physics chapter wise notes

HSSLIVE Plus One Physics Chapter 14: Oscillations Notes

by

Oscillations examines periodic motion, one of nature’s most common and important phenomena. Students learn to describe simple harmonic motion using mathematical tools that relate displacement, velocity, and acceleration. This chapter introduces the concepts of amplitude, period, frequency, and phase, while analyzing energy transformations in oscillating systems. Applications range from pendulums and springs to electrical circuits and sound waves, providing a foundation for understanding more complex wave phenomena.

Chapter 14: Oscillations

1. Periodic and Oscillatory Motion

Periodic motion is any motion that repeats itself at regular time intervals. The smallest time after which motion repeats is called the period (T), and the number of repetitions per unit time is frequency (f = 1/T).

Oscillatory motion is a special type of periodic motion where an object moves to and fro about a fixed point called the equilibrium position. Common examples include:

  • Pendulum swings
  • Mass on a spring
  • Vibrating guitar strings
  • Alternating current in circuits

Oscillations can be:

  • Free oscillations: Occur without external forces (except initial displacement)
  • Forced oscillations: Maintained by external periodic force
  • Damped oscillations: Amplitude decreases with time due to resistive forces

2. Simple Harmonic Motion (SHM)

Simple Harmonic Motion (SHM) is the simplest form of oscillatory motion where the restoring force is directly proportional to the displacement from equilibrium position and acts in the opposite direction:

  • F = -kx
  • Where k is force constant or spring constant (N/m)

Examples of SHM include:

  • Mass on a spring (horizontal or vertical)
  • Simple pendulum (for small angles)
  • Liquid in U-tube
  • Torsional pendulum

The negative sign indicates that the force is always directed opposite to the displacement (restoring nature). A restoring force that follows Hooke’s Law (F = -kx) leads to SHM.

3. SHM Equation

From Newton’s Second Law (F = ma) and Hooke’s Law (F = -kx):

  • ma = -kx
  • a = -(k/m)x = -ω²x
  • Where ω = √(k/m) is angular frequency

This second-order differential equation has the solution:

  • x = A sin(ωt + φ)
  • Where A is amplitude and φ is phase constant

The key parameters of SHM are:

  • Angular frequency: ω = √(k/m) rad/s
  • Period: T = 2π/ω = 2π√(m/k) seconds
  • Frequency: f = 1/T = ω/2π Hz

Importantly, the period and frequency of SHM are independent of amplitude, a property called isochronism.

4. Displacement in SHM

The equation x = A sin(ωt + φ) represents displacement as a function of time, where:

  • A is amplitude (maximum displacement from equilibrium)
  • ω is angular frequency
  • φ is phase constant (determined by initial conditions)
  • ωt + φ is phase at time t

If we set initial conditions:

  • At t = 0, if x = 0 and velocity is positive, then φ = 0
  • At t = 0, if x = A and velocity is zero, then φ = π/2

The phase constant φ essentially shifts the sine curve along the time axis. The value of φ depends on when we start measuring time (t = 0) relative to the oscillation.

5. Velocity and Acceleration in SHM

Velocity in SHM is obtained by differentiating displacement with respect to time:

  • v = dx/dt = Aω cos(ωt + φ)
  • Maximum velocity: v_max = Aω (at equilibrium position)

Acceleration is obtained by differentiating velocity:

  • a = dv/dt = -Aω² sin(ωt + φ) = -ω²x
  • Maximum acceleration: a_max = Aω² (at extreme positions)

The negative sign in acceleration indicates it’s always directed towards the equilibrium position. The proportionality between acceleration and negative displacement (a ∝ -x) is the defining characteristic of SHM.

6. Energy in SHM

In SHM, energy continuously transforms between kinetic and potential forms:

Kinetic energy:

  • K = (1/2)mv² = (1/2)mω²A² cos²(ωt + φ)
  • Maximum at equilibrium position (x = 0)
  • Zero at extreme positions (x = ±A)

Potential energy:

  • U = (1/2)kx² = (1/2)mω²A² sin²(ωt + φ)
  • Maximum at extreme positions (x = ±A)
  • Zero at equilibrium position (x = 0)

Total energy:

  • E = K + U = (1/2)mω²A² = constant
  • Independent of time
  • Proportional to square of amplitude

This constant total energy is a consequence of the conservative nature of the restoring force in SHM. In real systems, energy is gradually lost due to friction or resistance, leading to damped oscillations.

7. Simple Pendulum

A simple pendulum consists of a point mass suspended by a massless, inextensible string. For small oscillations (where sin θ ≈ θ), the motion is approximately SHM.

The period of a simple pendulum:

  • T = 2π√(L/g)
  • Where L is length and g is acceleration due to gravity

This equation is valid only for small angles (< 10°). For larger amplitudes, the period increases and can be approximated by:

  • T = 2π√(L/g)[1 + (1/4)sin²(θ₀/2) + …]
  • Where θ₀ is the maximum angular displacement

Applications of pendulums include:

  • Grandfather clocks
  • Seismographs
  • Determination of local value of g

8. Physical Pendulum

A physical pendulum is any rigid body that oscillates about a fixed axis passing through a point other than its center of mass. Its period is:

  • T = 2π√(I/mgd)
  • Where I is moment of inertia about the axis of rotation, m is mass, and d is distance from pivot to center of mass

A compound pendulum is a special case of physical pendulum, where the axis of rotation can be moved. For a given body, there exists a point (center of oscillation) such that if the body is suspended from this point, it will have the same period as if the body were suspended from the center of gravity.

9. Damped Oscillations

In real systems, resistive forces like friction and air resistance cause the amplitude of oscillations to decrease over time. This is called damping, and the resulting motion is damped oscillation.

The displacement equation for damped oscillation:

  • x = Ae⁻ᵇᵗsin(ω’t + φ)
  • Where b is damping constant, ω’ = √(ω² – b²) is angular frequency of damped oscillation

Depending on the value of damping constant:

  • Underdamped (b < ω): Oscillations with decreasing amplitude
  • Critically damped (b = ω): System returns to equilibrium without oscillation in minimum time
  • Overdamped (b > ω): System returns to equilibrium without oscillation but takes longer time

Examples include shock absorbers in vehicles (designed to be nearly critically damped) and electromagnetic damping in galvanometers.

10. Forced Oscillations and Resonance

When an external periodic force is applied to an oscillator, the resulting motion is called forced oscillation. After initial transients die out, the oscillator vibrates with the frequency of the applied force.

The amplitude of forced oscillations depends on:

  • Natural frequency of the oscillator (ω₀)
  • Frequency of the driving force (ω)
  • Damping constant (b)

Resonance occurs when the driving frequency matches the natural frequency of the oscillator (ω = ω₀). At resonance:

  • Amplitude becomes maximum
  • Phase difference between force and displacement is 90°
  • Energy transfer from the driving force to the oscillator is most efficient

Examples of resonance include:

  • Bridge collapse due to resonant vibrations
  • Tuning of radio receivers
  • Resonance in musical instruments
  • Microwave heating of food

The sharpness of the resonance peak depends on damping – less damping gives a sharper peak. Resonance can be both useful (MRI scanners, microwave ovens) and destructive (bridge collapse, glass breaking).

 

Complete Chapter-wise Hsslive Plus One Physics Notes

Our Hsslive Plus One physics notes cover all chapters with key focus areas to help you organize your study effectively:

Leave a Comment