} \]
Acceleration: \[ a = -\omega^2 x = -(4\pi)^2 \times 0.06 \approx \boxed{-9.47 \text{ m/s}^2} \]

Given:

Solution:

(a) Period of oscillation:
\[ T = 2\pi\sqrt{\frac{m}{k}} = 2\pi\sqrt{\frac{0.5}{50}} = 2\pi \times 0.1 = 0.2\pi \text{ s} \approx \boxed{0.628 \text{ s}} \]

(b) Maximum velocity (at equilibrium):
\[ \omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{50}{0.5}} = 10 \text{ rad/s} \]
\[ v_{max} = A\omega = 0.08 \times 10 = \boxed{0.8 \text{ m/s}} \]

(c) Total energy:
\[ E = \frac{1}{2}kA^2 = \frac{1}{2} \times 50 \times (0.08)^2 = \boxed{0.16 \text{ J}} \]

(d) Velocity at \( x = 5 \) cm = 0.05 m:
\[ v = \pm\omega\sqrt{A^2 - x^2} = \pm 10\sqrt{0.0064 - 0.0025} \approx \boxed{\pm 0.625 \text{ m/s}} \]

Given:

Solution:

(a) Period on Earth:
\[ T_1 = 2\pi\sqrt{\frac{L}{g_1}} = 2\pi\sqrt{\frac{1}{9.8}} \approx \boxed{2.01 \text{ s}} \]

(b) Period on Moon:
\[ T_2 = 2\pi\sqrt{\frac{L}{g_2}} = 2\pi\sqrt{\frac{1}{1.63}} \approx \boxed{4.92 \text{ s}} \]
Ratio: \[ \frac{T_2}{T_1} = \sqrt{\frac{g_1}{g_2}} = \sqrt{\frac{9.8}{1.63}} \approx 2.45 \]
Period on moon is about 2.45 times longer

(c) Effect of length change:
If L increases by 21%, new length \( L' = 1.21L \)
\[ \frac{T'}{T} = \sqrt{\frac{L'}{L}} = \sqrt{1.21} = 1.1 \]
Period increases by factor of 1.1
Percentage increase = (1.1 - 1) × 100% = 10%

Given:

Solution:

(a) Spring constant:
\[ T = 2\pi\sqrt{\frac{m}{k}} \Rightarrow 2 = 2\pi\sqrt{\frac{0.2}{k}} \]
\[ k = 0.2\pi^2 \approx \boxed{1.97 \text{ N/m}} \]

(b) Total energy:
\[ E = \frac{1}{2}kA^2 = \frac{1}{2} \times 1.97 \times (0.05)^2 \approx \boxed{2.46 \text{ mJ}} \]

(c) At \( x = 3 \) cm = 0.03 m:
PE: \[ PE = \frac{1}{2}kx^2 \approx \boxed{0.887 \text{ mJ}} \]
KE: \[ KE = E - PE \approx \boxed{1.57 \text{ mJ}} \]

(d) Position where KE = PE:
When KE = PE, each is half the total energy:
\[ \frac{1}{2}kx^2 = \frac{1}{2}\left(\frac{1}{2}kA^2\right) \Rightarrow x = \frac{A}{\sqrt{2}} \]
\[ x = \frac{0.05}{\sqrt{2}} \approx \boxed{3.54 \text{ cm}} \]

❓ Conceptual Questions and Answers

📋 Summary of Important Formulas

Simple Harmonic Motion

Energy in SHM

Spring-Mass System

Simple Pendulum

Angular SHM (Torsional Pendulum)

Damped Oscillation

Resonance

🎯 Exam Tips and Important Points

High-Weightage Topics (Most Asked in NEB)

✍️ Derivation Writing Tips

• Start with diagrams: Always draw a labeled diagram for pendulum, spring-mass systems
• State assumptions: "For small angles," "massless string," "no friction"
• Show all steps: Don't skip algebraic steps in derivations
• Box final formulas: Highlight \( T = 2\pi\sqrt{L/g} \), \( E = \frac{1}{2}kA^2 \), etc.
• Use proper notation: Distinguish between \( \omega \) (angular frequency) and \( \omega_0 \) (natural frequency)

📊 Graph Skills

• Be able to sketch \( x \) vs \( t \), \( v \) vs \( t \), \( a \) vs \( t \) graphs for SHM
• Sketch KE, PE, and total energy vs displacement
• Draw resonance curves showing effect of damping
• Draw damped oscillation showing exponential decay envelope

🔢 Numerical Problem Strategy

• Write all given data with proper units
• Identify what formula to use
• If needed, find intermediate quantities (like \( \omega \) from \( T \))
• Substitute and calculate carefully
• Check if answer makes physical sense
• Write final answer with correct units and significant figures

🧠 Key Relationships to Remember

📚 Before the Exam

• Practice all four major derivations multiple times
• Solve at least 15-20 numerical problems
• Review past 5 years' NEB question papers
• Memorize all formulas in the summary section
• Understand physical meanings, not just formulas
• Practice sketching graphs without looking at notes

🏁 Conclusion

Periodic motion, particularly Simple Harmonic Motion, is one of the most fundamental concepts in physics. Its principles apply not just to springs and pendulums, but to countless phenomena in nature and technology—from atomic vibrations to electrical circuits, from sound waves to quantum mechanics.

Mastering this chapter requires:

• Conceptual understanding of what makes motion "simple harmonic"
• Mathematical proficiency in derivations and problem-solving
• Physical intuition about energy transformations and resonance
• Graphical skills for representing oscillatory motion
• Real-world connections to applications and examples

Practice regularly, understand deeply, and success will follow!
🍀 Best Wishes for Your NEB Examination!