1.1 Simple Harmonic Motion

SIMPLE HARMONIC MOTION AND WAVES

Definition of Key Terms

  • Oscillation or Vibration: A body oscillates or vibrates when it moves back and forth around a fixed or mean position.

Characteristics of Oscillatory Motion

  • Oscillatory Motion: Repeated back-and-forth or to-and-fro motion about a mean position.

    • Each complete trip from a point back to the same point constitutes one vibration, oscillation, or cycle.

  • Displacement: Distance from the mean position at any instant; measured in meters.

  • Amplitude (A): Maximum displacement from the mean position.

  • Time Period (T): Time taken to complete one vibration; measured in seconds.

  • Frequency (f): Number of vibrations per second, inverse of time period (f = 1/T); measured in Hertz (Hz).

SIMPLE HARMONIC MOTION

Definition

  • Simple Harmonic Motion (SHM): Motion where acceleration is proportional to displacement from equilibrium, directed towards it.

Features of SHM

  1. Vibration Around a Fixed Position: Body always vibrates around a mean position.

  2. Direction of Acceleration: Always directed towards the mean position.

  3. Magnitude of Acceleration: Proportional to displacement; zero at mean position, maximum at extremes.

  4. Velocity: Maximum at the mean position, zero at extremes.

Cases of SHM

  • Motion of Mass Attached to Spring

  • Ball and Bowl System

  • Motion of Simple Pendulum

MOTION OF MASS ATTACHED TO SPRING

Description

  • A mass 'm' on a frictionless surface attached to an elastic spring.

    • When displaced by a distance 'x', a restoring force 'F' acts on the mass.

Hook's Law

  • Restoring Force (F): Proportional to displacement 'x' (F = -kx).

  • Constant 'k' is the spring constant.

Relation to SHM

  • Equation of motion indicates SHM characteristics:

    • Acceleration (a) is proportional to displacement (x) and directed towards mean position.

    • Mathematically: a = -k/m * x.

Time Period (T) and Frequency (f)

  • Time period: T = 2π√(m/k)

  • Frequency: f = 1/(2π√(m/k))

BALL AND BOWL SYSTEM

Description

  • Represents an example of SHM.

    • Ball at mean position (equilibrium) experiences zero net force.

  • When disturbed, it moves toward mean position due to restoring force (weight).

  • Continues to oscillate until energy is lost through friction.

MOTION OF SIMPLE PENDULUM

Description

  • Consists of a mass 'm' on a string of length 'l'.

    • At mean position, forces cancel (T = mgcosθ).

  • Displaced to extreme position, restoring force acts towards the mean position.

Conditions of Motion

  • Speed maximum at mean position and decreases towards extreme positions.

  • Continues its oscillation until energy dissipates.

Time Period & Frequency of Pendulum

  • Time period: T = 2π√(l/g)

  • Frequency: f = 1/(2π√(l/g))

DAMPING OR DAMPED OSCILLATIONS

Definition

  • Damped oscillations occur in the presence of resistive forces.

    • Amplitude decreases over time.

Energy Loss

  • Energy lost due to work done against friction or resistance,

    • Ideal systems would continue indefinitely without damping.

Applications of Damping

  • Shock Absorbers in automobiles: Convert vibrational energy into heat energy, dampening vibrations when driving over bumps.