Simple Harmonic Motion

We see various motions like the motion of the clock, the motion of the cradle, the vibrating string of a guitar, swings in the park. What is the similarity in their motion? The motion is to and fro. Take an example of a swing; the swing moves forward from its mean position leading to some displacement. After some distance, it will come back to a mean position due to a restoring force opposite to the swing but directly proportional to the magnitude of the displacement. This motion of the string, which has the magnitude of restoring force opposite to the displacement, is Simple Harmonic Motion (SHM)

What is simple harmonic motion?

A body is said to be in simple harmonic motion if its motion is about its equilibrium position. At any given point, its acceleration is directly proportional in magnitude to its displacement but opposite in direction to displacement.

Simple harmonic motion definition– SHM is defined as the body’s motion in which the body oscillates about its equilibrium position, having an acceleration that is of the same magnitude of displacement but is antiparallel to the displacement. The unit of SHM is a meter. 

Conditions of a Body to be in Simple Harmonic Motion.

1. The body must have periodic motion and its mean position.
2. Acceleration and displacement should always be in an antiparallel direction.
3. Acceleration must be towards the fixed mean position.

Derivation of Equation of Simple Harmonic Motion

Let’s take any object of mass ‘m’ having SHM with mean position x0 and the displacement x. Now as per the definition of SHM,

Restoring force α – Displacement ( ‘-‘ denotes that restoring force and displacement are opposite) 

As per Newton’s 2nd law of motion,

Force= Mass × Acceleration

Therefore,

Mass × Acceleration α -displacement

m × a α –x

By differentiating the equation,

md2x / dt2 α –x

⇒ md2x / dt2 = –kx        …… (where k is force constant)

⇒d2x / dt2 = -k × x/m

Now let’s take k /m as ω2, the equation becomes

d2x / dt2 = – ω2 × x

⇒d2x /dt2+ ω2 × x = 0….. (Differential equation of SHM)

Multiply differential equation of SHM with 2dx / dt,

⇒2dx / dt × d2x / dt2+ ω2 × 2xdx/dt = 0

⇒d/dt (dx/dt)2  + ω2 d(x2)/dt =0

Integrating the above equation,

⇒ (dx/dt)2 + ω2 × x2 = A (constant)…….(1)

At maximum displacement, 

x = a (where a= amplitude), dx / dt = 0, therefore,

0 + ω2 × a2 = A 

⇒A= ω2 × a2

Substituting value of A in equation (1), 

(dx/dt)2 + ω2 × x2 = ω2 × a2

⇒ (dx/dt)2 = ω2 × x2 – ω2 × a2

⇒ (dx/dt)2 = ω2 (a2 – x2 )

By taking square root on both side, we get

Velocity= dx/dt = ω √(a2 – x2 )

⇒dx/ √(a2 – x2 ) = ωdt

On integrating,

sin-1 (x/a)= ωt + ø(constant)

x/a =sin(ωt + ø)

Displacement- The distance between the start point and the final point of the moving object.

Amplitude- Maximum displacement of the object having SHM from the equilibrium position.

Angular frequency- Angular displacement of an object per unit time(the object must be in oscillatory motion).

Phase in SHM- The phase of an SHM is an angular term representing the position of the particle performing SHM at a specific or any instant.

Phase Difference- The difference between the two-phase angles of SHM.

Types of simple harmonic motion 

1. Linear SHM 

In Linear SHM, the restoring force of the body in linear periodic motion is always directed to its mean position, and the magnitude of the restoring force is always equal to the displacement of the body from its mean position.

2. Angular SHM

In Angular SHM, the body in oscillatory motion has torque for angular acceleration, directly proportional to the magnitude of angular displacement. Its direction is opposite to the angular displacement.

Example of Simple Harmonic Motion

1. Mass of the string
2. Oscillatory motion.
3. Mass of a simple pendulum
4. Uniform circular motion.
5. Scotch yoke.

Applications of Simple Harmonic Motion 

1. Swing
2. Pendulum
3. Guitar
4. Bungee Jump
5. Cradle
6. The process of hearing.
7. Diving board
8. Metronome.
9. Earthquake-proof buildings.

Conclusion

In SHM, the acceleration of the body is always equal to the magnitude of the displacement, and both the quantities are opposite each other. The overall energy during the Simple Harmonic Motion is conserved. 

SHM is a periodic motion. According to the velocity equation, i.e. v = ω √(a2 – x2 ) in SHM, velocity will be maximum at the mean position, i.e. v= ωa, and zero at the extreme position.

K.E maximum at the mean position and zero at the extreme position. P.E maximum at the extreme position and zero at the mean position.