Simple Harmonic Motion Simulator

Introduction to Oscillations and Waves

Oscillatory motion is one of the most important types of motion in physics, occurring in mechanical systems, electrical circuits, and even in quantum mechanics. This simulator helps you understand the principles of Simple Harmonic Motion (SHM), which is the most basic form of oscillatory motion.

In this learning module, we will explore concepts from Unit 1, Module 2: Oscillations and Waves as outlined in the CAPE Physics curriculum.

Harmonic Motion

Simple Harmonic Motion (SHM) occurs when a body is subjected to a restoring force that is proportional to the displacement from equilibrium and in the opposite direction.

Period of Simple Pendulum

For a simple pendulum, the period is given by:

T = 2π√(L/g)

Where L is the length of the pendulum and g is the acceleration due to gravity.

Period of Mass on a Spring

For a mass on a spring, the period is given by:

T = 2π√(m/k)

Where m is the mass and k is the spring constant.

Equation of Motion

The displacement of a body undergoing SHM can be described by:

x = A cos(ωt + φ)

Where A is the amplitude, ω is the angular frequency, and φ is the phase constant.

Angular Frequency

The angular frequency ω is related to the period T by:

ω = 2π/T = √(k/m)

Energy in Simple Harmonic Motion

In SHM, there is a continuous interchange between kinetic and potential energy while the total energy remains constant (in the absence of damping).

Kinetic Energy

The kinetic energy of a body in SHM is given by:

KE = ½mv² = ½mω²A² sin²(ωt + φ)

Potential Energy

The potential energy is given by:

PE = ½kx² = ½kA² cos²(ωt + φ)

Total Energy

The total energy remains constant:

E = KE + PE = ½kA² = ½mω²A²

Damped Oscillations and Resonance

In real systems, energy is gradually dissipated due to friction and other resistive forces, leading to damped oscillations.

Equation of Damped Motion

The displacement in damped oscillation is given by:

x = Ae^-bt/2m cos(ω't + φ)

Where b is the damping constant and ω' is the angular frequency of the damped oscillation.

Types of Damping

Light damping: Oscillation amplitude gradually decreases
Critical damping: System returns to equilibrium in the shortest time without oscillation
Heavy damping: System returns slowly to equilibrium without oscillation

Real-life Examples

Damping is utilized in many real-life situations, such as:

Shock absorbers in vehicles
Door closers
Electrical circuits with resistors

Properties of Waves

Waves are disturbances that transfer energy from one point to another without the transfer of matter.

Wave Terminology

Displacement: The distance of a point from its equilibrium position
Amplitude: The maximum displacement from the equilibrium position
Period: The time taken for one complete oscillation
Frequency: The number of oscillations per unit time
Wavelength: The distance between two consecutive points in phase

Types of Waves

Waves can be classified into two main types based on the direction of particle movement relative to the direction of wave propagation:

Transverse waves: Particles move perpendicular to the direction of wave propagation
- Example: Light waves, water waves
Longitudinal waves: Particles move parallel to the direction of wave propagation
- Example: Sound waves, compression waves in springs