# Oscillatory Motion

### Introduction

If a body moves to and fro about a mean position repeatedly, then that motion is called oscillatory motion. Periodic motion is a motion that repeats itself after a fixed interval of time. It is not necessary that in periodic motion the body moves in to and fro motion.

Oscillatory motion is always periodic but vice-versa is not true. For example: motion of wheel of car is only periodic because it moves in a circular motion. Examples of oscillatory motion are: oscillations of simple pendulum, vibrations of the mass attached to a string etc.

### PERIOD, FREQUENCY AND DISPLACEMENT

As we know Periodic motion is a motion that repeats itself after a fixed interval of time. So, the shortest time taken by an object to repeat its motion is called Time Period of that motion represented by ‘T’.

The reciprocal of the time period tells us the number of repetitions of the motion in a unit time. It is called Frequency of the periodic motion represented by ‘v’. The unit of frequency is hertz:

$\mathrm{V}=1 / \mathrm{T}$

Now consider a block attached to a spring on one end. The other end of spring is fixed to a wall. Now suppose the surface is frictionless then the motion of the block can be described in terms of displacement ‘x’ of the wall.

Similarly, in a pendulum the motion can be described in terms of the angle of the string from the wall which is called angular displacement from the vertical.

### PERIODIC FUNCTIONS

A function that repeats its value on regular interval or periods is called a periodic function. In the graph of a periodic function we can observe that a single pattern is being repeated over and over again. Example: trigonometric functions etc.

The simplest periodic function can be written as:

$\mathrm{F}(\mathrm{t})=\mathrm{A} \cos (\omega \mathrm{t})$

### SIMPLE HARMONIC MOTION

When the restoring force is directly proportional to the displacement from equilibrium, the resulting periodic motion is called simple harmonic motion (SHM). The object in a harmonic motion moves in a uniform path with a variable force acting on it. The magnitude of force is proportional to the displacement of the mass and the force is always opposite in direction to the displacement direction.

If a particle is oscillating along the y-axis, its location on the y-axis at any given instant of time t, measured from the start of the oscillation is given by the equation

$y=A \sin (2 \pi f t)$

now as we know velocity is the first derivative of displacement thus velocity at time t is,

$v=2 \pi f A \underbrace{\cos (2 \pi f t)}$

and acceleration is second derivative of the displacement hence at time t it is equal to,

$a=-(2 \pi f) 2[A \underline{\sin (2 \pi f t)}]$

Here we will notice that both velocity and acceleration are also sinusoidal. But velocity has a $90^{\circ}$ or $\pi / 2$ phase difference while the acceleration function has a $180^{\circ}$ or $\pi$ phase difference relative to the displacement function.

This means that if displacement is positive max. then the velocity is zero and the acceleration is negative maximum.

### FORCE LAW FOR SIMPLE HARMONIC MOTION

In a block and spring example there is a force acting on the block that tries to return the block to its equilibrium position. This force is the restoring force and the force law for simple harmonic motion is based on this force. Let the displacement of the block from its equilibrium position be x and the restoring force be F.

We know restoring force is directly proportional to displacement but in opposite direction hence,

$F=-k x$

here k is a constant known as the force constant and its unit is N/m in S.I. system.

Now we can also substitute in equation $(1), \mathrm{a}=\mathrm{F} / \mathrm{m}$ and we will get:

$a=-k x / m=-\omega^{2} \underline{x}\left(\text {where } k / m=\omega^{2}\right)$

So, equation (1) and (2) give us the force laws of the simple harmonic motion.

### ENERGY IN SIMPLE HARMONIC MOTION

Suppose there is a pendulum, at its extreme position it is at rest so it only possesses potential energy and at the lowest point it has max speed with max kinetic energy.

Hence energy of simple harmonic motion can be calculated by combining both potential as well as kinetic.

As we know kinetic energy is energy dur to motion. The instantaneous velocity of the particle performing S.H.M.  at a distance x from the mean position is given by

$v^{2}=\omega^{2}\left(\underline{a}^{2}-x^{2}\right)$. Hence,

Kinetic energy

$=\frac{1}{2} m v^{2}=\frac{1}{2} m \omega^{2}\left(a^{2}=x^{2}\right)$

Similarly, energy possessed due to position is potential energy. You know the restoring force acting on the particle at a displacement x is $\mathrm{F}=-\mathrm{kx}$ where k is the force constant.

Now, the particle is given further infinitesimal displacement dx against the restoring force F. Let the work done to displace the particle be dW. Therefore, the work done dW during the displacement is

$d W=-f d x=-(-k x) d x=k x d x$

Therefore, the total work done to displace the particle now from 0 to x is

$\int d W=\int k x d x=k \int X d X$

Hence Total work done $=\frac{1}{2} \mathrm{kx}^{2}=\frac{1}{2} \mathrm{m} \omega^{2} \mathrm{x}^{2}$

The total work done here is stored in the form of potential energy.

So, this gives us the total energy of simple harmonic motion at any point. The sum of both the energies gives us the total energy of the simple harmonic motion.

Hence, $\mathrm{T.E.}=\mathrm{E}=\frac{1}{2} \mathrm{m\omega}^{2} \mathrm{a}^{2}$

The law of conservation of energy states that energy can neither be created nor destroyed. Therefore, the total energy in simple harmonic motion will always be constant. However, kinetic energy and potential energy are interchangeable.

### DAMPED SIMPLE HARMONIC MOTION

When an external force acts on the motion of an oscillator its motion gets damped. So, these periodic motions with amplitude decreasing gradually is known as damped harmonic motion. Generally, the forces that are the reason of damping are frictional forces.

Damping force is directly proportional to velocity of object and acts in the opposite direction of velocity. It depends on the surroundings and medium. If the damping force is $F_{d}$, we have,

$F_{d}=-b v(\mathrm{I})$

where the constant b depends on the properties of the medium (viscosity, for example) and size and shape of the block.

The expression of damped harmonic motion is given by equation as:

$m\left(d^{R} x / d t^{2}\right)+b(d x / d t)+k x=0$

Now we can obtain expression of position and energy at any time in a damped motion by solving equation (1).

The expression for the position is,

$x(t)=A e_{w}^{-b t / 2 m} \cos (\omega t+\varnothing)$

where A is the amplitude and $\omega$ is the angular frequency of damped simple harmonic motion given by,

$\omega=\underline{V}\left(K / m-b^{2} / 4 m^{2}\right)$

and similarly, energy expression of the same would be given by,

$E(t)=1 / 2 k A e^{-b t / 2 m}$

### SIMPLE PENDULUM

if we displace the pendulum by a very small angle $\Theta$, then it performs the simple harmonic motion.

As gravity points down, we need to take the component of gravity which is parallel to its motion.   Only the component of gravity which is parallel to the direction of motion will do work.   In this case, the force on a pendulum can be given as

$F=m g \sin \theta$

Now let $\sin \theta=\theta$ be the assumption for very small $\theta$ the we will get,

$F=-m g \theta=-m g x / L$

So, we can say that for small displacement the simple pendulum performs simple harmonic motion.

Now time period of the simple pendulum is given by,

$T=2 \pi / \omega=2 \pi /$ Vacceleration per unit displacement

And acceleration per unit displacement $=a / x=g / L$

Hence the formula becomes,

$T=2 \pi / \sqrt{g} / L=2 \pi \sqrt{L / g}$

Similarly, the frequency of a simple pendulum is given by,

$f=1 / T=1 / 2 \pi \sqrt{g} / L$

### FORCED OSCILLATIONS AND RESONANCE

In the previous section we read about damping motion. Now if we apply some external force so that the damping force cancels out and the object keeps on oscillating. The motion that the system performs under this external agency is known as Forced Simple Harmonic Motion. The external force is itself periodic with a frequency ωd which is known as the drive frequency.

Suppose the force applied to damped oscillator is given by,

$F(t)=F_{0} \cos \omega_{d} t$

Now as we know acceleration is second derivative of displacement thus the damping equations modifies to,

$m\left(d^{2} x / d t^{2}\right)+b(d x / d t)+k x=F_{0} \cos \omega_{d} t$

the equation of an oscillator of mass m on which a periodic force of frequency $\omega_{d}$ is applied.

The phenomenon of driving a system with a frequency equal to its natural frequency is called resonance. As the driving frequency gets progressively higher than the resonant or natural frequency, the amplitude of the oscillations becomes smaller, until the oscillations nearly disappear

The following graph shows the resonant frequency and different shapes for different type of damped oscillations.

### CONCLUSION

At first, we study about oscillations and then we distinguished between oscillatory and periodic motion. Then we study about different quantities like time period, frequency ad displacement of a periodic motion. One of the most important periodic motion is simple harmonic motion. The chapter tells you about the velocity and acceleration of an object under SHM and also talks about its energy. Then we discuss about the spring mass system and Hooke’s law. We get to know that simple pendulum also undergoes SHM. This all was going on when there were no external forces by the surroundings. Then we talked about damping and to overcome damping we got to know about forced oscillations and resonance.