Simple Harmonic Motion (SHM)

Simple harmonic motion (SHM) is defined as the periodic motion of a particle whose acceleration and restoring force are directed towards a fixed (equilibrium) point and are directly proportional to its displacement from the equilibrium position.

Examples of Simple Harmonic Motion

• The motion of a simple pendulum swinging back and forth.
• The motion of a weight moving up and down on one end of a spiral spring.
• The motion of the prong of a tuning fork.
• The up-and-down movement of a loaded test tube when pressed down and released in a liquid.
• Vibration of electric and magnetic waves in an electromagnetic field.
• Vibration of atoms in a solid above their equilibrium position.

Terms Used in Describing Simple Harmonic Motion

• Amplitude (A): The maximum vertical displacement of a particle from its mean or rest position. It is denoted by ‘A’ and measured in meters (m).
• Wavelength (λ): The distance between two successive crests or troughs of a wave. It is measured in meters (m).
• Period (T): The time taken to complete one full oscillation. It is denoted by T and measured in seconds (s).
• Frequency (F): The number of cycles a wave completes in one second. Its S.I. unit is Hertz (Hz).
• Cycle/Oscillation/Vibration: The to-and-fro movement of a particle from one extreme position to the other and back.

Formulas for the Period of Simple Harmonic Motion

• Period of a Spiral Spring:

  $$ T = 2π\sqrt{\frac{m}{k}} $$

  • m = Mass of the attached body
  • k = Force constant of the spring
• Period of a Simple Pendulum:

  $$ T = 2π\sqrt{\frac{l}{g}} $$

  • l = Length of the pendulum
  • g = Acceleration due to gravity
• Period of a Loaded Test Tube in a Liquid:

  $$ T = 2π\sqrt{\frac{m}{pgA}} $$

  • ρ = Density of the liquid
  • g = Acceleration due to gravity
  • A = Area

Speed and Acceleration in S.H.M.

Linear Displacement (\(\theta\)):

This is the angular distance (\(\theta\)) and the linear distance (\(s\)) covered when a particle is moving around a circle of radius (\(r\)). It is given as:

\[ \theta = \frac{s}{t} \]

The displacement in S.H.M. is given as:

\[ x = r \sin(\omega t) \]

Angular Velocity (\(\omega\)):

This is the angle turned through per second or the rate of change of angular displacement. It is given as:

\[ \omega = \frac{\theta}{t} \]

Its unit is \(\text{rad s}^{-1}\).

In relation to linear velocity:

\[ v = \omega r \]

The angular velocity in S.H.M. is given as:

\[ v = \omega r \cos(\omega t) = \pm \omega \sqrt{r^2 - x^2} \]

At maximum velocity:

\[ v_{\text{max}} = \omega r \]

Angular Acceleration (\(\alpha\)):

This is the angle turned through per second squared or the rate of change of angular velocity. It is given as:

\[ \alpha = \frac{\omega}{t} \]

In relation to linear acceleration:

\[ a = \frac{v^2}{r} = \omega^2 r \]

The angular acceleration in S.H.M. is given as:

\[ a = -\omega^2 r \sin(\omega t) \]

At maximum acceleration:

\[ a_{\text{max}} = \omega^2 r \]

Free, Damped, and Forced Vibration

A vibration is said to be free if the total energy of a vibrating object remains constant. No energy is lost, keeping the amplitude and frequency unchanged.

A vibration is said to be damped if energy is continuously lost as the body vibrates. This energy loss causes a decrease in both amplitude and frequency.

A forced vibration occurs when energy is continuously supplied to the vibrating body by an external periodic force. Examples include:

• Vibration of table surfaces when a sounding tuning fork is pressed on the tabletop.
• Vibration of a piston and the connecting rod in an engine.

Resonance and Energy Transformation in Simple Harmonic Motion (S.H.M.)

Resonance

Resonance occurs when a body is set into vibration at its natural frequency by another vibrating body due to an impulse received, causing them both to vibrate at the same frequency.

Examples of resonance include:

• Shattering of fragile glass by directing high-pitch sound at it.
• Barton's pendulum experiment.
• Collapse of a bridge when forced to vibrate at its natural frequency.
• Television and radio tuning circuits that resonate at a frequency corresponding to the transmitting station.

Natural frequency is the frequency at which a body tends to vibrate if left undisturbed.

During resonance, the following occurs:

1. The frequency of the applied external force equals the natural frequency of the vibrating body.
2. The body vibrates with maximum amplitude, with the gained energy replacing lost energy over the same interval.

Energy of Simple Harmonic Motion

The energy of simple harmonic motion can be derived by considering the motion of a loaded spiral spring. The following equations are used to solve problems involving a spiral spring and the energy of simple harmonic motion.

Key Equations

The period of oscillation for a mass-spring system is given by:

\[ T = 2\pi \sqrt{\frac{m}{k}} \]

Alternatively, the square of the period can be expressed as:

\[ T^2 = \frac{4\pi^2 m}{k} \]

The force acting on the spring follows Hooke’s Law:

\[ F = k e \quad \text{or} \quad mg = k e \]

Another form of the period equation using extension is:

\[ T = 2\pi \sqrt{\frac{e}{g}} \]

The angular frequency of the oscillation is:

\[ \omega = 2\pi f = \sqrt{\frac{k}{m}} \]

The total energy stored in the spring or the total work done is given by:

\[ W = \frac{1}{2} k A^2 = \frac{1}{2} m \omega^2 A^2 \]

Definitions of Variables

• T = Period of oscillation (seconds, s)
• m = Mass of the body (kilograms, kg)
• k = Force constant of the spring (N/m)
• e = Extension of the spring (meters, m)
• g = Acceleration due to gravity (m/s²)
• F = Force acting on the spring (Newtons, N)
• \(\omega\) = Angular velocity (rad/s)
• f = Frequency of oscillation (Hz or s⁻¹)
• W = Total energy stored in the spring (Joules, J)
• A = Amplitude of motion (meters, m)