Lesson 7, Topic 4
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# Simple Harmonic Motion and Uniform Circular Motion

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Remember: A particle is said to be in Simple Harmonic Motion if it moves to and fro about its mean position, such that, its acceleration is directly proportional to displacement in magnitude but opposite in direction and is always directed towards the mean position.

Consider a particle P moving around a reference circle with the centre, O, radius, A = CZ = CY, and uniform angular velocity, Ï‰, in an anti-clockwise direction.

BD and YZ are the diameters of the circle.

The angular velocity, Ï‰, is related to the speed v of particle, P, by the equation:

v = Ï‰A

Let Q be the projection of P on the diameter YZ. (This is a perpendicular line from P to YZ)

As the particle moves on the circumference of the circle, Q moves to and fro about a fixed point along the diameter YZ. It moves with a maximum speed as it passes through O the centre of diameter YZ, and is momentarily at rest at Y and Z.

While the particle P moves around the circle with a constant velocity v, the motion of Q along the diameter is therefore said to be simple harmonic motion.

### Equation of Displacement of the Particle:

Let x be the horizontal displacement of the projection Q from the centre O at any point in time.

From triangle OPQ

Cos Î¸ = $$\frac{x}{A}$$

x = A cos Î¸

### Movement of the Particle:

As the particle completes one revolution from Z through B, Y, D and back to Z; The angle Î¸ swept through by the radius vector from Z to P varies from:

• 0Âº at Z
• to 90Âº at B
• to 180Âº at Y
• to 270Âº at D
• and 360Âº or 0Âº at Z

cos Î¸ varies from 1 to 0 to -1 to 0 and to 1
as x varies from A to 0 to -A to 0 and then to A again.

Therefore the displacement x from the centre O varies from A to -A. This can also be written as;

$$\scriptsize -A \leq x \leq A$$

for example if A = 2cm, then x can be any value between +2 and -2
i.e $$\scriptsize -2 \leq x \leq +2$$

Consider the figure above;

A is the maximum value of x which is called the amplitude of motion.

Amplitude A: This is the maximum displacement of a body performing Simple harmonic motion from its equilibrium position, O.

The time taken by Q, the projection of particle P, to move from Z to Y and back to Z is known as the period of the simple harmonic motion. It is also equal to the time taken to move from centre, O to Z then to Y and then back to O.

Period, T: This is the total time taken by a vibrating body to make one complete revolution about a reference point. It is measured in seconds, s.

Frequency: This is the number of complete revolutions per second made by a vibrating body. It is measured in Hertz (Hz) = $$\frac{1}{s}$$.

1 Hz = 1 cycle per second

f = $$\frac{1}{T}$$ or f = $$\frac{n}{T}$$

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