# 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.##### MISSING IMAGE!

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- 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,##### MISSING IMAGE!

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**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 .##### MISSING IMAGE!

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Example:

[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