Definition

Simple harmonic motion is a special case of periodic motion where the force acting on an object is proportional to the displacement from equilibrium and always directed towards the equilibrium position. Any system exhibiting this behavior follows the condition that acceleration is proportional to displacement and directed opposite to displacement.

Mathematical Description

For a mass attached to a spring on a frictionless surface, applying Hooke's law and Newton's second law:

$F = -kx = ma$

$a = -\frac{k}{m}x$

This shows acceleration $a$ is proportional to displacement $x$, with the constant $\frac{k}{m}$ representing the angular frequency squared:

$a = -\omega^2 x$

where $\omega = \sqrt{\frac{k}{m}}$

The displacement of a particle executing SHM is given by:

$y = A\sin\omega t = A\sin(2\pi ft)$

where $A$ is amplitude, $f$ is frequency, and $\omega$ is angular velocity.

Period and Frequency

The period $T$ (time for one complete oscillation) is:

$T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}$

The frequency $f = \frac{1}{T}$

Velocity and Acceleration

Velocity is the derivative of displacement:

$v = \frac{dy}{dt} = A\omega\cos\omega t = \pm\omega\sqrt{A^2 - y^2}$

Maximum velocity occurs at equilibrium ($y = 0$):

$v_{max} = \omega A$

Acceleration:

$a = \frac{dv}{dt} = -\omega^2 y = -\omega^2 A\sin\omega t$

Maximum acceleration occurs at maximum displacement:

$a_{max} = \omega^2 A$

Energy in SHM

The total mechanical energy remains constant:

$E = \frac{1}{2}m\omega^2A^2 = \frac{1}{2}kA^2$

Potential energy: $U = \frac{1}{2}kx^2$

Kinetic energy: $K = \frac{1}{2}mv^2 = \frac{1}{2}m\omega^2(A^2 - x^2)$

At equilibrium ($x = 0$), kinetic energy is maximum; at maximum displacement, potential energy is maximum.

Examples of SHM

Vertical oscillations of a loaded spring: $T = 2\pi\sqrt{\frac{m}{k}}$

Simple pendulum (for small angles): $T = 2\pi\sqrt{\frac{l}{g}}$

Liquid in a U-tube: $T = 2\pi\sqrt{\frac{h}{g}}$

where $h$ is the height difference.

Worked Example

A particle executes SHM with period 4 s and amplitude 2 cm. Find: (a) Maximum velocity (b) Velocity at half maximum displacement

Solution

(a) $v_{max} = \omega A = \frac{2\pi}{T} \times A = \frac{2\pi}{4} \times 0.02 = 0.0314$ m/s

(b) At $y = \frac{A}{2}$: $v = \omega\sqrt{A^2 - (A/2)^2} = \frac{2\pi}{4}\sqrt{(0.02)^2 - (0.01)^2}$ $v = 0.0272 \text{ m/s}$

Damped and Forced Vibrations

Free oscillations: Occur at natural frequency with constant amplitude (ideal case).

Forced oscillations: When an external periodic force maintains oscillation. When driving frequency equals natural frequency, resonance occurs with maximum amplitude.

Damped oscillations: Amplitude decreases over time due to friction or resistance. Types include:

• Under-damped: Oscillates with decreasing amplitude
• Critically-damped: Returns to equilibrium without oscillating (fastest approach)
• Over-damped: Returns to equilibrium slowly without oscillating

Progressive (Travelling) Waves

A progressive wave transfers energy without transporting matter. Two fundamental types exist:

Transverse waves: Particle displacement perpendicular to wave propagation direction (e.g., light, waves on a string).

Longitudinal waves: Particle displacement parallel to wave propagation direction (e.g., sound waves).

The wave equation for a progressive wave moving along the x-direction:

$y(x,t) = A\sin(\omega t - kx)$

where $k = \frac{2\pi}{\lambda}$ is the wave number.

Wave velocity: $v = f\lambda = \frac{\omega}{k}$

Stationary (Standing) Waves

Formed when two progressive waves of same frequency and amplitude travel in opposite directions. Key features:

• Nodes: Points of zero amplitude (permanent rest)
• Antinodes: Points of maximum amplitude

The equation for stationary wave:

$y(x,t) = 2A\cos(kx)\sin(\omega t)$

For a string fixed at both ends, wavelengths for harmonics:

$\lambda_n = \frac{2L}{n}$

Frequencies:

$f_n = \frac{nv}{2L}$

where $n = 1, 2, 3,...$

Velocity of Waves on a String

$v = \sqrt{\frac{T}{\mu}}$

where $T$ is tension and $\mu$ is linear mass density.

Superposition Principle

When two or more waves pass through the same medium, the resultant displacement equals the vector sum of individual displacements:

$y = y_1 + y_2 + ...$

Constructive interference: Path difference = $n\lambda$ (waves in phase)

Destructive interference: Path difference = $(n + \frac{1}{2})\lambda$ (waves out of phase)

Beats: Produced by two waves of slightly different frequencies. Beat frequency:

$f_b = |f_1 - f_2|$

Diffraction is the spreading of waves when passing through small apertures or around edges. It occurs noticeably when the obstacle size is comparable to the wavelength.

Single Slit Diffraction (Fraunhofer)

For a slit of width $a$, dark fringes occur at:

$a\sin\theta = n\lambda$

where $n = 1, 2, 3,...$

Diffraction Grating

For a grating with slit separation $d$, principal maxima occur at:

$d\sin\theta = n\lambda$

The number of lines per meter: $N = \frac{1}{d}$

Maximum order: $n_{max} = \frac{d}{\lambda}$

Young's Double Slit Experiment

Light from a single source passes through two narrow slits, producing interference patterns.

For bright fringes: $y_n = \frac{n\lambda D}{d}$

For dark fringes: $y_n = \frac{(n + \frac{1}{2})\lambda D}{d}$

Fringe width (spacing): $\beta = \frac{\lambda D}{d}$

Thin Film Interference

Interference occurs due to reflections from front and back surfaces of thin films. Conditions:

• Bright reflected: $2t = (n + \frac{1}{2})\lambda$
• Dark reflected: $2t = n\lambda$

Newton's Rings

Produced by interference between light reflected from a convex lens and glass plate. For dark rings: $D_n^2 = 4a\lambda n$

where $a$ is the radius of curvature.

Polarization restricts vibrations to a single plane perpendicular to propagation direction. Light can be polarized by:

Methods of Polarization

1. Polaroid filters: Absorb one plane of vibration
2. Reflection: At Brewster's angle, reflected light is completely polarized
3. Double refraction: Calcite crystals split into two polarized rays
4. Scattering: Air molecules scatter polarized light

Brewster's Law

At the polarization angle $i_p$: $\tan i_p = \mu$

where $\mu$ is the refractive index.

Malus's Law

For polarized light passing through an analyzer: $I = I_0\cos^2\theta$

where $\theta$ is the angle between polarization directions.

Apparent frequency change due to relative motion between source and observer.

For Sound

Source moving towards stationary observer: $f' = \frac{v}{v - u_s}f$

Source moving away: $f' = \frac{v}{v + u_s}f$

Observer moving towards stationary source: $f' = \frac{v + u_o}{v}f$

Observer moving away: $f' = \frac{v - u_o}{v}f$

Both moving: $f' = \frac{v \pm u_o}{v \mp u_s}f$

For Light

Doppler shift in light indicates whether celestial bodies are approaching or receding:

• Red shift: Wavelength increases (moving away)
• Blue shift: Wavelength decreases (approaching)

$\frac{\Delta\lambda}{\lambda} = \frac{v}{c}$

A train whistle has frequency 400 Hz. Calculate the frequency heard when: (a) Train approaches at 10 m/s (speed of sound = 340 m/s) (b) Train recedes at 10 m/s

Solution

(a) $f' = \frac{340}{340 - 10} \times 400 = 412.4$ Hz

(b) $f' = \frac{340}{340 + 10} \times 400 = 388.6$ Hz

In Tanzania, understanding vibrations and waves has practical applications in construction engineering—seismologists use knowledge of wave propagation to study earthquake behavior and design earthquake-resistant buildings. Additionally, mobile phone networks rely on electromagnetic wave principles for signal transmission, while healthcare facilities use ultrasound technology (based on sound wave principles) for diagnostic imaging. Radio stations broadcasting across the country use wave superposition and interference concepts to optimize signal coverage.