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# Simple Harmonic Motion And Circular Motion

July 31, 2022 written by Adeel Abbas

There is a relation between Simple Harmonic Motion And circular motion. A body performing the simple harmonic motion will be in circular motion also.

## Simple Harmonic Motion

An indirect geometrical method can be used to find mathematical expressions for describing the simple harmonic motion. Let a mass m attached to the end of a vertical spring vibrates simply harmonically with period T, frequency f, and amplitude X0.

The motion of the mass is displayed by the pointer P1 on the line BC with A as the mean position and B and C as the extreme position.

At t= 0, when no force is acting on, the pointer P1 is at means position A.

The pointer moves between extreme positions B and C in a way that it moves from A to B to A, A to C, and then back to A at time instants T/4, T/2, 3T/4, and T respectively. This will complete one cycle of vibration. During this time, the amplitude of vibration will be x0.

## Simple Harmonic Motion And circular motion

The concept of Simple Harmonic Motion And circular motion is introduced by considering that a point P moves on a circle of radius X0, with uniform angular frequency ω= 2 π /T. The radius of the circle is the same as the amplitude of vibration of the pointer’s motion. Draw a perpendicular PN on the vertical diameter of the circle and point N is called the projection of the point P on the diameter DE which is parallel to the line of vibration of the pointer.

The points D and E are at the same level as the points B and C. As point P describes the uniform circular motion with a constant angular velocity ω, N oscillates to and fro on the diameter DE with the time period T. The position of P at t= 0 is 01 on the circle and N is at the mean position O.

After time T/4 T/2 3T/4 and T the point N will be at D, O, E, and O respectively.

Comparing the motions of N and pointer P1, it is clear that both are identical.

### Instantaneous Displacement

It is the distance of the projection of point N from the mean position 0 at any instant. As it is clear from the figure that point P makes angle <O1OP= θ then angle NPO is also θ. ON/OP=sin θ

X/X0=sin θ

x=x0 sin θ

x=x0sin ω t

This will also be the displacement of the pointer P1at the instant t.

### The phase of Vibration:

The value of x is the function of θ. it draws a similar waveform that was traced experimentally but here it has been traced theoretically by linking SHM with circular motion.

When θ =0 radian then x= 0

When θ =900= π /2radianthen x=x0

When θ =180° = π radian then x = 0

When θ =2700=3 π/2 radian then x=-x0

When θ= 360°= 2 π radian then x = 0

Here θ is called the phase of vibration. This phase is also related to the circular motion aspect of SHM.

### Instantaneous velocity

The linear velocity of point p at any instant of time t will be directed along the tangent to the circle and its magnitude will be

vp =r ω

r=x0

vp=x0 ω

At the mean position where x=0, velocity is maximum

V= ω √xo2-x2= ω √xo2-0= ω √xo2= ω xo

At the extreme position where x=xo , velocity is minimum

V= ω √xo2-x2= ω √xo2-xo2= ω √xo2=0

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