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/aÂ
Or,                      Â
Now,Â
In ΔONP,
                CosΘ = ON/OP = x./a
                Or,Â
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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,                           Â
                                            Â
Shm in Circular Motion
The velocity of a particle at an instant is the rate of change of displacement.
From (1),Â
                    y = a Sinωt
Differentiating both sides:
        d(y)/dt = a d(Sinωt)/dtÂ
          V = a ω Cosωt = a ω \[\sqrt{(1-Sin²ωt)}\]                      Â
             = ω \[\sqrt{(a² - y²)}\] Â
At mean position, y = 0, thenÂ
                  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.
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 360° 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 ω 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 body’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.