Simple Harmonic Motion

We see various types of motion in our day-to-day life, such as the motion of the blades of a fan, motion of the hands of a wristwatch, motion of the wheels of a car, etc. We find the motion of these objects keeps repeating themselves. Such motions are periodic by nature. One such type of periodic motion is simple harmonic motion or SHM. The simple harmonic motion definition is that it is a periodic motion in which the restoring force on the object is directly proportional to the displacement of the object from the mean position. The restoring force of the simple harmonic motion is always directed towards the mean position.

Here, we will derive the simple harmonic motion formula. Along with this, we will go through various examples of simple harmonic motion. So, let us learn the concept of SHM!

What is Simple Harmonic Motion?

In physics, a simple harmonic motion is the repetitive back and forth movement of an object (spring) through an equilibrium, or mean position, so that the maximum displacement on one side of this position remains equal to the maximum displacement on the other side. 

Also, the time interval of each complete vibration remains the same. The force accountable for the motion is always directed toward the equilibrium position and is directly proportional to the distance from it. 

Simple Harmonic Motion Formula

Observe a fixed spring, i.e. the spring with its one end fixed. When you don’t apply any force on the spring, it remains at its equilibrium position. Now, if you pull it outwards, the force exerted on the spring, which is directed towards the equilibrium position, is displaced by a distance ‘x’.

On the other hand, if you push the spring inwards, the force exerted on the spring directs towards the equilibrium position. Therefore, you can see that the force imposed either outer or inner by the spring is always directed towards the equilibrium position. This force is known as the restoring force.

Let the force be F and the displacement of the string from the equilibrium position be x.

Therefore, the restoring force is given by,

\[F=-kx\]

Or,

\[F(t)=-kx(t)\]...(1)

Here,

F is the force, 

x is the displacement, and 

k is a constant or force constant. Its unit is N/m

This relation is called Hooke’s law. We also call this equation the Simple Harmonic Motion Equation.

The acceleration of a particle under simple harmonic motion (SHM) is given by,

\[a(t)=-\omega ^{2}x(t)\]....(3)

Here,

a = acceleration of the particle

ω = angular velocity of the particle.

Force Law for Simple Harmonic Motion

Additionally, the force law states that the force acting on simple harmonic motion is proportional to the displacement and is always towards the center of motion.

We know that,

\[F(t)=ma\]….(2)

Comparing eq (1) and (2), we get:

\[-kx(t)=ma(t)\]

From eq (3)

\[-Kx(t)=m(-\omega ^{2}x(t))\]

So, \[k=m\omega ^{2}\]

Or, 

\[\omega =\sqrt{\frac{k}{m}}\]…(4)

Also, we know that  ω  = \[\frac{2\pi }{T}\],  substituting this in eq (4):

\[\frac{2\pi }{T}=\sqrt{\frac{k}{m}}\]

Or,

\[T=\frac{1}{2\pi}\sqrt{\frac{k}{m}}\]…(5)

Here, eq (5) is the formula for the time period of simple harmonic motion.

Now, let us derive the formula for Angular SHM.

Angular Simple Harmonic Motion Formula

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From the above diagram, let θ be the angular displacement of the  circular disc and its resultant torque be , then

\[\vec{\tau } \alpha \vec{\theta }\]

\[\Rightarrow \vec{\tau }=-\kappa \vec{\theta}\]

Here,

is the restoring torsion constant, which is defined as the torque per unit angular displacement. 

Here, if ‘I’  is  the moment of inertia of the body and is the angular acceleration then

\[\vec{\tau}=\vec{I\alpha}=-\kappa \vec{\theta}\]

We know that \[\vec{\alpha }=\frac{d^{2}\vec{\theta }}{dt^{2}}\] 

So,\[\frac{d^{2}\vec{\theta}}{dt^{2}}=-\frac{\kappa }{I}\theta\]

(This differential equation is just like the simple harmonic motion equation in differential form)

The angular frequency will be:

\[\Rightarrow \omega =\sqrt{\frac{\kappa }{I}}rad/s\]

The frequency will be:

\[\Rightarrow f=\frac{1}{2\pi}\sqrt{\frac{\kappa}{I}}Hz\]

Time period of angular simple harmonic motion will be:

\[\Rightarrow T=2\pi \sqrt{\frac{I}{\kappa}}s\]

Examples of Simple Harmonic Motion

We see various examples of simple harmonic motion in our everyday life, such as the pendulum, simple harmonic motion, motion of a spring, etc. 

Besides these, we have two specific examples of SHM that are as follows:

• Simple harmonic oscillator

• Musical instruments

Now, let us discuss the science of SHM behind the two.

The Motion of a Simple Harmonic Oscillator

A particular example of a simple harmonic oscillator is the vibration of a mass connected to a vertical spring, the other end of which is fixed in a ceiling. At the maximum displacement of − x, the spring is under its greatest tension, which forces the mass upward. At the maximum displacement of + x, the spring reaches its greatest compression, which forces the mass to back downward again. 

At both ends of maximum displacement, the force is greatest and is directed in the direction of the equilibrium position, the velocity (v) of the mass is zero, its acceleration is at most, and the mass changes direction. At the equilibrium position, the velocity is at its maximum and the acceleration has declined to zero. 

Characteristics of simple harmonic motion include the changing acceleration, which is always towards the equilibrium position and is proportional to the displacement from the equilibrium position.

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Moreover, the interval of time for each whole vibration is constant and does not depend upon the size of the maximum displacement. In some forms, therefore, simple harmonic motion is at the heart of timekeeping.

SHM in Musical Instruments

The motion of musical instruments is simple harmonic because musical instruments make such vibrations that in turn cause corresponding sound waves in the air. Musical sounds are simply a mixture of many simple harmonic waves, wherein the vibrating parts of a musical instrument like a guitar oscillate in sets of superimposed SHM and with frequencies that are multiples of the lowest fundamental frequency. 

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In fact, any regularly repetitive motion and any wave, irrespective of how complicated its form, can be treated as the sum of a series of simple harmonic motions or waves, a discovery first published in 1822 by the French mathematician Joseph Fourier.

Now, let us define linear SHM.

Define Linear Simple Harmonic Motion

Here’s how to define linear SHM: It is one of the simplest kinds of oscillatory motion in which a body when displaced from its mean position, oscillates 'to and fro' about mean position and the restoring force is always towards its mean position and its magnitude is proportional to the displacement from the mean position.

The derivation of linear SHM  is as follows:

Assume that a particle P moves 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 reference circle. As the particle P moves from X to Y, its projection on diameter YOY’ shifts from O to Y. When the particle further moves from Y to X’, its projection shifts from Y to O. Similarly, when particle P moves on the circle from X’ to Y’ and finally to X, its projection also shifts from O to Y’, and then from Y’ to O.

Thus, the time during which P completes one revolution, its projection given by ‘N’, oscillates about the point O along the diameter YOY’ and completes one vibration. Since the projection of the reference particle is in simple harmonic order, the projection of M on diameter YOY’ is also SHM.

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

Now, again from Fig. 1, assume that a reference particle is moving on a circle of reference with radius, ‘a’ with uniform angular velocity, ‘ω’.

Let the particle at time t = 0, starts from point X, and makes an angle Θ (angular displacement) in time ‘t’ with angular velocity ω, then equal θ = ωt.

Assume that the projection of the particle P on diameter YOY’ is N. Then the displacement in SHM at time t is

ON = y

In ΔOPN,

Sin θ = ON/OP = y/a 

y = aSinΘ  = aSinωt ….(1)

Now, 

In ΔONP,

Cos θ = ON/OP = x/a 

x = aCos θ = aCosωt…(2)

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Assume that A is the starting position of the reference particle. Then,

∠AOX = ф₀, 

∠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 SHM.

The phase is a physical quantity that we use to represent the position and direction of motion of the particle at an instant regarding time represented by a sine or cosine function.

Consider B as the starting position of the particle of reference. Then,

∠BOX = ф₀,

∠BOP = ωt, and

∠XOP =  ωt + ф₀

Now, from eq (1) and (2), we have the following:

y  = a Sin (ωt + ф₀)

x =  a Cos (ωt + ф₀)

Here, + ф₀ is called the initial phase of SHM.

So, from the simple harmonic definition, we understand that SHM is a repetitive back-and-forth motion of an object, directed towards an equilibrium, or central, position so that the maximum displacement on either of the positions is equal to the maximum displacement on their opposite sides. 

Thus, F = - kx, where F is the force, x is the displacement, and k is a constant or the ‘Force Constant,’ whose unit is N/m.’

Interesting Facts About SHM

• Several physical systems exhibit SHM (assuming no energy loss): the electrons in a wire transmitting alternating current, the vibrating particles of the medium in a sound wave, and other assemblages incorporating comparatively small oscillations about a position of stable equilibrium.

• Just like the force, acceleration in SHM also directs towards the mean position.

• We can represent the SHM by both sine and cosine functions.