Oscillations

INTRODUCTION TO OSCILLATIONS

**Oscillatory motion** refers to any motion in which an object moves repeatedly back and forth about a mean (equilibrium) position. Unlike rectilinear or projectile motion, oscillations are repetitive in nature. Common examples include a swing, a pendulum, a boat on waves, and pistons in engines.

**Key distinction**: Every oscillatory motion is periodic, but not every periodic motion is oscillatory. For example, uniform circular motion is periodic but not oscillatory because there is no fixed equilibrium position about which the object oscillates.

**Real-life applications**:

Vibrating strings in musical instruments (sitar, guitar, violin)

Vibrations of air molecules enabling sound propagation

Atomic vibrations in solids determining temperature

AC voltage oscillations in electrical circuits

Drumheads and speaker diaphragms

The study of oscillations is fundamental because the concepts form the basis for understanding waves and many physical phenomena.

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PERIODIC AND OSCILLATORY MOTIONS

Definition and Characteristics

**Periodic motion** is any motion that repeats itself at regular intervals of time. The motion follows a recognizable pattern that returns to its initial state after a fixed duration.

Examples shown in typical periodic motions include:

An insect climbing a ramp and falling repeatedly

A child climbing and descending stairs repeatedly

A ball bouncing between palm and ground (parabolic paths given by Newton's equations)

**Important distinction**: In oscillatory motion, the body has an **equilibrium position** within its path. When displaced from this position, a restoring force acts to bring it back, creating oscillations. For instance, a ball in a bowl experiences equilibrium at the bottom; when displaced, gravity and normal force combine to create a restoring force.

**Oscillations vs. Vibrations**: There is no physical difference. The terms differ only in frequency convention:

**Oscillation**: Used when frequency is low (e.g., tree branch oscillations)

**Vibration**: Used when frequency is high (e.g., musical string vibrations)

Period and Frequency

**Period (T)**: The smallest interval of time after which a motion repeats itself exactly. SI unit: **second (s)**.

**Frequency (ν)**: The number of complete oscillations or repetitions occurring per unit time. SI unit: **hertz (Hz) = s⁻¹**.

**Mathematical relation**:

$$ν = \frac{1}{T}$$

Therefore: $$T = \frac{1}{ν}$$

**Important notes**:

1 Hz = 1 oscillation per second

Frequency need not be an integer

Period and frequency are reciprocals

**Example 13.1**: A human heart beats 75 times in 60 seconds.

Frequency: ν = 75/(60 s) = 1.25 Hz = 1.25 s⁻¹

Period: T = 1/1.25 = 0.8 s

Displacement

In oscillatory motion, **displacement** refers to the deviation of any physical quantity from a reference value, measured as a function of time.

**General definition**: Displacement is not restricted to position alone. It can represent:

Position x from equilibrium (block on spring)

Angular displacement θ from vertical (pendulum)

Voltage across capacitor (AC circuit)

Pressure variations (sound wave propagation)

Electric and magnetic field variations (light waves)

**Key properties**:

Displacement can be positive or negative

For periodic motion, displacement is a periodic function of time

Most commonly represented by sinusoidal or cosinusoidal functions

Mathematical Representation of Periodic Functions

**Fundamental periodic functions**:

$$f(t) = A \cos ωt \quad \text{or} \quad f(t) = A \sin ωt$$

where A is amplitude and ω is angular frequency.

**Period relationship**: If f(t) = A cos ωt has period T, then:

$$ωT = 2π$$

$$T = \frac{2π}{ω}$$

**General periodic function**:

$$f(t) = A \sin ωt + B \cos ωt$$

This can be rewritten as:

$$f(t) = D \sin(ωt + φ)$$

where:

$$D = \sqrt{A^2 + B^2} \quad \text{and} \quad \tan φ = \frac{B}{A}$$

**Fourier's Principle**: Any periodic function can be expressed as a superposition of sine and cosine functions of different periods with suitable coefficients. This is fundamental to the analysis of complex periodic phenomena.

**Example 13.2 Analysis**:

(i) **sin ωt + cos ωt** = √2 sin(ωt + π/4)

Periodic with period T = 2π/ω

(ii) **sin ωt + cos 2ωt + sin 4ωt**

sin ωt has period 2π/ω

cos 2ωt has period π/ω

sin 4ωt has period π/(2ω)

Overall period is LCM = 2π/ω (the largest period)

(iii) **e^(-ωt)**

Non-periodic; decreases monotonically to zero

Does not represent physical oscillation

(iv) **log(ωt)**

Non-periodic; increases monotonically to infinity

Cannot represent physical displacement

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SIMPLE HARMONIC MOTION

Definition and Mathematical Form

**Simple Harmonic Motion (SHM)** is a special type of oscillatory motion in which the displacement of the particle from equilibrium varies sinusoidally with time.

**Mathematical equation**:

$$x(t) = A \cos(ωt + φ)$$

where:

**x(t)**: displacement as function of time

**A**: amplitude (maximum displacement magnitude)

**ω**: angular frequency (rad/s)

**φ**: phase constant or initial phase (rad)

**ωt + φ**: instantaneous phase (rad)

**Alternative form**: The same motion can be expressed as:

$$x(t) = A \sin(ωt + φ') \quad \text{where } φ' = φ - \frac{π}{2}$$

Both representations are equivalent; choice depends on initial conditions.

Amplitude (A)

**Definition**: The magnitude of maximum displacement from equilibrium position.

**Key characteristics**:

Always taken as positive by convention

Fixed for a given SHM

Displacement varies between +A and –A

Different SHMs can have same ω and φ but different amplitudes

**Physical meaning**: Amplitude determines the total extent of oscillation and relates to the energy of the system.

Phase and Phase Constant

**Instantaneous phase (ωt + φ)**: The argument of the cosine function at time t, determines the state of motion (position and velocity) at that instant.

**Phase constant (φ)**: The value of phase at t = 0.

$$φ = \arccos\left(\frac{x(0)}{A}\right)$$

If displacement at t = 0 is known, φ can be determined.

**Phase difference**: SHMs with same A and ω but different φ are out of phase. Time lag between two oscillations is related to phase difference Δφ by:

$$Δt = \frac{Δφ}{ω}$$

Angular Frequency (ω)

**Definition**: The rate of change of phase with time, measured in radians per second.

**Relationship with period**:

$$ω = \frac{2π}{T}$$

Therefore:

$$T = \frac{2π}{ω}$$

**Relationship with frequency**:

$$ω = 2πν$$

**Physical interpretation**: Angular frequency measures how quickly the oscillation completes its cycles. It is 2π times the ordinary frequency.

**SI unit**: rad/s or simply s⁻¹

**Example 13.3**: Determine which functions represent SHM.

(1) **sin ωt – cos ωt** = √2 sin(ωt – π/4)

This is SHM with amplitude A = √2, period T = 2π/ω, and phase constant φ = –π/4

(2) **sin² ωt** = ½ – ½cos 2ωt

Periodic but NOT SHM (oscillates about ½, not zero; contains cos 2ωt term)

Period T = π/ω

Graphical Representation

**Displacement-time graph characteristics**:

Smooth sinusoidal curve

Oscillates between +A and –A

Completes one full cycle in period T

Returns to same position and velocity after time T

**Key positions in one period**:

t = 0: x = A cos φ

t = T/4: x = A cos(ωT/4 + φ) = A cos(π/2 + φ)

t = T/2: x = A cos(πt + φ) = –A cos φ

t = 3T/4: x = A cos(3π/2 + φ)

t = T: x = A cos(2π + φ) = A cos φ (returns to initial value)

**Velocity behavior**: Maximum speed occurs at equilibrium (x = 0); zero speed at extreme positions (x = ±A).

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SIMPLE HARMONIC MOTION AND UNIFORM CIRCULAR MOTION

Connection Between Circular Motion and SHM

There is a fundamental geometric relationship between uniform circular motion and SHM: **the projection of uniform circular motion onto any diameter yields simple harmonic motion**.

Geometric Derivation

**Setup**: Consider a particle P moving with constant angular speed ω on a circle of radius A in the horizontal plane.

**At time t = 0**: The position vector OP makes angle φ with the positive x-axis.

**At time t**: The particle has rotated through additional angle ωt, so OP makes angle (ωt + φ) with the x-axis.

**Projection on x-axis**: The x-coordinate of point P is:

$$x(t) = A \cos(ωt + φ)$$

This is precisely the equation for SHM.

**Projection on y-axis**: Similarly, the y-coordinate is:

$$y(t) = A \sin(ωt + φ)$$

This is SHM with phase difference π/2 from the x-projection.

Physical Interpretation

When observing a particle in uniform circular motion from the side (edge-on viewing), the particle appears to oscillate back and forth along a diameter. This visual observation corresponds to viewing only the projection of the circular motion.

**Reference circle and reference particle**:

**Reference particle**: The particle moving uniformly on the circle

**Reference circle**: The circle on which it moves

These are mathematical constructs helping visualize SHM as projection of circular motion

Important Distinction

Despite this geometric connection, the forces are completely different:

**Circular motion**: Requires centripetal force directed toward center (perpendicular to velocity)

**Linear SHM**: Requires restoring force directed toward equilibrium, proportional to displacement

**Example 13.4**: Obtaining SHM equations from circular motion.

For a particle with initial angular position π/4, rotating with period T = 4 s:

Position vector makes angle: (π/4) + 2πt/T

x-projection: $$x(t) = A \cos\left(\frac{2πt}{4} + \frac{π}{4}\right) = A \cos\left(\frac{πt}{2} + \frac{π}{4}\right)$$

This is SHM with amplitude A, period 4 s, and initial phase π/4 rad.

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VELOCITY AND ACCELERATION IN SIMPLE HARMONIC MOTION

Velocity in SHM

**Definition**: Velocity is the time derivative of displacement.

Starting with displacement:

$$x(t) = A \cos(ωt + φ)$$

**Velocity equation**:

$$v(t) = \frac{dx}{dt} = -Aω \sin(ωt + φ)$$

This can also be written as:

$$v(t) = Aω \cos\left(ωt + φ + \frac{π}{2}\right)$$

**Maximum velocity** (amplitude of velocity):

$$v_{max} = Aω$$

This occurs when sin(ωt + φ) = ±1, i.e., when x = 0 (at equilibrium position).

**Minimum velocity**: v = 0 when x = ±A (at extreme positions).

**Phase relationship**: Velocity leads displacement by phase π/2 radians.

Acceleration in SHM

**Definition**: Acceleration is the time derivative of velocity.

$$a(t) = \frac{dv}{dt} = \frac{d^2x}{dt^2}$$

$$a(t) = -Aω^2 \cos(ωt + φ) = -ω^2 x(t)$$

**Critical result**:

$$a = -ω^2 x$$

This shows that **acceleration is directly proportional to displacement and always directed opposite to it** (restoring in nature).

**Maximum acceleration**:

$$a_{max} = Aω^2$$

This occurs at extreme positions where x = ±A.

**Zero acceleration**: a = 0 when x = 0 (at equilibrium position).

**Phase relationship**: Acceleration is 180° (π radians) out of phase with displacement.

Important Velocity-Displacement Relationship

Eliminating time from velocity and displacement equations:

From $v = -Aω \sin(ωt + φ)$ and $x = A \cos(ωt + φ)$

$$v^2 = A^2ω^2 \sin^2(ωt + φ) = A^2ω^2[1 - \cos^2(ωt + φ)]$$

$$v^2 = A^2ω^2 - ω^2x^2$$

$$v = ±ω\sqrt{A^2 - x^2}$$

**Physical interpretation**:

At equilibrium (x = 0): v = ±Aω (maximum)

At extreme positions (x = ±A): v = 0 (minimum)

This relationship is independent of time and useful for solving problems without knowing time explicitly.

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FORCE LAW FOR SIMPLE HARMONIC MOTION

Restoring Force

**Definition**: The force that acts to restore the particle toward its equilibrium position is called the restoring force.

From Newton's second law and the acceleration in SHM:

$$F = ma = m \cdot (-ω^2x)$$

$$F = -mω^2 x$$

**Key characteristic**: The force is proportional to displacement and directed opposite to it.

Condition for SHM

For a particle to execute SHM, the restoring force must satisfy:

$$F = -kx$$

where k is the force constant (stiffness).

Comparing with F = ma = –mω²x:

$$k = mω^2$$

Therefore:

$$ω = \sqrt{\frac{k}{m}}$$

Spring-Mass System

**Setup**: A mass m attached to a spring with spring constant k oscillating on a frictionless surface.

**Hooke's Law**: The spring force is F = –kx, where x is displacement from equilibrium.

**Angular frequency**:

$$ω = \sqrt{\frac{k}{m}}$$

**Period and frequency**:

$$T = 2π\sqrt{\frac{m}{k}}$$

$$ν = \frac{1}{2π}\sqrt{\frac{k}{m}}$$

**Physical insight**:

Stiffer springs (larger k) → higher frequency

Larger mass (larger m) → lower frequency

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ENERGY IN SIMPLE HARMONIC MOTION

Kinetic Energy

For a particle in SHM with velocity $v = -Aω \sin(ωt + φ)$:

**Kinetic energy**:

$$KE = \frac{1}{2}mv^2 = \frac{1}{2}m A^2 ω^2 \sin^2(ωt + φ)$$

**Maximum kinetic energy** (at x = 0):

$$KE_{max} = \frac{1}{2}mA^2ω^2$$

**Minimum kinetic energy** (at x = ±A):

$$KE_{min} = 0$$

Potential Energy

The potential energy associated with restoring force F = –kx = –mω²x is:

$$PE = \frac{1}{2}kx^2 = \frac{1}{2}mω^2 x^2 = \frac{1}{2}mω^2 A^2 \cos^2(ωt + φ)$$

**Maximum potential energy** (at x = ±A):

$$PE_{max} = \frac{1}{2}kA^2 = \frac{1}{2}mω^2A^2$$

**Minimum potential energy** (at x = 0):

$$PE_{min} = 0$$

Total Mechanical Energy

**Total energy**:

$$E = KE + PE$$

$$E = \frac{1}{2}mω^2A^2\sin^2(ωt + φ) + \frac{1}{2}mω^2A^2\cos^2(ωt + φ)$$

$$E = \frac{1}{2}mω^2A^2[\sin^2(ωt + φ) + \cos^2(ωt + φ)]$$

$$E = \frac{1}{2}mω^2A^2 = \frac{1}{2}kA^2$$

**Critical result**: Total mechanical energy in SHM is **constant** and proportional to the square of amplitude.

Energy Conservation

**Key principle**: In undamped SHM, total mechanical energy is conserved. Energy continuously converts between kinetic and potential forms:

When particle passes through equilibrium: KE = maximum, PE = minimum

When particle is at extreme positions: KE = minimum, PE = maximum

At any other point: both KE and PE are non-zero

**Energy diagram**: The total energy line is horizontal; KE and PE curves are mirror images about E/2.

Average Energy

Over one complete period:

**Average KE**: $\langle KE \rangle = \frac{E}{2} = \frac{1}{4}mω^2A^2$

**Average PE**: $\langle PE \rangle = \frac{E}{2} = \frac{1}{4}mω^2A^2$

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THE SIMPLE PENDULUM

Definition and Setup

A **simple pendulum** consists of:

A small mass (bob) of mass m

A light inextensible string of length L

Suspended from a fixed point

Oscillating under gravity

**Equilibrium position**: The bob hangs vertically downward.

**Restoring force**: Arises from the component of gravitational force tangent to the circular arc.

Equation of Motion for Small Angles

For small angular displacement θ from vertical (in radians):

**Restoring force component** (tangential):

$$F = -mg \sin θ ≈ -mgθ$$ (for small θ)

where the approximation uses sin θ ≈ θ.

**Arc length displacement**: x ≈ Lθ

**Force equation**:

$$F = -mg \cdot \frac{x}{L} = -\frac{mg}{L}x$$

Comparing with F = –kx:

$$k = \frac{mg}{L}$$

Angular Frequency and Period

**Angular frequency**:

$$ω = \sqrt{\frac{k}{m}} = \sqrt{\frac{mg/L}{m}} = \sqrt{\frac{g}{L}}$$

**Period of simple pendulum**:

$$T = 2π\sqrt{\frac{L}{g}}$$

**Frequency**:

$$ν = \frac{1}{2π}\sqrt{\frac{g}{L}}$$

Important Properties

**Independence of mass**: The period is independent of the mass of the bob. This remarkable result was discovered experimentally by Galileo.

**Dependence on length and gravity**:

Longer pendulum → longer period (√L relationship)

Stronger gravity → shorter period (1/√g relationship)

**Small angle approximation**: The formula is valid only for small amplitude oscillations (θ < 15° or about 0.26 rad).

Practical Applications

**Timekeeping**: Pendulums were used in clocks for centuries

**Gravity measurement**: Measuring g using a simple pendulum

**Seismic detection**: Pendulums detect ground vibrations

Energy in Simple Pendulum

For small oscillations, the bob undergoes SHM:

**Total energy**:

$$E = \frac{1}{2}kA^2 = \frac{1}{2}mω^2A^2 = \frac{1}{2}m\left(\frac{g}{L}\right)A^2$$

At the lowest point (maximum KE):

$$KE_{max} = mgh_{max}$$

where h is vertical height difference from lowest point.

Simple Pendulum vs. Mass-Spring System

| Property | Simple Pendulum | Mass-Spring |

|----------|-----------------|-------------|

| Restoring force | Gravity component | Spring force |

| Equation | T = 2π√(L/g) | T = 2π√(m/k) |

| Mass dependence | Independent | Dependent |

| Validity | Small angles only | All amplitudes (ideal) |

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DAMPED AND FORCED OSCILLATIONS

Damping in Real Oscillations

In real systems, oscillations decay over time due to:

Friction with surrounding medium

Air resistance

Internal friction in materials

Energy dissipation through heat

**Damping force**: Typically proportional to velocity: F_d = –bv, where b is damping coefficient.

Damped Harmonic Motion

**Equation of motion**:

$$m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0$$

**Solution** (for underdamped case):

$$x(t) = Ae^{-bt/2m}\cos(ω't + φ)$$

where ω' < ω is the damped angular frequency.

**Key features**:

Amplitude decreases exponentially: A(t) = A₀e^(–bt/2m)

Frequency decreases slightly

Eventually comes to rest at equilibrium

Energy is continuously dissipated

Forced Oscillations and Resonance

When an external periodic force is applied:

$$F_{ext} = F_0 \cos(Ωt)$$

where Ω is the driving frequency.

**Steady-state response**: The oscillator eventually oscillates at the driving frequency Ω, not its natural frequency ω.

**Resonance**: When driving frequency equals natural frequency (Ω = ω), the amplitude reaches a maximum.

**Practical importance**:

Bridges resonating with wind or marching troops

Microwave ovens using resonance of water molecules

Radio tuning using resonance of LC circuits

Musical instruments exploiting resonance

**Conditions for practical oscillations**: External driving force must be maintained to overcome damping and sustain oscillations.

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SUMMARY OF KEY FORMULAS AND RELATIONSHIPS

| Quantity | Formula | SI Unit |

|----------|---------|---------|

| Period | T = 1/ν = 2π/ω | s |

| Frequency | ν = 1/T | Hz = s⁻¹ |

| Angular frequency | ω = 2πν = 2π/T | rad/s |

| Displacement (SHM) | x = A cos(ωt + φ) | m |

| Velocity (SHM) | v = –Aω sin(ωt + φ) | m/s |

| Acceleration (SHM) | a = –ω²x | m/s² |

| Velocity-displacement | v² = ω²(A² – x²) | — |

| Force (SHM) | F = –kx = –mω²x | N |

| Kinetic energy | KE = ½mω²(A² – x²) | J |

| Potential energy | PE = ½mω²x² | J |

| Total energy | E = ½kA² = ½mω²A² | J |

| Spring-mass period | T = 2π√(m/k) | s |

| Simple pendulum period | T = 2π√(L/g) | s |

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EXAMINATION IMPORTANT POINTS

**Must-know definitions**: Periodic motion, oscillatory motion, equilibrium position, amplitude, period, frequency, phase, simple harmonic motion

**Must-know relationships**: ν = 1/T, ω = 2πν, x = A cos(ωt + φ), v = –Aω sin(ωt + φ), a = –ω²x

**Must-know equations**: T = 2π√(m/k) for spring-mass, T = 2π√(L/g) for simple pendulum, E_total = ½kA² for energy

**Common mistakes to avoid**:

Confusing period with angular frequency

Using degrees instead of radians in trigonometric functions

Forgetting negative sign in restoring force

Applying simple pendulum formula for large angles

Not recognizing that frequency is independent of amplitude

**Numerical problem types**:

Finding period/frequency from given data

Determining amplitude and phase from initial conditions

Calculating maximum velocity and acceleration

Energy conversions in oscillating systems

Finding length of pendulum for given period

Determining motion parameters from graphs