Comprehensive Notes on Oscillations and Simple Harmonic Motion

Periodic Motion

• Definition: If a body moves in such a way that it crosses a certain point from the same direction after a certain period of time, the motion is called periodic motion.
• Examples:
  • Earth's motion around the sun.
  • The motion of the hands (hour and minute) of a clock.
  • The orbits of planets in the solar system.

Oscillation

• Definition: A periodic back-and-forth motion where the body moves for half of its time period in one direction and the other half in the opposite direction.
• Examples:
  • Motion of a simple pendulum.
  • Vibration of a tuning fork.
  • Motion of a mass on a spring.

Classification of Oscillations

• Oscillation can be categorized into various types:
  • Simple Harmonic Motion (SHM)
  • Damped Harmonic Motion
  • Free Oscillation
  • Maintained Oscillation
  • Forced Oscillation

Simple Harmonic Motion (SHM)

• Definition: If an oscillation is such that the oscillating body experiences a restoring force when displaced from its equilibrium position, and that restoring force is proportional to the displacement, the motion is called Simple Harmonic Oscillation. The system is known as a Simple Harmonic Oscillator.
• Criteria: The force acts to return the body to equilibrium and varies linearly with distance.
• Examples:
  • Mechanical waves.
  • Motion of a spring.
  • Oscillations of a liquid column in a U-Tube.
  • A child swinging on a playground.
  • A simple pendulum.
  • An electron in a wire carrying Alternating Current ($AC$).

Characteristics of Simple Harmonic Motion

• The motion is inherently periodic.
• At a particular fixed time interval, the direction of motion becomes opposite.
• The motion occurs along a straight line.
• The acceleration is directly proportional to the displacement ($a \propto -x$).
• Acceleration always acts in the direction opposite to displacement.
• Acceleration always points toward the mean (equilibrium) position of the object.

Parameters of an Oscillator

• Equilibrium Position: The specific point where an oscillating object experiences zero ($0$) resultant forces.
• Complete Oscillation: An oscillation is defined as complete if a vibrating or oscillating object, starting from a specific point, returns to that same point while moving in the same direction.
• Amplitude ($x_m$): The maximum displacement on both sides of an object from its equilibrium position. It occurs when $\cos(\omega t + \phi) = \pm 1$. The SI unit is the meter ($m$). The amplitude points are at $+x_m$ and $-x_m$.
• Phase: The state of motion of a vibrating particle at any instant. It is determined by displacement, velocity, and acceleration at that time and is represented by the term $(\omega t + \phi)$.
• Phase Constant / Epoch ($\phi$): Defines the initial state (position at $t=0$) of the particle.
  • Case 1: Counting from the mean position ($x=0$ at $t=0$). This leads to $\phi = \pi/2$. The equation becomes $x = x_m \sin(\omega t)$.
  • Case 2: Counting from the extreme position ($x=x_m$ at $t=0$). This leads to $\phi = 0$. The equation remains $x = x_m \cos(\omega t)$.

Displacement Equations

• Consider a particle moving in a circular path of radius $x_m$ at angular velocity $\omega$. At $t=0$, the position is $P$, and at $t=t$, the position is $Q$. A projection $QN$ on the X-axis gives the displacement $x$.
• X-axis Displacement: $x = x_m \cos(\omega t + \phi)$.
• Y-axis Displacement: $y = y_m \sin(\omega t + \phi)$.
• Where:
  • $x$: Displacement from origin $O$ to $N$.
  • $x_m$: Radius of the circle (Amplitude).
  • $\omega t + \phi$: Phase.
  • $\phi$: Phase constant.

Velocity and Acceleration in SHM

Velocity ($v$)

• Derivation from displacement: $v = \frac{dx}{dt} = \frac{d}{dt} [x_m \cos(\omega t + \phi)] = -\omega x_m \sin(\omega t + \phi)$.
• In terms of displacement: $v = \omega \sqrt{x_m^2 - x^2}$.
• Equilibrium ($x=0$): Velocity is maximum, $v_{max} = \omega x_m$.
• Extreme Position ($x=\pm x_m$): Velocity is zero, $v = 0$.

Acceleration ($a$)

• Derivation from velocity: $a = \frac{dv}{dt} = \frac{d}{dt} [-\omega x_m \sin(\omega t + \phi)] = -\omega^2 x_m \cos(\omega t + \phi)$.
• Simplified: $a = -\omega^2 x$.
• Equilibrium ($x=0$): Acceleration is zero, $a = 0$.
• Extreme Position ($x=x_m$): Acceleration is maximum, $a = -\omega^2 x_m$.

Time Period and Frequency

• Time Period ($T$): The time taken for one complete oscillation. Unit: Second ($sec$).
  • $T = \frac{2\pi}{\omega}$.
  • Displacement repeats such that $x(t + T) = x(t)$.
• Frequency ($f$): Total number of oscillations per second. Unit: $sec^{-1}$ or Hertz ($Hz$).
  • $f = \frac{1}{T} = \frac{\omega}{2\pi}$.
  • $f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$.

Differential Equation of SHM

1. Restoring Force: $F = -kx$, where $k$ is the force constant.
2. Newton's 2nd Law: $F = m \frac{d^2 x}{dt^2}$.
3. Combining equations: $m \frac{d^2 x}{dt^2} + kx = 0$.
4. Standard Form: $\frac{d^2 x}{dt^2} + \frac{k}{m} x = 0$.
5. Substituting $\omega^2 = \frac{k}{m}$: $\frac{d^2 x}{dt^2} + \omega^2 x = 0$.

Mechanical Energy ($E$) in SHM

Mechanical energy is the sum of Potential Energy ($U$) and Kinetic Energy ($K$): $E = U + K$.

Potential Energy ($U$)

• Work done against restoring force for displacement $x$:
  • $dW = F dx = kx dx$.
  • $U = W = \int_0^x kx dx = \frac{1}{2} k x^2$.
• As a function of time: $U = \frac{1}{2} k x_m^2 \cos^2(\omega t + \phi)$.

Kinetic Energy ($K$)

• $K = \frac{1}{2} m v^2 = \frac{1}{2} m [-\omega x_m \sin(\omega t + \phi)]^2 = \frac{1}{2} m \omega^2 x_m^2 \sin^2(\omega t + \phi)$.
• Since $\omega^2 = \frac{k}{m}$, $K = \frac{1}{2} k x_m^2 \sin^2(\omega t + \phi)$.

Total Energy ($E$)

• $E = \frac{1}{2} k x_m^2 [\cos^2(\omega t + \phi) + \sin^2(\omega t + \phi)] = \frac{1}{2} k x_m^2$.
• The total mechanical energy is constant or conserved as both $k$ and $x_m$ are constant.

Average Energies over a Cycle

• Average Kinetic Energy ($\langle K \rangle$): $\langle K \rangle = \frac{1}{T} \int_0^T \frac{1}{2} k x_m^2 \sin^2(\omega t + \phi) dt = \frac{k x_m^2}{4}$.
• Average Potential Energy ($\langle U \rangle$): $\langle U \rangle = \frac{1}{T} \int_0^T \frac{1}{2} k x_m^2 \cos^2(\omega t + \phi) dt = \frac{k x_m^2}{4}$.
• Thus, $\langle U \rangle = \langle K \rangle = \frac{1}{2} E$. The average energy is half of the total mechanical energy.

Parameter Values at Key Points in the SHM Cycle

Analogies: Translational vs. Rotational Motion

Torsional Pendulum

• Definition: A disk suspended by a wire attached to its center of mass, with the other end fixed to a rigid support.
• Restoring Torque: When twisted by angle $\theta$, wire exerts $\tau \propto -\theta \rightarrow \tau = -\kappa \theta$, where $\kappa$ (kappa) is the torsional constant.
• Equation of Motion: $I \frac{d^2 \theta}{dt^2} = -\kappa \theta \rightarrow \frac{d^2 \theta}{dt^2} + \frac{\kappa}{I} \theta = 0$.
• Angular Velocity: $\omega = \sqrt{\frac{\kappa}{I}}$.
• Time Period: $T = 2\pi \sqrt{\frac{I}{\kappa}}$.

Types of Oscillations (Detailed Descriptions)

• Free Oscillations: Occur when a body vibrates with its own natural frequency without external influence (e.g., a tuning fork, stretched string, simple pendulum).
• Damped Oscillations: Oscillations that fade over time due to dissipative forces like friction or air resistance. Most real-world oscillations are damped.
• Maintained Oscillations: Oscillations where energy loss is precisely compensated by an external source to maintain a constant amplitude (e.g., a swing being regularly pushed).
• Forced Oscillations: Produced by an external periodic driving force with a frequency different from the natural frequency (e.g., sound boards in stringed instruments).

Resonance

• Condition: Occurs in forced vibrations when the driving frequency matches the system's natural frequency, causing maximum amplitude.
• Advantages:
  • Determining the frequency of a tuning fork.
  • Selecting frequencies in Radio/TV tank circuits.
• Disadvantages:
  • Earthquake disasters where building frequencies match seismic oscillations, leading to collapse.
  • A singer shattering glass by hitting its resonant frequency.

Damped Harmonic Oscillations Math

• Damping Force ($D$): $D = -bv$, where $b$ is the damping coefficient.
• Total Force: $\sum F = -kx - bv$.
• Differential Equation: $m \frac{d^2 x}{dt^2} + b \frac{dx}{dt} + kx = 0$.
• Standard Form: $\frac{d^2 x}{dt^2} + \frac{b}{m} \frac{dx}{dt} + \omega_0^2 x = 0$, where $\omega_0^2 = \frac{k}{m}$.

Mathematical Problems

1. Particle Amplitude 15cm, Frequency 4 Hz: Compute (i) max acceleration and velocity, (ii) acceleration and velocity at displacement $9\,cm$.
2. SHM Velocity/Displacement Correlation: Particle displacement is $8\,cm$ when velocity is $6\,cm/s$, and displacement is $6\,cm$ when velocity is $8\,cm/s$. Calculate amplitude ($A$), frequency ($f$), and time period ($T$).
3. SHM object amplitude 0.01m, frequency 12Hz: Find velocity at displacement $0.005\,m$ and max velocity.
4. SHM amplitude 3cm, max velocity 6.24cm/s: Determine the time period.
5. Spring problem: A spring is stretched $0.02\,m$ by a $4\,N$ force. A $2\,kg$ mass is then attached and pulled $0.04\,m$ from equilibrium. Find: (a) force constant, (b) force before release, (c) period and frequency, (d) amplitude, (e) max velocity, (f) mechanical energy.