2.1 Simple Harmonic Motion (SHM)

2.1 Simple Harmonic Motion (SHM)

Definition:

Simple Harmonic Motion (SHM) is a type of oscillatory motion where the restoring force acting on an object is directly proportional to its displacement from the equilibrium position and acts in the direction opposite to the displacement. Mathematically, it can be expressed as:

F = -kx

where:
- F is the restoring force,
- k is the spring constant or force constant, and
- x is the displacement from the equilibrium position.

In SHM, the motion of the object is sinusoidal in time and demonstrates a periodic behavior. The most common examples of SHM include the motion of a mass on a spring and the oscillation of a simple pendulum.

Equation of Motion:

The displacement x(t) of an object in SHM as a function of time is given by:

x(t) = A cos(ωt + φ)

where:
- A is the amplitude (maximum displacement),
- ω is the angular frequency (in radians per second),
- t is the time, and
- φ is the phase constant (determined by initial conditions).

The velocity and acceleration as functions of time can be derived as:

v(t) = -Aω sin(ωt + φ)

a(t) = -Aω² cos(ωt + φ)

Example Problem:

A mass of 0.5 kg is attached to a spring with a spring constant of 200 N/m. If the mass is displaced 0.1 m from its equilibrium position and released, determine the frequency of the resulting oscillation.

Solution:

The angular frequency ω is given by:

ω = √(k/m)

ω = √(200 N/m / 0.5 kg) = √(400 rad²/s²) = 20 rad/s

The frequency f of the oscillation is:

f = ω / 2π ≈ 3.18 Hz