2. Energy of SHM and to oscillations

Simple Harmonic Motion and Uniform Circular motion

Tie a ball to the end of a string and make it move in a horizontal plane about a fixed point with a constant angular speed. The ball would then perform a uniform circular motion in the horizontal plane.

The displacement of the particle in a uniform circular motion can be represented by an oscillating function using either sine or cosine function. Like in the figure given below, displacement of the particle in a uniform circle is represented by sine function .x(t)= A sin(wt) for the left and x(t)= A cos (wt+ϕ)  for the right.

As the particle is moving in the same way the projections are also moving.

• When the particle is moving in the upper part of the circle then the projections start moving towards the left.
• When the particle is moving in the lower part of the circle then the projections are moving towards the right.
• We can conclude that the particle is swinging from left to right and again from right to left.  This to and fro motion is SHM.

Velocity and acceleration in SHM

Velocity in SHM

The speed v of a particle in a uniform circular motion, its angular speed ω times the radius of circle A. v= ωA

The direction of velocity v at a time t is along the tangent of the circle at the point where the particle is located at that instant. From the geometry of the given figure. It is clear that the velocity, of the projection particle P at time t.

When the displacement of the particle is given as

Where the negative sign shows the direction of v(t) is opposite to the +ve direction of the axis.

Acceleration in SHM: The instantaneous acceleration of the particle in SHM is given by  

Acceleration is always directed towards the equilibrium position.

The magnitude of the acceleration is minimum at equilibrium position and maximum at extremes.

|a|min =0   at equilibrium  position  ; |a|max =w2A  at extreme positions

Energy in SHM

The Kinetic and Potential energies in an SHM vary between 0 and their maximum values.

Kinetic energy, potential energy and the total energy is a function of time in the above graph. BothKinetic energy and potential energy repeats after time T/2.

Kinetic energy, potential energy and total energy is a function of displacement in the above graph.

The kinetic energy (K.E.) of a particle executing SHM can be defined as

 , where k=mw²= Force constant

)

The above expression is a periodic function of time, being zero when the displacement is maximum and maximum when the particle is at the mean position.

The potential energy (U) of a particle executing simple harmonic motion is,

The potential energy of a particle executing simple harmonic motion is also periodic, with period T/2, being zero at the mean position and maximum at the extreme displacements.

Total energy of the system always remains the same 

T.E.= P.E+ K.E. 

The above expression can be written as  

Total energy is always constant

Simple Pendulum

A simple pendulum is defined as an object that has a small mass (pendulum bob), which is suspended from a wire or inextensible string having negligible mass and suspended from a fixed support.    

• The vertical line passing through the fixed support is the mean position of a simple pendulum.
• The vertical distance between the point of suspension and the center of mass of the suspended body (when it is in the mean position) is called the length of the simple pendulum denoted by L.

• When the pendulum bob is displaced it oscillates on a plane about the vertical line through the support.
• Simple pendulum can be set into oscillatory motion by pulling it to one side of equilibrium position and then releasing it.

In the above image one end of a bob of mass, m is attached to a string of length L and another to rigid support executing simple harmonic motion. 

Time Period of Simple Pendulum

A point mass M is suspended from the end of a light inextensible string whose upper end is fixed to a rigid support. The mass is displaced from its mean position.

Assumptions:

• There is negligible friction from the air and the system
• The arm of the pendulum does not bend or compress and is massless
• The pendulum swings in a perfect plane
• Gravity remains constant

Free, Damped and forced oscillations

Free Oscillation   A system capable of oscillating is said to be executing free oscillations if it vibrates with its own natural frequency without the help of any external periodic force. For example Oscillation of a tuning fork when struck on a rubber band, the oscillation of a simple pendulum when displaced from its mean position etc.

Damped Oscillation: Damped oscillation refers to the type of vibration of a body whose amplitude keeps on decreasing with time.

• In this type of vibration, the amplitude decreases exponentially because of damping forces like frictional force, viscous force etc.
• Because of the decrease in amplitude, the energy of the oscillator also keeps on decreasing exponentially.

Forced Oscillation:

When a system (like a simple pendulum) is displaced from its equilibrium position and released. It oscillates with its natural frequency ‘w’ and the oscillations are called free oscillations. But they die out eventually due to damping force. However, they can be maintained by an external agency.

Forced oscillation refers to the type of vibration in which a body vibrates under the influence of an external periodic force.

Resonance

When a body oscillates with its own natural frequency f₀ with the help of an external periodic force whose frequency ‘f’ is equal to the natural frequency of the body, the oscillation of the body is called resonance.

f0 / f =1   ;    fo= f    this is the condition of resonance

• This is the condition of resonance. If accidentally the forced frequency happens to be close to one of the natural frequencies of the system, the amplitude of oscillation will shoot up (resonance), resulting in the possible destruction.
• This is the reason why soldiers go out of step while crossing a bridge else the frequency of steps may equal the natural frequency of the bridge which may result in the breaking of the bridge due to resonance