Module 7: Mechanical Waves

Periodic Motion

  • Motion that repeats on a regular cycle.

Harmonic Motion

  • Repetitive motion (oscillation) caused by a restoring force.

  • The restoring force tends to return the system to equilibrium even if it is displaced.

Quantities in Harmonic Motion

  1. Amplitude (A): Maximum magnitude of displacement from equilibrium.

  2. Period (T): Time for one cycle (seconds per cycle).

    T=1fT=\frac{1}{f}

  3. Frequency (f): Number of cycles in a unit of time (Hertz or cycles per second).

    f=1Tf = \frac{1}{T}

  4. Angular Frequency (ω): Equal to 2π2π times the frequency.

    ω=2πfω = 2πf

Simple Harmonic Motion (SHM)

  • When the restoring force is directly proportional to the displacement from equilibrium, the oscillation is called SHM.

  • Obeys Hooke’s Law: F=−kxF = -kx

Equations for SHM

  • Acceleration: a=−(km)x=−ω2xa = -(\frac{k}{m})x = -ω^2x

  • Angular frequency: ω=kmω = \sqrt{\frac{k}{m}}

  • Acceleration is not constant and depends on the displacement from equilibrium.

  • k is the force constant (spring constant), and m is the mass of the object in SHM.

  • An object in oscillation is also known as a harmonic oscillator (HO).

  • In real-life oscillations, Hooke’s Law does not apply if the body moves too far from equilibrium. However, small-amplitude oscillations can be approximated as SHM.

Frequency and Period in SHM

  • Frequency: f=ω2π=12πkmf = \frac{ω}{2π} = \frac{1}{2π} \sqrt{\frac{k}{m}}

  • Period: T=1f=2πmkT = \frac{1}{f} = 2π \sqrt{\frac{m}{k}}

  • Frequency and period do not depend on amplitude in SHM; they depend only on the values of k and m.

Simple Pendulum

  • A point mass suspended by a massless, unstretchable string swings in a circular path.

Equations for Simple Pendulum

  • Angular Frequency: ω=gLω = \sqrt{\frac{g}{L}}

  • Frequency: f=ω2π=12πgLf = \frac{ω}{2π} = \frac{1}{2π} \sqrt{\frac{g}{L}}

  • Period: T=1f=2πLgT = \frac{1}{f} = 2π \sqrt{\frac{L}{g}}

  • Where L is the radius of the circular path

  • Using this angular frequency, we can determine the frequency and the period of a simple pendulum

  • These equations are valid for small amplitudes only.