Simple Harmonic Motion SHM

What is Simple Harmonic Motion?

The motion of an object that moves to and fro about a mean position along a straight line is called simple harmonic motion. A pendulum undergoes simple harmonic motion. It swings to and fro about its mean position where the string and the bob undergo the motion. These movements of pendulums are called oscillations, which show simple harmonic motion.

Let us consider a spring that is fixed at one end. If no force is applied to the spring, it remains in the equilibrium position. Now,

  • If we pull the spring outwards, a restoring force is generated on the spring, which pulls it inwards towards the equilibrium position.

  • If we push the spring inwards, a force is generated to bring it to its equilibrium position.

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The force exerted by the spring is to attain its equilibrium position. This is called the restoring force. Now, if the force is ‘F’ and the displacement of the spring from its equilibrium position is ‘x’.

Then the restoring force will be, F = - kx.

The negative sign indicates that the force is exerted in the opposite direction.

Where, 

k is the force constant. The unit of spring constant/force constant is N/m in the S.I system and dynes/cm in the C.G.S. system.

We can calculate the energy in SHM simple harmonic motion.


Simple Harmonic Motion Equation

Consider a particle of mass ‘m’ exhibiting Simple Harmonic Motion along the path x O x. Let the mean position of the particle be O. Let the speed of the particle be ‘v0’ when it is at position p (at a distance x0 from the mean position O).

At t = 0, the particle is at point P (moving towards the right direction).

At t = t, the particle is at point Q (at a distance x from O).

With a velocity ‘v’.

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Simple Harmonic Motion Formula

Let us consider a spring fixed at one end. The spring remains in its equilibrium position when no force is applied to it. If we pull the spring outwards, then a force is exerted by the spring, which directs it towards the equilibrium position.

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If we push the spring inwards, a force is generated on it, which is directed towards the equilibrium position. In both cases, the force given by the spring is towards the equilibrium position. This is called the restoring force.

Let the force exerted by the spring be ‘F’, and the displacement of the spring from the equilibrium position be ‘x’.

The restoring force will be given by,

F= – kx.

The negative sign denotes that the restoring force acts in the opposite direction.

Here, k is the force constant.

Let the mass of the spring be ‘m’. The acceleration of the body will be:

a = F/m

a = – k . x/m

= – ω2x

Herek/m=ω2

The time taken by an object to finish one oscillation is called its time period. The frequency of an object exhibiting Simple Harmonic Motion is the number of oscillations that it undergoes per unit amount of time. The frequency of oscillatory motion is given by,

f = 1/T

Where,

a = acceleration

F = force

T = time period

m = mass

f = frequency

k = force constant

ω = angular frequency.


Simple Harmonic Motion Differential Equation

The solutions of simple harmonic motion differential equation are given below:

x = Asinωt, it is the solution for the particle when it is in its mean position point ‘O’ in figure (a).

x0​ = Asinϕ, it is the solution for the particle when it is in any other position but not in the mean position in figure (b).

x = Asin(ωt+ϕ) (When the particle is at point Q in figure ‘b’ shown above (at any time t).

These solutions can be easily verified by substituting the value of x in the differential equation.


Simple Harmonic Motion Examples

Any object which repeats its motion over a period of time, to and fro about a mean position, executes simple harmonic motion.

Examples:

  1. Simple pendulum.

  2. Mass spring system.

  3. A steel ball rolling in a curved dish.

  4. The motion of a swing.


Application of Simple Harmonic Motion

Some of the real-world applications of SHM are as follows:

  1. Clock

  2. Musical Instruments

  3. Car Shock Absorbers

  4. Bungee Jumping

  5. Diving Board

  6. The Process of Hearing

  7. Metronome

  8. Earthquake-proof buildings


1. Clock

Either a pendulum or vibrating quartz maintains time in a clock. The motion of pendulum or quartz must be periodic to ensure accurate time.

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2. Car Shock Absorbers

Springs attached to the wheels of cars are necessary for a smooth ride for the passengers. When the car passes through a bump in the road, the wheels rise up. 

If there are no shock absorbers in the car, the whole car will be thrown up and down, which makes it uncomfortable for the passengers. 


3. Musical Instruments

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Sound is produced from the oscillations of the air. In musical instruments like violin and guitar, bowing and plucking of the string provides the necessary force to make the string oscillate.

FAQ (Frequently Asked Questions)

Q1. State the Difference Between Periodic Motion, Oscillation, and Simple Harmonic Motion.

Ans: The difference between periodic motion, oscillation, and simple harmonic motion is given below.


i. Periodic Motion

A motion that continues itself in a rhythmic way after equal intervals of time is called periodic motion. For example, uniform circular motion.


ii. Oscillation Motion

The to-and-fro motion of an object about a mean position is known as oscillatory motion. It is a kind of periodic motion that occurs between two extreme points. 

For example, the oscillation of the pendulum and spring-mass system.


iii. Simple Harmonic Motion or SHM

It is a special case of oscillatory motion in which the object oscillates between two extreme points in a straight line.

Q2. In SHM, the Motion is to-and-fro with Periodic Reason(R): 

Here, the Velocity of the Particle V = ω√A2 – x2 Where x is Displacement as Measured from the Extreme Position Chose the Right Answer:


a. Both A and B are Correct & R is the Right Explanation of A.


b. A is False, and R is True.


c. Both A and B are Correct & R is not the Right Explanation of A.


d. A is True, and R is False.

Ans: c) A is correct, and R is false. In V = ω√A2 – x2, x is measured from the mean position, not from the extreme position, and SHM involves to-and-fro periodic motion.

Q3. What is Amplitude in Simple Harmonic Motion?

Ans: The amplitude is the highest displacement the particle exhibiting SHM, undergoes from the equilibrium state. Therefore, the equation x = Acosωt applies to the state of the particle that experiences SHM. 

Here, the dimension of the state of the particle experiences a uniform circular motion.

Q4. What is Damped SHM?

Ans: If the motion of an oscillator is reduced due to some external force, the motion is said to be damped. For example, in the spring, though, the damping is smaller, but the frictional force acting on its mass, brings it to rest.