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:
As it moves from X to P, there is an angular displacement (an arc) which is equal to [theta] 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, [theta] = 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 OF. This circle is considered as a circle of reference with particle P as the particle of reference.
Here, if you look at Fig.1, XOX’ and YOU’ 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 [theta] in time ‘t’ with angular velocity ω, equal to ωt.
Now, let the projection of the particle P on diameter YOU’ be at M.
Then the displacement in SHM at time t is given by,
OM = y
In ΔOPM,
Sin [theta] = OM/OP = y/a
or,
|
y = aSin[theta] = aSinωt ….(1) |
Now,
In ΔONP,
Cos[theta] = ON/OP = x./a
or,
|
x = aCos [theta] = aCosωt…(2) |
From Fig. 3(a)
Now, if A is the starting position of the reference particle.
Here, ∠AOX = ф₀ and ∠AOP = ωt and [theta] = 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-Sin^{2}ωt)}]
= ω[sqrt{(a^{2} – y^{2})}]
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.
Modulus of Elasticity
Let us understand the modulus of elasticity using examples in this post. The modulus of elasticity is a measurement of the object’s stress–strain relationship. The modulus of elasticity is the most important factor in calculating the deformation response of concrete when stress is applied.
Elastic constants are the constants that govern the deformation caused by a specific stress system operating on a material.
Uniform Circular Motion Projection and Simple Harmonic Motion of a Spring
Connect the Ball-and-Spring set to a ring stand. Next to the spring, place the vertical-plane spinning motor. Keep the illuminated slide projector at a position approximately a meter away and then the images of the ball undergoing oscillations must be projected on the spring and the revolving ball on the wall. Begin the mass’s oscillation so that it is in phase with the driven ball.
The movement of an ite
m along a circular path at a constant pace is described as uniform circular motion. This motion’s one-dimensional projection may be represented as simple harmonic motion.
Examples
A point P traveling on a circular route with constant angular velocity is in uniform circular motion. Its x-axis projection exhibits simple harmonic motion. This can be compared to the projection of an oscillating mass’s linear vertical motion on a spring.
If we attach a stone to the end of a string and move it in a horizontal plane around a fixed point at a constant angular speed, the stone will travel in a uniform circular motion in the plane. If we look at the stone from the side, it appears to move back and forth along the horizontal line, with the other end of the string serving as the halfway point.
Similarly, the stone’s shadow or projection of motion would appear to move in a to and fro motion perpendicular to the circle’s plane. Galileo witnessed a similar instance when he found that Jupiter’s four main moons moved back and forth relative to the planet in a simple harmonic motion.
If a particle moves in a uniform circular motion, its projection moves in a simple harmonic motion, where the axis of oscillation is the diameter of the circle, or in other words, we can say that SHM is nothing but the projection of uniform circular motion along the circle’s diameter.
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More About SHM and its Similarities to Circular Motion
In SHM the maximum displacement on one side of this position equals the maximum displacement on the other side. Each full vibration is characterized by the same time interval. The force that is responsible for driving the motion exhibits a direction that is towards the equilibrium position and is proportionate to the distance from it. That is, F = kx, where F denotes force, x denotes displacement, and k denotes a constant. A force applied in the opposite direction of displacement to return the system to its rest position. The size of the force is solely determined by displacement, like in Hooke’s law.
Uniform circular motion is defined as the constant-speed movement of an item in a circle. When an item travels in a circle, its direction is continually changing. The item is always traveling tangent to the circle. Due to the fact that the velocity vector’s direction is equal in value as the direction of the motion of the object, the velocity vector is also tangent to the circle.
An item traveling in a circle is gaining speed. Objects that accelerate are those that change their velocity – either the speed (i.e., magnitude of the velocity vector) or the direction. It is seen that any object that undergoes uniform circular motion, is always moving at a constant pace. Despite this, it is speeding as a result of its shift of course. The acceleration is pointing inwards.
The net force is the ultimate motion characteristic of an item moving in a uniform circular motion. The net force acting on such an item is directed towards the circle’s center. The net force is referred to as an inward force or centripetal force. Without such an internal force, an object would travel in a straight line, never straying from its intended path. However, because the inward net force is oriented perpendicular to the velocity vector, the item is continually changing direction and accelerating inward.