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Simple harmonic motion is a special kind of periodic motion, in which a particle moves to-and-fro repeatedly about a mean or an equilibrium position under a restoring force that is directed towards the mean position.

Consider a particle placed on the circumference of a circle.

Initially, the particle is at point X as you can see in the figure below:

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As it moves from X to P, there is an angular displacement (an arc) which is equal to Î˜ and at time = t, the particle reaches from point OX to P.

The motion is along the circle with a constant angular velocity Ï‰.

So, the angle subtended by a particle,Â Î˜ = Ï‰t

The mean position of the particle is at point O.

Now, we draw a perpendicular from P to a certain point on the diameter XOXâ€™.

So the displacement from O to a certain point is, â€˜xâ€™.

The instantaneous acceleration will be directly proportional to this displacement.

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â a Î± x

Now, if we multiply m on both sides, we get

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ma Î± mx

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Or, Â Â F Î± x

So, we concluded that one-dimensional motion of a particle in a uniform circular motion about its mean position is in simple harmonic motion.

Shm as a Projection of Uniform Circular Motion on Any Diameter

Consider a particle P moving with uniform speed along the circumference of a circle with radius a, having center O. This circle is considered as a circle of reference with particle P as the particle of reference.

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Here, if you look at Fig.1, XOXâ€™ and YOYâ€™ are perpendicular diameters of the circle of reference.

As the reference particle moves from X to Y, its projection on diameter YOYâ€™ moves from O to Y.

As this reference particle moves from Y to Xâ€™, its projection moves along the diameter from Y to O.

Similarly, when the reference particle moves on the circle from Xâ€™ to X via Yâ€™, its projection moves along the diameter from O to Yâ€™ and then from Yâ€™ to O.

Thus, during the time the particle P goes around the circle and completes one revolution, its projection, â€˜Mâ€™ oscillates about the point O along the diameter YOYâ€™ and completes one vibration. Since the projection of the reference particle is in SHM, and the projection of M on diameter YOYâ€™ is also a simple harmonic motion.

Therefore, simple harmonic motion is defined as the projection of uniform circular motion on any diameter of a circle of reference.

Shm as projection of uniform circular motion

Consider a reference particle moving on a circle of reference with radius, â€˜aâ€™ with uniform angular velocity, â€˜Ï‰â€™

From Fig.1

Let the particle at time t = 0, start from point X, and sweep an angular displacement Î˜ in time â€˜tâ€™ with angular velocity Ï‰, equal to Ï‰t.

Now, let the projection of the particle P on diameter YOYâ€™ be at M.

Then the displacement in SHM at time t is given by,

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â OM = y

In Î”OPM,

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â SinÎ˜ = OM/OP = y/aÂ

Or,Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Now,Â

In Î”ONP,

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â CosÎ˜ = ON/OP = x./a

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Or,Â

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From Fig. 3(a)

Now, if A is the starting position of the reference particle.

Here, âˆ AOX = Ñ„â‚€ andÂ âˆ AOP = Ï‰t and Î˜Â = Ï‰t - Ñ„â‚€

From eq(1) and (2)

Â Â Â Â Â Â Â Â Â Â Â Â Â yÂ = a Sin(Ï‰t - Ñ„â‚€)

Â Â Â Â Â Â Â Â Â Â Â Â Â x =Â a Cos(Ï‰t - Ñ„â‚€)

Â Here, - Ñ„â‚€ is called the initial phase of S.H.M.

Here, the phase is a physical quantity that is used to express the position and direction of motion of the particle at an instant concerning time represented by a sine or cosine function.

From Fig 3(b)

If we consider B as the starting position of the particle of reference.

IfÂ âˆ BOX = Ñ„â‚€ andÂ âˆ BOP = Ï‰t

Then, âˆ XOP =Â Ï‰t + Ñ„â‚€

Now, from eq(1) and (2), we get

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â yÂ = a Sin(Ï‰t + Ñ„â‚€)

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â x =Â a Cos(Ï‰t + Ñ„â‚€)

Here, + Ñ„â‚€ is called the initial phase of S.H.M.

Shm Circular Motion

A reference particle moving along the circumference of a circle of reference makes a displacement.

Where the maximum displacement of a particle from its position is called the amplitude denoted by, â€˜Aâ€™. It is equal to the radius of a circle.

If S is the span of S.H.M. Then,Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Shm in Circular Motion

The velocity of a particle at an instant is the rate of change of displacement.

From (1),Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â y = a SinÏ‰t

Differentiating both sides:

Â Â Â Â Â Â Â Â d(y)/dt = a d(SinÏ‰t)/dtÂ

Â Â Â Â Â Â Â Â Â Â V = a Ï‰ CosÏ‰t =Â a Ï‰ \[\sqrt{(1-SinÂ²Ï‰t)}\]Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â Â Â Â = Ï‰ \[\sqrt{(aÂ² - yÂ²)}\]Â Â

At mean position, y = 0, thenÂ

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â V = a Ï‰

At extreme position, y = a

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â V = 0

Thus, the maximum velocity in shm for a body in uniform circular motion is called the velocity amplitude.

FAQ (Frequently Asked Questions)

Q1: What is a Circular Motion and Periodic Motion?

Ans: Â A body that makes a circular locus can make a full circle or even a part of the circle while the body repeats itself after a fixed interval of time.

Such a circular motion can be a periodic motion if it repeats its motion after each 360Â° revolution.Â

Q2:Â Is Circular Motion SHM?

Ans: Yes, one-dimensional circular motion can be described as simple harmonic motion. A particle P moves along the circle of reference with a constant velocity Ï‰ undergoing a uniform circular motion. Its projection on the x-axis undergoes a simple harmonic motion.

Q3: Is Rotational and Circular Motion the Same?

Ans: The difference between circular motion and rotational motion is outlined below:

The motion of a body around its center of mass is called the rotational motion,

while circular motion is a special case of rotational motion, where the distance between the bodyâ€™s center of mass and the axis of rotation remains fixed.

Q4: Write Circular Motion Examples.

Ans: There are certain real-life examples of circular motion as outlined below:

The spinning of a ball on a string

Revolution of planets around the sun.

Turning a car through a curve in a race track.