CBSE NOTES CLASS 11 CHAPTER 14

OSCILLATIONS

Periodic Motion

A motion that repeats itself at regular intervals of time is called periodic motion, For example, orbital motion of the earth around the sun, motion of arms of a clock, motion of a simple pendulum etc.

Oscillatory Motion

A motion taking place to and fro or back and forth about a fixed point is called oscillatory motion, e.g., motion of a simple pendulum, motion of a loaded spring etc.

Assuming that there is no loss of mechanical energy, every oscillatory motion is periodic motion but every periodic motion is not oscillatory motion.

Harmonic Oscillation

The oscillation which can be expressed in terms of single harmonic function, i.e., sine or cosine function is called harmonic oscillation.

Simple Harmonic Motion

A harmonic oscillation of constant amplitude and of single frequency under a restoring force whose magnitude is proportional to the displacement and always acts towards mean position or equilibrium position is called Simple Harmonic Motion (SHM).

Simple harmonic motion can be defined as the projection of a uniform circular motion on any diameter of a circle of reference.

When the point P on the circle moves around, its projection M moves in a simple harmonic motion.

SOME TERMS RELATED TO SHM

Time Period of Oscillation

The smallest interval of time after which the motion is repeated is called its period.

Time taken by the body to complete one oscillation is known as time period.

It is denoted by T.

Frequency of Oscillation

The number of oscillations completed by the body in one second is called frequency. It is denoted by ν.

Its SI unit is ‘hertz’ or ‘s-1.

Angular Frequency Oscillation

The angle subtended in one second is called by the oscillating body is called angular frequency.

The product of frequency with factor 2π is called angular frequency. It is denoted by ω.

Angular frequency (ω) = 2πν

Its SI unit is rad s-1.

Displacement in Oscillation

The change of position of the oscillating object from the mean position at a particular instant is called displacement. It is denoted by x.

Amplitude of Oscillation

The maximum displacement on either direction from mean position is called amplitude. It is denoted by A.

Displacement in SHM

Displacement in SHM at any instant is given by

x = A sin ωt

or x = A cos ωt

where

A = amplitude and

ω = angular frequency.

Since the motion is repeating after every T seconds,

x (t + T) = a sin ω(t + T) and

x (t + T) = a cos ω(t + T)

are also SHM.

Principle of Superposition

Any linear combination of sine and cosine functions is also SHM, that is

x = A sin ωt + B cos ωt

Proof

Let us consider superposition of two SHMs,

x1 = A sin ωt and

x2 = B cos ωt

then we need to prove that x = A sin ωt + B cos ωt is also an SHM.

Let us assume that,

A = D cos ϕ and

B = D sin ϕ

So,

, and

Now, x = D (cos ϕ sin ωt + sin ϕ cos ωt)

Using identity sin (α + β) = sin α cos β + cos α sin β

x = D sin (ωt + ϕ)

Hence any periodic function can be expressed as a superposition of sine and cosine functions of different time periods with suitable coefficients

Phase

A time dependent physical quantity which expresses the position and direction of motion of an oscillating particle, is called phase (ωt + ϕ).

Phase Constant

The phase at t = 0 is called phase constant and is represented by ϕ

Velocity of Simple Harmonic Motion

We can describe the SHM as projection, of a particle undergoing uniform circular motion, on one of the diameter s of the circle being described by it.

The magnitude of velocity, v, with which the reference particle P is moving in a circle is related to its angular speed ω, as

v = ω A

where A is the radius of the circle described by the particle P.

The magnitude of the velocity vector v of the projection particle is ω A; therefore its projection on the x -axis at any time t is

v(t) = – ωA sin (ωt + ϕ)

The velocity, v (t), of the particle P′ is the projection of the velocity v of the reference particle, P.

The negative sign appears because the velocity component of P is directed towards the left, in the negative direction of x

Also by calculus

Or v(t) = –ωA sin (ωt + ϕ)

Acceleration of Simple Harmonic Motion

At mean position y = 0, vmax = ± Aω, a = 0

At right extreme position y = ± A, v = 0, a = $\mp$ amax = $\mp$2

Force Law for SHM

• Simple harmonic motion is the motion executed by a particle subject to a force, which is proportional to the displacement of the particle and is directed towards the mean position. That is,

F(t) = - k x

Where k = m ω2

Therefore, ω = $\sqrt{\frac{\mathrm{k}}{\mathrm{m}}}$

Time period of the SHM is given by

Phase Difference between two SHMs

If x1 = A1 cos (ωt + ϕ1) and x2 = A2 cos(ωt + ϕ2)

Then ϕ1 - ϕ2 is called the phase difference.

For a particle executing SHM, the phase difference between

(i) Instantaneous displacement and instantaneous velocity

= $\frac{\mathrm{\pi }}{2}$ rad

(ii) Instantaneous velocity and instantaneous acceleration

=$\frac{\mathrm{\pi }}{2}$ rad

(iii) Instantaneous acceleration and instantaneous displacement

Energy of Particle in SHM

A particle executing simple harmonic motion has kinetic and potential energies, both varying between the limits, zero and maximum.

Kinetic Energy of Particle under SHM

We can see that, KE is also a periodic function of time, being zero when the displacement is maximum and maximum when the particle is at the mean position.

Since the sign of v does not matter, the period of K is $\frac{\mathrm{T}}{2}$.

Potential Energy of Particle under SHM

The concept of potential energy is possible only for conservative forces.

Since F = – kx

Total energy TE = K + U

Also, K = TE – U

From this expression we can see that, the kinetic energy is maximum at the centre (x = 0) and zero at the extremes of oscillation (x = ± A).

F(t) = - k x

Where k = m ω2

Therefore,

Time period of the SHM is given by

vmax = max speed = Aω

Spring Pendulum

A point mass suspended from a massless (or light) spring constitutes a spring pendulum. If the mass is once pulled downwards so as to stretch the spring and then released, the system oscillated up and down about its mean position simple harmonically.

• Time period and frequency of oscillations are given by

Or,

• If the spring is not light but has a definite mass ms, then it can be easily shown that period of oscillation will be

• When two springs of force constants k1 and k2 are connected in parallel to mass m as shown in figure,

Then, effective force constant of the spring combination

k = k1 + k2

Time period

• When two springs of force constant k1 and k2 are connected in series to mass m as shown in figure,

Then, effective force constant of the spring combination

Simple Pendulum

A simple pendulum consists of a heavy point mass (bob) suspended from a rigid support by means of an elastic inextensible light string.

Let the mass of the bob be m and length of the string be L.

Let us consider a point, when the string makes an angle θ with the vertical.

The forces acting on the bob are the force T, tension in the string and the gravitational force Fg = mg,

We can resolve the force Fg into a radial component Fg cos θ and a tangential component Fg sin θ.

The radial component is cancelled by the tension, since there is no motion along the length of the string.

The tangential component produces a restoring torque about the pendulum’s pivot point.

The restoring torque τ is given by,

The central location is called the equilibrium position (when θ = 0)

For rotational motion we have,

where I is the pendulum’s rotational inertia about the pivot point and α is its angular acceleration about that point.

Therefore,

Substituting Fg = mg, we get,

If θ is small, sin θ ≈ θ

Now acceleration

and

$\mathrm{x/L \approx \theta }$

Comparing with the standard SHM equation,

we have

Since all the mass is concentrated at the end of the string,

Hence

• If the effective length L of simple pendulum is very large and comparable with the radius of earth (R), then its time period is given by

• For a simple pendulum of infinite length (L >> R)

• For a simple pendulum of length equal to radius of earth,

• Effect of temperature on time period of simple pendulum

If the bob of the simple pendulum is suspended by a metallic wire of length L, having coefficient of linear expansion α, then due to increase in temperature by dθ, effective length becomes

Therefore,

Fractional increase in time period

• When a bob of simple pendulum of density ρ oscillates in a fluid of density ρoo < ρ), then increased time period

• When simple pendulum is in a horizontally accelerated vehicle, then its time period is given by

where a = horizontal acceleration of the vehicle.

• When simple pendulum is in a vehicle sliding down an inclined plane, then its time period is given by

Where θ = inclination of plane.

• If a simple pendulum is moving up with acceleration a

• If a simple pendulum is moving down with acceleration a

Second’s Pendulum

A simple pendulum having time period of 2 second is called second’s pendulum.

The effective length of a second’s pendulum is 99.992 cm or approximately 1 meter on earth.

Conical Pendulum

If a simple pendulum is fixed at one end and the bob is rotating in a horizontal circle, then it is called a conical pendulum.

If θ is the angle made by string with the vertical, then in equilibrium

where T = tension

Therefore,

Torsional Pendulum

Time period of torsional pendulum is given by

where,

I = moment of inertia of the body about the axis of rotation and

C = restoring couple per unit twist, called torsional constant of the material.

Physical Pendulum

When a rigid body of any shape is capable of oscillating about an axis (may or may not be passing through it). It constitutes a physical pendulum.

Where,

I is the moment of inertia of the body about axis of rotation

L= distance of the CG of the body from the axis of rotation

Oscillations of Liquid in a U – tube

If a liquid of density ρ is filled up to height L in both limbs of a U-tube and now liquid is depressed upto a small distance x in one limb and then released, then liquid column in U-tube start executing SHM.

Let us work out the relations for frequency and time period.

When the fluid is depressed on one side, it rises on the other.

Let the area of cross section of the tube be A.

The difference of mass of the liquid in two columns

This difference in the height between two sides result in a restoring force equal to the mass of the liquid in two columns, i.e.,

Hence the restoring force,

Therefore, the liquid column will perform SHM.

Comparing this equation, with the equation of SHM, i.e.,

We get,

Now total mass of the liquid is oscillating. Total mass of liquid in the tube,

Therefore,

Alternatively,

Let the area of cross section of the tube be A.

The Mass of the fluid which is above the equilibrium position

The potential energy of the fluid is given by

Total mass of liquid in the tube

This mass is moving with a velocity v (say)

Hence KE of the system

Total energy = Potential energy + Kinetic Energy

Now, assuming there is no energy loss due to friction, using the principle of conservation of energy, we get,

Taking

We get,

Comparing this equation with the equation of SHM, i.e.,

We get,

The time period of oscillation is given by

Now, if x = $\frac{\mathrm{L}}{2}$, then,

Oscillations of a floating cylinder in liquid

In equilibrium gravitational force mg is balanced by the upthrust of the liquid.

Now, let us push the cylinder slightly inside the liquid by a distance y and then released.

F = Net upthrust

= - Weight of the liquid column of height y

Let A be the area of cross section of the cylinder, ρ be the density of cylinder and σ be the density of the liquid.

Then weight of the displaced liquid = A y σ g

Therefore,

i.e.,

Hence the cilynder executes SHM with

If l is the total length of the cylinder, the mass of the cylinder will be,

m = A l ρ

Time period of the oscillation

Oscillation of an object dropped in a tunnel along the diameter of earth

We can consider earth to be sphere of radius R. Let O be the centre of the earth. A straight tunnel is dug along the diameter of the earth. An object of mass m is dropped in the tunnel. Suppose the body is at point P, at a depth ‘d’ below the surface of the earth. If gd is the acceleration due to gravity at depth d, then we can write,

${\mathrm{g}}_{\mathrm{d}}=\mathrm{g}\left(1-\frac{\mathrm{d}}{\mathrm{R}}\right)$

$=\mathrm{g}\frac{\left(\mathrm{R}-\mathrm{d}\right)}{\mathrm{R}}$

$=\mathrm{g}\frac{\mathrm{y}}{\mathrm{R}}$

Where, y = R-d = distance of the object from the centre of the earth.

Gravitational force on the object,

Therefore the object performs SHM, with

$\mathrm{k}=\frac{\mathrm{m}\mathrm{g}}{\mathrm{R}}$

The time period of oscillation

Damped Oscillations

Damping is an influence within or upon an oscillatory system that has the effect of reducing, restricting or preventing its oscillations. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation.

Examples include viscous dragin mechanical systems, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators.

• Oscillations with decreasing amplitude with time are called damped oscillations.

Consider a block of mass m oscillating vertically on a spring with spring constant k. The block is connected to a vane through a rod (the vane and the rod are considered to be massless).

The vane is submerged in a liquid. As the block oscillates up and down, the vane also moves along with it in the liquid.

The up and down motion of the vane displaces the liquid, which in turn, exerts an inhibiting drag force (viscous drag) on it and thus on the entire oscillating system.

With time, the mechanical energy of the block spring system decreases, as energy is transferred to the thermal energy of the liquid and vane.

The damping force is proportional and opposite to the velocity of the particle in the viscous medium,

Where b is the damping constant, which depends on the characteristics of the fluid and the body that oscillates in it.

Also the spring force

So the total force

Now if a is the acceleration of the mass, then

Since

And

We have,

The solution of this differential equation has been found to be of the form,

which can be written as

Where

Time period,

The mechanical energy E of the damped oscillator at an instant t is given by

Free Oscillations or Un-damped Oscillations

When a body which can oscillate about its mean position is displaced from mean position and then released, it oscillates about its mean position. These oscillations are called free oscillations and the frequency of oscillations is called natural frequency ω.

Oscillations with constant amplitude with time are called un-damped oscillations.

Forced or Driven Oscillations

Oscillations of any object with a frequency different from its natural frequency under a periodic external force are called forced or driven oscillations. Let the frequency of the driving force be ωd.

Therefore,

The motion of a particle under the combined action of a linear restoring force, damping force and a time dependent driving force is given by,

The oscillator initially oscillates with its natural frequency ω. When we apply the external periodic force, the oscillations with the natural frequency die out, and then the body oscillates with the (angular) frequency of the external periodic force.

Its displacement, after the natural oscillations die out, is given by

)

where t is the time measured from the moment when we apply the periodic force.

The amplitude A is a function of the forced frequency ωd and the natural frequency ω.

And

For small damping factor

For ωd ≈ ω,

Resonant Oscillations

The phenomenon of increase in amplitude when the frequency of driving force is close to the natural frequency of the oscillator is called resonance.

The resonant amplitude is larger for smaller damping.