Examples Of Simple Harmonic Motion

a. Simple Harmonic motion & Hook's Law

i. Harmonic motion &Hook's Law Any motion that repeats itself in equal intervals of time is called periodic motion. A special form of periodic motion is called Simple Harmonic Motion (SHM). Simple Harmonic Motion is defined as oscillatory motion in which the resultant force on the oscillating body at any instant is directly proportional to its displacement from the rest position and opposite in direction to its motion. For a spring system, this can be written as F = -kx where F is the resultant force on the object attached to the spring, x is the displacement of the object from equilibrium, k is a constant called the spring constant. The force is a restoring

The period T of the oscillating system does not depend on the displacement from rest as long as the spring is not overstretched. The period is the time it takes for as system to go through one full oscillation and return to its starting position.

ii. Mass-spring oscillator

Consider a compact mass (m) that slides over a frictionless horizontal surface. Suppose that the mass is attached to end of a light horizontal spring whose other end is anchored in an immovable wall. At time (t), let X(t) be the extension of the spring: that is, the difference between the spring's actual length and its unstretched length. X(t) can also be used as a coordinate to determine the instantaneous horizontal displacement of the mass. then the mass experiences a horizontal force given by Hooke's law,

F =

Since v = R, we can rewrite this as

Let's set the origin at the center of the circle so the position vector R is along the radius. Notice that the acceleration vector a is always in the direction opposite the position vector R . Since , the vectors a and R are related by . The x-component of this vector equation is: . If we write Rx = x , then we have , which is the SHM equation. Done.

iv. Simple Harmonic motion

A pendulum, a mass on a spring, and many other kinds of oscillators exhibit a special kind of oscillatory motion called Simple Harmonic Motion (SHM).

SHM occurs whenever :

i. there is a restoring force proportional to the displacement from equilibrium: F  x ii. the potential energy is proportional to the square of the displacement: PE  x2 iii. the period T or frequency f = 1 / T is independent of the amplitude of the motion. iv. the position x, the velocity v, and the acceleration a are all sinusoidal in