Simple harmonicmotion

Simple Harmonic Motion

Definition and Form

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is proportional to the displacement, but in the opposite direction.
A body is said to be executing SHM when its acceleration, a, is proportional to its displacement, x, and always acts towards a fixed point (usually the origin) in the direction opposite to the displacement. This is captured in the equation a = -ω²x.
ω is the angular frequency of the motion and is always positive. It determines how fast the oscillation happens.

Differential Equation of SHM

The differential equation of simple harmonic motion is d²x/dt² = -ω²x. This equation is derived from Newton’s second law of motion.
The solution to this equation provides the displacement of the body as a function of time.

General Solution of SHM

The general solution of the above differential equation (with ω not equal to zero) is x(t) = Acos(ωt + φ), where A is the amplitude, ω is the angular frequency and φ is the phase angle.
The cosine function can be replaced with a sine function depending on the initial conditions.
The period T of the motion is given by 2π/ω, and the frequency f is given by 1/T.

Energy in SHM

The potential energy U and kinetic energy K of a system executing simple harmonic motion can be expressed in terms of displacement x and velocity v respectively. The total energy E of the system remains constant: E = K + U.
Kinetic energy is maximum when the system is at the equilibrium position (displacement zero) and potential energy is maximum when the system is at the extremes (displacement equals amplitude).

Remember, understanding the equations of SHM and how to solve them can help you analyse various physical situations in which oscillations and waves are involved, such as pendulum motion, spring systems, and more.