Waves and Oscillations, Exercises of Physics

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Oscillations

Waves and

Oscillations

Waves and

(SECOND EDITION)

Former Professor and Head

Department of Physics, Visva-Bharati

Santiniketan, West Bengal

R.N. Chaudhuri

Ph.D.

Copyright © 2010, 2001, New Age International (P) Ltd., Publishers Published by New Age International (P) Ltd., Publishers

All rights reserved.

No part of this ebook may be reproduced in any form, by photostat, microfilm, xerography, or any other means, or incorporated into any information retrieval system, electronic or mechanical, without the written permission of the publisher. All inquiries should be emailed to [email protected]

ISBN (13) : 978-81-224-2842-

PUBLISHING FOR ONE WORLD

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It is a great pleasure for me to present the second edition of the book after the warm response of the first edition. There are always important new applications and examples on Waves and Oscillations. I have included many new problems and topics in the present edition. It is hoped that the present edition will be more useful and enjoyable to the students.

I am very thankful to New Age International (P) Ltd., Publishers for their untiring effort to bringing out the book within a short period with a nice get up.

R.N. Chaudhuri

Preface to the Second Edition

The purpose of this book is to present a comprehensive study of waves and oscillations in different fields of Physics. The book explains the basic concepts of waves and oscillations through the method of solving problems and it is designed to be used as a textbook for a formal course on the subject. Each chapter begins with the short but clear description of the basic concepts and principles. This is followed by a large number of solved problems of different types. The proofs of relevant theorems and derivations of basic equations are included among the solved problems. A large number of supplementary problems at the end of each chapter serves as a complete review of the theory. Hints are also provided in the case of relatively complex problems.

The topics discussed include simple harmonic motion, superposition principle and coupled oscillations, damped harmonic oscillations, forced vibrations and resonance, waves, superposition of waves, Fourier analysis, vibrations of strings and membranes, Doppler effect, acoustics of buildings, electromagnetic waves, interference and diffraction. In all, 323 solved and 350 supplementary problems with answers are given in the book.

This book will be of great help not only to B.Sc. (Honours and Pass) students of Physics, but also to those preparing for various competitive examinations.

I thank Professor K.C. Gupta for going through the manuscripts carefully and for suggesting some new problems for making the book more interesting and stimulating.

R.N. Chaudhuri

Preface to the First Edition

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1.1 PERIODIC MOTION

When a body repeats its path of motion back and forth about the equilibrium or mean position, the motion is said to be periodic. All periodic motions need not be back and forth like the motion of the earth about the sun, which is periodic but not vibratory in nature.

1.2 THE TIME PERIOD (T)

The time period of a vibrating or oscillatory system is the time required to complete one full cycle of vibration of oscillation.

1.3 THE FREQUENCY ( ννννν )

The frequency is the number of complete oscillations or cycles per unit time. If T is the time for one complete oscillation.

ν =

T

1.4 THE DISPLACEMENT (X ORY )

The displacement of a vibrating body is the distance from its equilibrium or mean position. The maximum displacement is called the amplitude.

1.5 RESTORING FORCE OR RETURN FORCE

The mass m lies on a frictionless horizontal surface. It is connected to one end of a spring of negligible mass and relaxed length a 0 , whose other end is fixed to a rigid wall W [Fig. 1.1 ( a )].

If the mass m is given a displacement along the x -axis and released [Fig. 1.1 ( b )], it will oscillate back and forth in a straight line along x -axis about the equilibrium position O. Suppose at any instant of time the displacement of the mass is x from the equilibrium position. There is a force

a (^0) W m x O (a)

x

m

a 0 x W

O (b) Fig 1.

Simple Harmonic Motion

11111

2 WAVES AND OSCILLATIONS

tending to restore m to its equilibrium position. This force, called the restoring force or return force, is proportional to the displacement x when x is not large:

F = – k^ x^ i

^ ...(1.2)

where k , the constant of proportionality, is called the spring constant or stiffness factor, and i ^ is the unit vector in the positive x -direction. The minus sign indicates that the restoring force is always opposite in direction to the displacement.

By Newton’s second law Eqn. (1.2) can be written as m && x^ = – kx or, && x + ω^2 x = 0 ...(1.3)

where ω^2 = k/m = return force per unit displacement per unit mass. ω is called the angular frequency of oscillation.

1.6 SIMPLE HARMONIC MOTION (SHM)

If the restoring force of a vibrating or oscillatory system is proportional to the displacement of the body from its equilibrium position and is directed opposite to the direction of displace- ment, the motion of the system is simple harmonic and it is given by Eqn. (1.3). Let the initial

conditions be x = A and (^) x & = 0 at t = 0, then integrating Eqn. (1.3), we get

x ( t ) = A cos ω t ...(1.4)

where A , the maximum value of the displacement, is called the amplitude of the motion. If T is the time for one complete oscillation, then

x ( t + T ) = x ( t )

or A cos ω( t + T ) = A cos ω t

or ω T = 2π

or T =

2 π ω = 2 π^

m k

and ν =

T

ω 2 π

or, ω = 2 πν.

The general solution of Eqn. (1.3) is x ( t ) = C cos ω t + D sin ω t ...(1.6)

where C and D are determined from the initial conditions. Euqation (1.6) can be written as

x ( t ) = A cos (ω t – φ) ...(1.7)

where C = A cos φ and D = A sin φ. The amplitude for the motion described by Eqn. (1.7) is now A = ( C^2 + D^2 ) 1/2^ and the angular frequency is ω which is uneffected by the initial conditions. The angle φ called the phase angle or phase constant or epoch is given by

φ = tan –1^ ( D/C ), where φ is chosen in the interval 0 ≤ φ ≤ 2 π.

1.7 VELOCITY, ACCELERATION AND ENERGY OF A SIMPLE HARMONIC

OSCILLATOR

From Eqn. (1.7), we find that the magnitude of the velocity v is

v = |– A ω sin(ω t – φ)| = A ω(1 – x^2 / A^2 ) 1/

or v = ω( A^2 – x^2 ) 1/2^ ...(1.8)

4 WAVES AND OSCILLATIONS

about the centre O is OB = the radius of the circle = A. Suppose Q is at B at time t = 0 and it takes a time t for going from B to Q and by this time the point P moves form B to P. If ∠ QOB = θ, t = θ/ω or, θ= ω t , and x = OP = OQ cos θ = A cos ω t.

A

Q

x P B

y

B′ O

θ (^) x

Fig. 1.

When Q completes one revolution along the circular path, the point P executes one complete oscillation. The time period of oscillation T = 2π/ω. If we choose the circle in the xy plane, the position of Q at any time t is given by

r (^) = A cos ω t i ^

• A sin ω t j

^ .

1.9 THE SIMPLE PENDULUM

The bob of the simple pendulum undergoes nearly SHM if its angle of swing is not large. The time period of oscillation of a simple pendulum of length l is given by

T = 2 π l g ...(1.13)

where g is the acceleration due to gravity.

1.10 ANGULAR SIMPLE HARMONIC MOTION (TORSIONAL PENDULUM)

A disc is suspended by a wire. If we twist the disc from its rest position and release it, it will oscillate about that position in angular simple harmonic motion. Twisting the disc through an angle θ in either direction, introduces a restoring torque

Γ = – C θ …(1.14)

and the period of angular simple harmonic oscillator or torsional pendulum is given by

T = 2 π I C …(1.15)

where I is the rotational inertia of the oscillating disc about the axis of rotation and C is the restoring torque per unit angle of twist.

SIMPLE HARMONIC MOTION 5

SOLSOLSOLSOLSOLVED PRVED PRVED PRVED PRVED PROBLEMSOBLEMSOBLEMSOBLEMSOBLEMS

1. A point is executing SHM with a period π s. When it is passing through the centre of its path, its velocity is 0.1 m/s. What is its velocity when it is at a distance of 0.03 m from the mean position?

Solution When the point is at a distance x from the mean position its velocity is given by Eqn. (1.8):

v = ω( A^2 – x^2 ) 1/^. Its time period, T = 2π/ω = π; thus ω = 2 s–1^. At x = 0, v = A ω = 0.1; thus A = 0.05 m. When x = 0.03 m , v = 2 [(0.05) 2 – (0.03) 2 ] 1/2^ = 0.08 m/s.

2. A point moves with simple harmonic motion whose period is 4 s. If it starts from rest at a distance 4.0 cm from the centre of its path, find the time that elapses before it has described 2 cm and the velocity it has then acquired. How long will the point take to reach the centre of its path?

Solution Amplitude A = 4 cm and time period T = 2π/ω = 4 s. The distance from the centre of the path x = 4–2 = 2 cm. Since x = A cos ω t , we have 2 = 4 cos ω t. Hence t = 2/3 s and the

velocity v = ω (^) A^2 − x^2 = π/2 (^4 2) − 22 = π 3 cm/s. At the centre of the path x = 0 and ω t

= π/2 or, t = 1 s.

3. A mass of 1 g vibrates through 1 mm on each side of the middle point of its path and makes 500 complete vibrations per second. Assuming its motion to be simple harmonic, show that the maximum force acting on the particle is π^2 N.

Solution A = 1 mm = 10–3^ m, ν = 500 Hz and ω = 2πν. Maximum acceleration = ω^2 A. Maximum force = m ω^2 A = 10 –3^ × 4π^2 (500) 2 × 10–3^ = 2 π 2 N.

4. At t = 0, the displacement of a point x (0) in a linear oscillator is –8.6 cm, its velocity v (0) = – 0.93 m/s and its acceleration a (0) is + 48 m/s^2_. (a) What are the angular frequency_ ω and the frequency ν? (b) What is the phase constant? (c) What is the amplitude of the motion?

Solution ( a ) The displacement of the particle is given by x ( t ) = A cos(ω t + φ) Hence, x (0) = A cos φ = – 8.6 cm = – 0.086 m v (0) = – ω A sin φ = – 0.93 m/s a (0) = – ω^2 A cos φ = 48 m/s^2

Thus, ω = − =

a x

b g b g.

= 23.62 rad/s

ν = ω/2π =

π

= 3.76 Hz