# Simple Harmonic Motion

An analysis of Simple Harmonic Motion.

**Contents**

## Simple Harmonic Motion.

If a particle moves in a straight line in such a way that its acceleration is always directed towards a fixed point on the line and is proportional to the distance from the point, the particle is said to be moving in Simple Harmonic Motion. Let O be a fixed point on the line X'X and x the distance of the particle from O at any time t. Also, let the acceleration of the particle along OX as is a positive constant. No matter whether x is positive or negative the acceleration will be directed towards O. If v is the velocity at time t, the acceleration will be in the direction OX and will be given by the differential relationship :- This can be expressed as: or, In the initial stage of the motion v is negative as the particle is moving towards O and, When t = 0 x = a and = 0 and hence K = 0 and, When cos nt = 0, and thus a particle starting from A moving towards O arrives in a time with a velocity of . It will continue along the straight line and its velocity will be zero when and . It will then return to O arriving when with a velocity and reach A in a time with zero velocity. The motion is then repeated indefinitely unless destroyed by some force. Note:- The time is called the
**Period**of the oscillation and is the time for one complete cycle. - If the frequency is
*f*and the period , then .

- If the period of the motion is known, the motion is completely determined.
- The Period maybe written down at once if the magnitude of the acceleration for some value of x is known.
- The amplitude is determined by the initial displacement.

## Other Initial Conditions

If the the motion is started by giving the particle a velocity when its distance from O is , the type of motion is unchanged and the time is measured from this instant, instead of the instant when x = a. In this case the value of x at any instant is given by :- where is a constant] Now, Also when Then, And, The Constant is called the**Epoch**of the motion. The

**Phase**of the motion at time t is the time which has elapsed since the particle was at the positive end of its path. Thus the phase is less a multiple of the period. Also, In particular if , i.e. the particle starts from O and the amplitude is Since the acceleration at any instant is This is characteristic of Simple Harmonic Motion and its solution is given by:- may be written down if the amplitude and the epoch are known.

## The Relation To Uniform Motion In A Circle.

If a particle is describing a circle of radius with uniform angular velocity , its orthogonal projection on a diameter of the circle moves on the diameter in simple harmonic motion of amplitude and period . Let be the angle which the radius initially makes with the diameter X'OX . Then after a time t the angle made by the radius to the particle is . Hence, if P is the position of the particle at time t and N the foot of the perpendicular from P on OX, then:- and the point moves with Simple Harmonic Motion of amplitude and periodExample:

[imperial]

##### Example - Simple Harmonic Motion

Problem

A particle moves with Simple Harmonic Motion in a straight line. Find the time of a complete
oscillation if the acceleration is 4 ft/sec

^{2}, when the distance from the centre of the oscillation is 2 ft. If the Velocity with which the particle passes through the centre of oscillations is 8 ft./sec. find the amplitude.Workings

If the acceleration is at a distance

*x*from the centre then: Hence the period is: If the phase is zero when where*a*is the amplitude. Then And the value of*v*at the centre of oscillation is andSolution

The period is
and amplitude