Simple Harmonic Motion and Uniform Circular Motion

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 Θ and at time = t, the particle reaches from point OX to P.

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

So, the angle subtended by a particle,  Θ = ωt

The mean position of the particle is at point O.

Now, we draw a perpendicular from P to a certain point on the diameter XOX’.

So the displacement from O to a certain point is, ‘x’.

The instantaneous acceleration will be directly proportional to this displacement.

                              a α x

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

                              ma α mx

                            Or,     F α 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, XOX’ and YOY’ are perpendicular diameters of the circle of reference.

As the reference particle moves from X to Y, its projection on diameter YOY’ moves from O to Y.

As this reference particle moves from Y to X’, its projection moves along the diameter from Y to O.

Similarly, when the reference particle moves on the circle from X’ to X via Y’, its projection moves along the diameter from O to Y’ and then from Y’ to O.

Thus, during the time the particle P goes around the circle and completes one revolution, its projection, ‘M’ oscillates about the point O along the diameter YOY’ and completes one vibration. Since the projection of the reference particle is in SHM, and the projection of M on diameter YOY’ 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, ‘a’ with uniform angular velocity, ‘ω’

From Fig.1

Let the particle at time t = 0, start from point X, and sweep an angular displacement Θ in time ‘t’ with angular velocity ω, equal to ωt.

Now, let the projection of the particle P on diameter YOY’ be at M.

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

                                 OM = y

In ΔOPM,

                            SinΘ = OM/OP = y/a 

Or,                       

Now, 

In ΔONP,

                CosΘ = ON/OP = x./a

                Or, 

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From Fig. 3(a)

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

Here, ∠AOX = ф₀ and  ∠AOP = ωt and Θ  = ωt - ф₀

From eq(1) and (2)

             y  = a Sin(ωt - ф₀)

             x =  a Cos(ωt - ф₀)

 Here, - ф₀ 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.

If  ∠BOX = ф₀ and  ∠BOP = ωt

Then, ∠XOP =  ωt + ф₀

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

                        y  = a Sin(ωt + ф₀)

                        x =  a Cos(ωt + ф₀)

Here, + ф₀ 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, ‘A’. It is equal to the radius of a circle.

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

Shm in Circular Motion

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

From (1), 

                    y = a Sinωt

Differentiating both sides:

        d(y)/dt = a d(Sinωt)/dt 

          V = a ω Cosωt =  a ω \[\sqrt{(1-Sin²ωt)}\]                       

             = ω \[\sqrt{(a² - y²)}\]  

At mean position, y = 0, then 

                  V = a ω

At extreme position, y = a

                   V = 0

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