Chapter 14: Oscillations

Introduction

In our daily life, we come across various kinds of motions. You have already learned about some of them, e.g., rectilinear motion and motion of a projectile. Both these motions are non-repetitive. We have also learned about the uniform circular motion and orbital motion of planets in the solar system. In these cases, the motion is repeated after a certain interval of time, that is, it is a periodic motion.

The study of oscillatory motion is basic to Physics. In musical instruments, like the sitar, guitar and violin. We come across vibrating strings that produce pleasing sounds. The vibration of air molecules makes the propagation of the sound possible.

In this chapter, we will learn about oscillatory motion or oscillations. Any motion which repeats itself at regular intervals of time is known as periodic motion. If a body moves back and forth repeatedly about its mean position then it is said to be in oscillatory motion. For example, The to and fro movement of the pendulum, jumping on a trampoline, a child swinging on a swing.

Periodic and Oscillatory Motion

Periodic Motion: A motion is called periodic motion when it repeats itself after equal intervals of time. The interval of time is called the Time period of periodic motion. Example: The rotational motion of the earth about its axis is periodic motion with a time period of 24 hours.

Oscillatory Motion:  An oscillatory or vibratory motion is defined as a periodic and bounded motion about a fixed point. In other words, Oscillations are defined as to and fro motion which repeat itself after regular intervals of time.In oscillations, the frequency of vibrations is comparatively less.

For example, the Motion of the Pendulum of the wall clock, the motion of the bob of a simple pendulum displaced once from its mean position.

Every oscillatory motion is periodic motion, that is every oscillatory motion repeats itself after a fixed interval of time. But every periodic motion is not oscillatory.For e.g.:- Motion of planets around the sun is periodic but is not oscillatory motion.

Simple Harmonic Motion

Simple harmonic motion is the simplest form of oscillatory motion. This motion 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.

Simple harmonic motion can be considered as a specific type of oscillatory motion, in which:

• The particle moves in a single dimension
• The particle oscillates to and fro about a fixed mean position (where Fnet=0).
• The net force on the particle always gets directed towards the equilibrium position
• The magnitude of the net force is always proportional to the displacement of the particle from the equilibrium position at that instant.

Equilibrium Position

Oscillating bodies come to rest at their equilibrium positions. When a bob is suspended from rigid support it goes to extreme positions and then comes to its mean position which is also known as equilibrium position.

Equilibrium Position is that position where an object tends to come at rest when no external force is applied.

To and fro motion of the pendulum oscillating from its mean position B to its either extreme positions A and C respectively.

Period/Time period (T)

The time is taken by an oscillating body to complete one cycle of oscillation. This means the to and fro motion of the body gets repeated after a fixed interval of time.

It is denoted by T. and the unit is second.

The above image describes the motion of the pendulum, it goes from B to A and then back to B from A. Similarly The motion of pendulum from B to C.

Frequency (f): The number of repetitions in one second of a periodic motion is called Frequency (ν).  Its unit is Hertz (Hz). The reciprocal of T gives the number of repetitions that occur per unit of time. The relation between  f and T   is    f= 1/T

Displacement: We defined the displacement of a particle as the change in its position vector. Displacement in periodic motion can be represented by a function which is periodic which repeats after a fixed interval of time.

In the above image, we can see that motion of an oscillating simple pendulum can be described in terms of angular displacement θ from the vertical.  And In the above right image, we can see that there is a block whose one end is attached to a spring and another is attached to a rigid wall.x is the displacement from the wall.

SIMPLE HARMONIC MOTION
Simple Harmonic Motion (SHM) is a periodic motion in the body that moves to and fro about its mean position. The restoring force on the oscillating body is directly proportional to its displacement and is always directed towards its mean position.

In the above image, we can see that a particle is vibrating to and fro within the limits –A and +A.
Mathematically,  F _net = ma=- kx
So, acceleration 
where k is known as force constant
However ,In SHM we know that acceleration a= d ² x/d x ² =- w ² x
This equation is known as the differential equation of S.H.M.
where ω is known as angular frequency here  
The general expression for solution   satisfying the equation d 2 x/d x 2 =- w 2 x  is: x(t)=Asin(ωt+ϕ)
The oscillatory motion is said to be SHM if the displacement x of the particle from origin varies with time t:

The above graph shows displacement as a continuous function of time.
Now let's consider this  x (t) = A cos (ωt + Φ)   Where
x (t) : displacement x as a function of time
A= amplitude, It is defined as the magnitude of maximum displacement of the particle from its mean position.
ωt +Φ = phase angle (time-dependent)  , ω= angular frequency and  Φ = phase constant
SHM is a periodic motion in which displacement is a sinusoidal function of time.
If we plot the graph between displacement versus time we can conclude that displacement is a continuous function of time.
Phase
It is that quantity that determines the state of motion of the particle.
1. Its value is (ωt + Φ)
2. It is dependent on time.
The value of phase at time t=0 is termed as Phase Constant. When the motion of the particle starts it goes to one of the extreme positions at that time phase is considered as 0.
Let x (t) = A cos (ωt) where we are taking (Φ = 0)
1. Mean Position (t= 0)
2. x (0) = A cos (0) = A (cos 0=1)
3. t=T/4, t= T/2, t=3T/4, t=T and t=5T/4

The above figures depict the location of the particle in SHM at different values of t=0, T/4, T/2, 3T/4, T, and 5T/4.The time after which motion repeats is T. The speed is maximum for zero displacements (x=0) and zeroes at the extremes of motion.

In the above graph the curves (3) and (4) are for φ = 0 and -π/4 respectively but the amplitude is the same for both.

Angular Frequency (ω)

Angular frequency refers to the angular displacement per unit time. It can also be defined as the rate of change of the phase of a sinusoidal waveform (e.g., in oscillations and waves). Angular frequency is larger than frequency f (in cycles per second, also called Hz), by a factor of 2π.

Mathematically ω=2πf =2π/T . The S.I unit of angular frequency is rad/s.

Oscillations due to spring

Consider a block if it is pulled on one side and is released, and then it executes to and fro motion about a mean position.

In the above image a block, is on a frictionless surface when pulled or pushed and released, executes simple harmonic motion.

F (x) = –k x (expression for restoring force)

‘K’ is known as spring constant and its value is governed by the elastic properties of the spring.

• The above expression is the same as the force law for SHM and therefore the system executes a simple harmonic motion. Therefore,
• Angular frequency ‘w’ is given by
• The time period of oscillation is 

Vertical spring ( loaded spring).

When the spring is suspended vertically from a fixed point and carries the block at its other end as shown, the block will oscillate along the vertical line.

• In first figure we have an unstretched spring of length L
•  In the second figure, we load a mass ‘m’ with the spring and it gets stretched by a distance s, this is its equilibrium position when gravity (mg) and restoring force ( ks ) balance each other.      mg= Ks   ⇒ s= mg/k
• In the third one we displaced the mass ‘m’ from its equilibrium position by a distance ‘x’. And then the loaded pendulum will start oscillating about its mean/ equilibrium position.

Time period of oscillation   = 

Combination of springs:

1. Springs in series:

Consider two springs of force constants  K1 and K2 respectively, connected in series as shown. They are equivalent to a single spring of force constant K which is given by 

2. Spring in parallel : For a parallel combination as shown, the effective spring constant is  K=K1+ K2

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