A motion that repeats itself at regular intervals of time is called periodic motion. Simple harmonic motion is the simplest form of oscillatory motion which is a kind of periodic motion bounded between two extreme points. Every oscillatory motion is periodic, but every periodic motion need not be oscillatory, ie., Circular motion is a periodic motion, but it is not oscillatory

Simple Harmonic Motion Definition 

Simple harmonic motion (SHM) defined as the motion that arises when the force on the oscillating body is directly proportional to its displacement from the mean position, which is also the equilibrium position. Further, at any point in its oscillation, this force is directed towards the mean position.

What is simple harmonic motion?

Simple harmonic motion is a periodic motion of a point along a straight line, such that its acceleration is always towards a fixed point in that line and is proportional to its distance from that point

Simple Harmonic Motion Examples :

A particle oscillating back and forth about the origin of an x-axis between the limits +A and – A as shown below. This oscillatory motion is said to be simple harmonic if the displacement x of the particle from the origin varies with time as :

x (t) = A cos ( ωt + f )

where A, ω and f are constants

representation of simple harmonic motion

X versus t graph for simple harmonic motion : 

X versus t graph for simple harmonic motion

Amplitude(A) : The amplitude A of SHM is the magnitude of maximum displacement of the particle.

Displacement : The distance of the particle from it’s equilibrium position. Equilibrium is where no net force acts on the particle.

Types of Simple Harmonic Motion : 

Simple Harmonic Motion can be classified into two types:

  • Linear SHM
  • Angular SHM

Linear SHM 

In linear SHM  a particle moves to and fro around a fixed equilibrium point along with a straight line.

Example : spring-mass system

Linear simple harmonic motion spring mass system

Angular SHM 

When a system oscillates angular with respect to a fixed axis then its motion is angular simple harmonic motion.

Example : simple pendulum 

Simple pendulum

Velocity, Acceleration and Time in Simple Harmonic Motion

Velocity in SHM 

The direction of velocity v at a time t is along the tangent to the circle at the point where the particle is located at that instant.

Particle P moving in a circle with angular speed ω.

Velocity analysis in simple harmonic motion

Velocity of the particle in SHM can be given by 

V = dx/dt

Since, x = A sin (ωt + Φ)

So, v = d(A sin (ωt + Φ)/ dt = ωAcos(ωt+ϕ)

Thus,

Acceleration in SHM :

Particle P moving in a circular motion is exerting a radial acceleration a towards the centre 

Acceleration analysis in simple harmonic motion

It’s projection on x acid at time is 

a = dv/dt 

   =  d(Aωcosωt+ϕ)/dt 

   = −ω² Asin(ωt+ϕ) 

Hence, amaximum = −ω² A

Thus the magnitude of radial acceleration of P is −ω² A

The acceleration of a particle in SHM is proportional to displacement. thus whatever the value of x between –A and A, the acceleration a(t) is always directed towards the center.

Time Period of SHM

Time period is given by

T = 2πr/v

but v=rω. So, r/v = 1/ω

Hence, T = 2π/ω

Force in Simple Harmonic Motion

Now by using Newton’s second law of motion, and the expression for acceleration of a particle undergoing SHM, the force acting on a particle of mass m in SHM is

F (t) =  ma

        =–mω2 × (t)     

F(t) = -K × (t)

Where K = –mω2

force is always directed towards the mean position hence it is sometimes called the restoring force in SHM.

ENERGY IN SIMPLE HARMONIC MOTION :

Both kinetic and potential energies of a particle in SHM vary between zero and their maximum value. 

Total energy = Kinetic energy + potential energy 

Kinetic energy of a Particle in SHM

Kinetic Energy(KE) = ½ mv²

Since v² : A²ω²cos²(ωt+ϕ) 

So KE = ½ mA²ω²cos²(ωt+ϕ)

           = ½ mω² ( A² – x²)

Therefore, the kinetic energy is ½ mω² ( A² – x²)

Kinetic energy is zero when the displacement is maximum and is maximum when the particle is at the mean position.

Potential energy of a Particle in SHM 

Potential energy (PE) = ½ kx²

Since X² = ω²sin²(ωt+ϕ) 

So, PE = ½ kω²sin²(ωt+ϕ)

Thus, potential energy is ½ kω²sin²(ωt+ϕ)

Therefore, Potential energy is zero at the mean position and maximum at the extreme displacements.

Total Mechanical Energy of the Particle in SHM :

Total energy = kinetic energy + potential energy 

Tota energy = ½ mA²ω²cos²(ωt+ϕ) + ½ kω²sin²(ωt+ϕ)

Total energy = ½ mA²ω² {cos²(ωt+ϕ) + sin²(ωt+ϕ) }

Since, cos²(ωt+ϕ) + sin²(ωt+ϕ) = 1 

So total energy = ½ mA²ω²

Thus, total energy is of particle doing SHM is  ½ mA²ω²

Graphical representation of energy :

  • Potential energy, kinetic energy and total energy as function of time t :
Potential energy, kinetic energy and total energy as function of time t
  • Potential energy, kinetic energy and total energy as function of position x :
Potential energy, kinetic energy and total energy as function of position x

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