QUESTION BANK (FIRST YEAR)
MECHANICS, WAVES, AND OSCILLATIONS
CHAPTER 1: VECTOR ANALYSIS:
SHORT ANSWER QUESTIONS
1) Define Gradient, Divergence and Curl. Explain their physical significance.
2) Explain the physical significance of divergence with two examples.
3) Explain the physical significance of gradient of a scalar field with two examples.
4) Explain Curl of a vector field. Give its physical significance.
5) Explain the terms divergence and curl of a vector field. Give examples from physics where
these ideas are used.
6) State and prove Stokes theorem.
7) State and prove Gauss divergence theorem.
8) Explain curl of a vector field with two examples.
9) State Gauss divergence theorem and its mathematical form.
10) Explain the physical meaning of a divergence of a vector field and define the curl of a
vector.
11) State and prove Green’s theorem.
12) Explain the physical significance of line integral.
13) What are scalar and vector fields? Give examples.
14) Prove that
a) The Curl of gradient of a scalar is Zero and
b) The divergence of Curl of a vector is Zero.
15) What is meant by the gradient of a scalar field? If r is the position vector of any point,
evaluate
Grade (1/r).
16) Prove that the gradient of a scalar field is a vector and explain the physical significance of
Gradient
17) Define the Curl of a vector field and explain its physical point of view. What is the
Significance when the curl is Zero.
18) Explain physical significance of Curl.
LONG ANSWER QUESTIONS:
1) Define Gradient divergence and Curl. Explain their physical significance with examples.
2) State and prove Gauss divergence theorem.
3) Define the line Integral, surface integral and volume integral of a vector quantity.
Prove Gauss’s theorem of divergence.
4) State and prove Stoke’s theorem. By Stoke’s theorem, prove that the Curl of gradient of
scalar function Φ is Zero.
PROBLEMS
1) If A=icy(x2+y2) +k (yz+zx) then find div A at point (1,-2, 3);
2) Evaluate div F, where
F=2x2zi-xy2zj+3y2xk
3) If r is the position vector of a point, then show that
A) Div r=3
And (b) Del (r.A) =A
4) Find the value of constant c for which the vector
A=i(x+3y) +j(y-2z) +k(x+CZ) is solenoid
1)If A=icy(x2+y2) +k (yz+zx), then find curl A at point (2, 2,-2).
2) If r is the position vector of a point, prove that curl r=0.
3) Calculate the work done by the force experienced by a charge q moving with a velocity v in a
magnetic field B.
4) Show that A=e-x (-yzi+zj+YK) is irrotational.
5) Find constants a, b, c so that the vector A=i(x+2y+ax) +j (bx-3y-z) +k (4x+cy+2z) is irrotational.
6) If ǿ(x, y, z) =3x2y-y3z2, find the value of ǿ at that point (1,-2,-1).
7) If A=x2zi-2y3z2j+xy2zk, find delA (gradeA) at that point (1,-1, 1).
CHAPTER 2: MECHANICS OF PARTICLES:
SHORT ANSWER QUESTIONS
1. What is meant by variable mass system? Give 2 examples?
2. Explain the principle of rocket motion.
3. Formulate Newton second Law for System of variable mass.
4. Calculate thrust on rocket.
5. Describe the principle of motion of a rocket as a system of variable mass.
6. Describe equation of motion of a system of variable mass.
7. Explain the motion of rocket under constant gravitational speed.
8. Write note on impact parameter.
9. Distinguish between elastic and elastic collision.
10. Define impact parameter and scattering cross section.
LONG ANSWER QUESTIONS:
1) Explain the motion of rocket and derive an expression for velocity of variable mass system.
Derive various stages of the rocket in motion.
2) Discuss the motion of rocket. Derive an expression for the velocity of rocket at any instant of time
moving under gravitational field.
3) What is a collision? Derive an expression for the final velocities of two spheres which are colliding
elastically in two dimensions.
4) Explain elastic and inelastic collisions. Obtain an expression for K.E in the case of an inelastic
oblique collision.
5) Define the term “Impact paramter”.Derive an expression of Ruther Ford’s scattering cross section.
6) Distinguish between elastic and inelastic collisions. Describe two dimensional (oblique) collisions
bin detail.
7) Write an expression for the force on a rocket moving under the influence of earth’s gravitational
field. Derive an expression for the velocity of a rocket at any given time.
8)Derive an expression for the Rutherford scattering cross section for the collision of α-particles with
heavy nucleic.
PROBLEMS:
1) A rocket burns 0.02 kg of fuel per second ejecting it as a gas with a velocity of 10,000 m/sec. What
force does the gas expert on the rocket?
2) A rocket of mass 20kg has 180kg fuel. The exhaust velocity of the fuel is 1.6 km/sec.Calculate the
minimum rate of consumption of fuel so that the rocket may rise from the ground.
3) An empty rocket weighs 6000 kg and contains 44000 kg of fuel. If the exhaust velocity of gases is 1
km/s.Find the maximum velocity attained by the rocket.
4) A rocket having initial mass 240 kg ejects fuel ay the rate of 6 kg/s with velocity 2 km/s vertically
downward relative to itself .Calculate its velocity 25 secs after start , taking initial velocity to be zero
and neglecting gravity.
5) An empty rocket 5000 kg and contains 40,000 kg of fuel. If the exhaust velocity of the fuel is 2.0
km/sec .Find the maximum velocity gained by the rocket. (Given that loge 10=2.3, log 10 3 =0.4771)
6) A rocket of mass 20 kg has 180 kg of fuel. The exhaust velocity of the fuel is 1.6 km/sec. calculate
the minimum rate of consumption of fuel so that the rocket may rise from the ground. Also calculate
the ultimate vertical speed gained by the rocket, when the rate of consumption of the fuel is 2 kg/sec.
7) A rocket of mass 40 kg has got a fuel of mass 360 kg inside it. The exhaust velocity of the fuel is 2
km/sec. The fuel is burning at the rate of 4 kg/s. Find the final velocity of rocket.
8) The stages of two stage rocket separately weigh 100 and 10 kg and contain 800 kg and 90 kg fuel
respectively .What is the final velocity that can be achieved with an exhaust velocity of 1.5 km/sec.
9) An α-particles with K.E 6.0 x 10 -14 joules is scattered at an angle of 60o by coulomb of a stationary
nucleus. Find the impact parameter.
10) An α-particle of energy 5 MeV approaches has a copper nucleus (Z=32) in the head on collision.
Calculate the distance of nearest approach.
CHAPTER 3: MECHANICS OF RIGID BODIES
SHORT ANSWER QUESTIONS
1)
2)
3)
4)
5)
6)
7)
8)
Define rigid body. Distinguish between pure translational and pure rotational motion.
Derive expression for rate of pression of a symmetric top.
Explain about moment of inertia tensor and mention some of its properties.
Explain the principle of working of gyroscope.
Explain working of gyroscope .Mention its applications
Explain about spinning atom and nuclei.
Write about precession of equinoxes
Explain the term precession and give 2 examples.
LONG ANSWER QUESTIONS:
1) What is symmetric top precession? Deduce an expression for the angular velocity of the precession
2)
3)
4)
5)
6)
7)
8)
of a symmetric top.
Derive Euler’s equations of a rotating rigid system fixed at end and hence deduce the conservation
of energy and angular momentum from Euler’s equation.
What is symmetric top and precessional motion? Deduce an expression for the precession velocity
of the top.
Explain the precession of a symmetric top .obtain an expression for its precessional frequently.
Derive torque. Give two examples where torqueses are acting. Deduce an expression for the
precession velocity of the symmetric top.
What is precessional motion? Find the angular velocity of precession of a spinning top. Show the
rate of precession is independent of mass, but depends upon distribution of mass.
What is symmetric top and its precessional motion? Derive an expression for precessional angular
speed in case of symmetric top moving with angular velocity with angle ‘Θ’ with the vertical
Write three properties of inertia tensor. Derive the relation between angular momentum and inertia
of motion.
PROBLEMS:
1) The speed of a body of mass 0.5kg moving along a circle of radius 0.6m increases at a constant
rate of 0.1ms-2.Find the torque on the body.
2) Calculate the angular momentum of the spherical earth rotating about its own axis.
(Mass of earth =6*1024kg and mean radius =6.4*106m).
3). Determine the moment of inertia of a uniform disk of mass 0.2kg and radius of 0.05m about an axis
passing through its diameter.
4).A gyroscope of wheel of mass 2 kg and having a radius of gyroscope 50cm is spinning about its
axis at a rate of 360rev/min. the axle is held horizontal and is supported by a point at one end that is
15cm from center of mass of the wheel. Calculate angular velocity of precession of the wheel
5) A couple of 20 N-m is applied to a flywheel of a mass 10kg and radius of gyration 0.5m. Find the
resultant angular acceleration.
6) A gyroscope wheel of mass 1.5kg and radius 0.30 m is spinning about its axle at a rate of 240
revolutions per minute. The axle is held horizontal and is supported by a pivot at one end which is 0.10
m from the centre of mass of the wheel. Find the precessional speed of the wheel.
CHAPTER 4: MECHANICS OF CONTINUOUS MEDIA
SHORT ANSWER QUESTIONS
1).Define bulk, young, rigidity moduli and Poisson’s ratio.
2). Give a brief account of classification of beams.
3) State Hooke’s law and define different moduli of elasticity.
4).Explain briefly about elastic constants.
5) Explain uniform and non-uniform bending beams.
6) Classify different types of beams.
LONG ANSWER QUESTIONS:
1) Derive an expression for the couple required to bend a uniform straight metallic strip into an arc of
a circle of small curvature.
2) A uniform beam is clamped horizontally at one end and located at the other end. Calculate the
depression at free end.
3) A beam of rectangular cross-section is resting on two knife edges and loaded at the middle point.
Find the expression for the depression produced.
4) Derive an expression for the depression of the mid point of a beam, supported at the ends and
loaded at the middle.
PROBLEMS
1) What is the minimum diameter of the brass rod if it is to support a 600 N load with out exceeding
the elastic limit? The elastic limit is 3.8*108pa.
2) A wire of 3.0 m long and 0.625 sq.cm in cross-section is found to stretch by 0.3 under the tension
of 1200 kg.what is the young’s modulus of material of the wire?
3) Calculate Poisson’s ratio for silver. Given its young’s modulus =7.25*1010N/m2 and bulk modulus
=11*1010 n/m2
4) A uniform steel wire of density ,7800 kg/m3 is 2.5 m long and mass 15.6*10-3 kg.It extends by
1.25 mm when loaded by 8 kg.calculate the value of Young’s modulus for steel .
5) The Young’s modulus for steel is Y=2*1011 N/m2 and its rigidity modulus η=8*1010 N/m2. Find
the Poisson’s ratio and Bulk modulus.
6) Calculate Poisson’s ratio for steel given young’s modulus is 2*1011 Pascal and rigidity modulus
8*1010 pa.
7) The modulus of rigidity and Poisson’s ratio of the material of a wire are 2.87*1010 N/m2 and
0.379 respectively. Find the value of Youngs modulus of the material of the wire.
8) A uniform rod of length 1 m is clamped horizontally at one end. A weight of 0.1 kg is attached at
the free end. Calculate the depression of the mid-point of the rod. The diameter of the rod is 0.02
m.Given Y=1010N/m2.
9) A cantilever of length 0.5 m has a depression of 15 mm at its free end. Calculate the depression at
a distance of 0.3 m from the fixed end.
10) The depression at the free end of a cantilever is 10 mm for a certain load. What is the depression
for
the same load at the free end of the cantilever of same material, but of twice length, twice
breadth, and Thrice width?
CHAPTER 5: CENTRAL FORCES
SHORT ANSWER QUESTIONS
1)
2)
3)
4)
What is central force? Give two examples and show it is conservative.
State Kepler’s laws of planetary motion.
State and explain Kepler’s laws of planetary motion.
Derive Kepler’s third law of planetary motion.
LONG ANSWER QUESTIONS:
1) What is central force? Show that if a particle moves under central force
(a) the angular momentum is conserved
(b) The areal velocity remains constant.
2) Distinguish between conservative and non-conservative forces. For motion in three dimensions
what the condition for force to be conservative. What is value of a if following force is
conservative
F= (2xy+z2) i+x2j+axzk
3) State Kepler’s laws of planetary motion and prove second law of motion.
4) State Kepler’s second law and show that it is a consequence of the law of conservation of angular
momentum.
5) Show that conservation of angular momentum of a system applied to planetary motion leads to law
of conservation of areal velocity.
6) Explain the terms ‘Gravitational field’,’ gravitational attraction’, and’Gravitational
potential’.Estabish relation between gravitational attraction and gravitational potential.
7) State Kepler’s laws of planetary motions. Derive the third law of planetary motion.
PROBLEMS
1) The periodic time of Venus is 224.7 day and that of earth is 365.26 days. Find the ratio of the major axes
of the orbits of Venus and earth.
2) If the mean distance lf mars from sun is 1.524 times that of earth, find period of revolution of mars about
sun.
3) If the earth be one-half of its present distance from the sun, what will be the number of days in a year?
4) The Jupiter’s period of revolution around the sun is 12 times that of earth. Assuming the planetary
orbital to be circular find how many times the distance between the Jupiter and sun exceeds that between the
earth and the sun.
5) The mean distance of mars from the sun is 1.54 times the distance of earth from the sun. Compute the
period of revolution of mars around the sun.
6) The semi-major axes of the orbits of Mercury and Mars are respectively 0.387 and 1.524 in astronomical
units. If the period of Mercury is 0.241 year, what is the period of mars?
7) The earth moves in an approximately circular orbit with radius 1.5*1011m Mars circles around sun in its
orbit in 687 days. What is radius of mars orbit around the sun.
CHAPTER 6: SPECIAL THEORY OF RELATIVITY
SHORTANSWERQUESTIONS:
1) Derive Einstein mass-energy relation.
2) Write about Lorentz transformations.
3) What is meant by Lorentz’s contraction? Write an expression for it.
4) Explain the concept of time dilation.
5) Get the relation between momentum and energy according to relative theory.
6) Obtain the equation of Lorentz-Fitgerald contraction.
LONGANSWERQUESTIONS:
1. State the postulates of the special theory of relativity. Obtain Lorentz
transformations equations.
2. What is the significance of Michelson-Morley experiment?
3. State the postulates of relativity. Explain length contradiction and time dilation.
4. State the fundamental postulates of special theory of relativity and deduce the Lorenz transformation
equations.
5. Describe the Michelson -Morley experiment and explain the physical significance of the negative result.
6. What was the aim of Michel- Morley experiment? Describe the construction of Michel-Morley
experiment with a sketch. Derive the expression for the fringe shift.
7. State the postulates of the special theory of relativity .Derive the mass-energy relation.
8. Describe Michelson’s-Morley experiment and critically comment on its negative result.
9. Describe Michelson’s –Morley experiment. What is the significance of the Michelson’s-Morley
experiment?
PROBLEMS:
Calculate the expected fringe shifts in the Michelsons –Morley experiment, if the distance of each plate
is 2m and the wavelength of monochromatic radiation is (a) 6000Ao and (b) 4000Ao
2) Calculate the velocity of the rod when its length will appear 90% of its proper length.
3) At what speed the mass of an object will be double of its value at rest.
4) The rest mass of electron is 9.1*10-31kg.What will be its mass if it were moving with (4/5) th the speed of
light.
5) A particle is moving with 90% of the velocity of light. Compare its relativistic mass with its rest mass.
6) Find the velocity with which a body should be moving such that its rest mass gets doubled.
1)
7) A clock showing correct time when at rest, loses 1 hour in a day when it is moving. What is its
velocity?
8) What will be the fringe shift in Michelsons-Morley experiment if the effective length of each path is
6m and light wavelength of 6000Ao is used? earth’s velocity is 3*104 m/s2 and c=3*108 m/s2
9) Calculate the velocity of rod when its length will appear 90% of its proper length.
10) Find the velocity with which the body should travel so that its length becomes half of the rest length?
11) At what speed mass of an object will be double of its value at rest?
12) Calculate the velocity of rod while its length will appear 80% of its proper length.(c= 3*108 m/s)
PART B
WAVES AND OSCILLATIONS
CHAPTER 7: FUNDAMENTALS OF VIBRATION
SHORTANSWERQUESTIONS:
1)Explain the physical characteristics of a simple harmonic motion.
2) (a) Define simple harmonic motion.
(b) Give the important characteristics of simple harmonic motion.
3) What are Lissajous figures? Give an example.
LONGANSWERQUESTIONS:
1)What is simple harmonic motion? What are the characteristics of simple harmonic motion?
2)Establish the equation of motion of a simple harmonic oscillator and solve it. Hence derive
expressions for its velocity, period and frequency. Discuss the relation between displacement, velocity
and accelarion in S.H.M.
3)Deduce expression for the total energy of a simple harmonic oscillator. Show that the total energy
remains independent of time and displacement.
4)Deduce the equation of motion of simple harmonic oscillator and show that frequency of oscillation is
independent of amplitude.
5)What is simplehormonic motion? Obtain an expression for the frequency of vibration of a loaded
spring taken its mass into consideration.
6)Find the resultant of two simple harmonic vibrations of the same frequency acting along the same line
but differing in phase.
7)Discuss the linear combination of simple harmonic oscillations of different frequencies.
What are Lissajous figures? Give some important uses of Lissajous figures.
8) (a)What are the characteristics of a simple harmonic oscillator?
(b) Obtain an expression for the total energy of a simple harmonic Oscillator; Show the total energy
is independent of the Instantaneous displacement of the body.
9)What is the simple harmonic motion? Write its characteristics. Establish the equation of motion of a
simple harmonic oscillator and solve it.
11) What are Lissajous figures? Find the resultant of two simple harmonic motions of the
same frequency acting along the same line but different phases.
12) Calculate the energy of vibration in simple harmonic motion.
PROBLEMS
1) A particle performing A.H.M has a maximum velocity of 0.4 m/s and a maximum acceleration of
0.8m/sec2.Calculate the amplitude and the period of the oscillator.
2) A particle under S.H.M has displacement of 0.4 m at the velocity 0.3m/s and at the displacement 0.3 m
at the velocity 0.4 m/s.calculate amplitude and frequency of the oscillation.
3) The displacement equation of a particle describing simple harmonic motion is
x=0.01sin100∏(t+0.005)meter, where x is the displacement of the particle at any instant t.calculate the
amplitude, periodic time, maximum velocity and displacement at that time of motion.
4) A particle executes S.H.M with a period of 0.002 sec and amplitude 10 cm .Find its acceleration where it
is 4 cm away from its mean position and also obtain its maximum velocity.
A particle of mass 5 gm executes S.H.M has an amplitude of 8 cm.If it makes 16 vibrations per sec.Find its
maximum velocity and energy at mean position.
5) A spring stretched by 8 cm by a force of 10 Newton. Find the force constant. What will be the period of
a 4 kg mass suspended by it.
6) The length of a weightless spring increases by 2 cm when a weight of 1.0 kg is suspended from it. The
weight is pulled down by 10 cm and released. Determine the (i) period of oscillation of spring (ii) K.E of
oscillation of spring.
7) A body of mass 4.9 kg hangs from a spring and oscillates with a period of 0.6 sec.How much will the
spring shorten when the body is removed.
8) A mass of 3 kg is hung from a vertical spring. When the mass of
0.5kg is gently added the spring
is further stretched by 5 cm.If the extra mass is removed and the first mass is set into oscillation, calculate
the period of oscillation.
9) A vertical spring extended by 0.25m when a mass of 5 kg loads it.
Calculate the frequency of oscillation if the spring is loaded with 4 kg
and allowed to oscillate.(g=9.8m/s2)
10) The period of oscillation of spring is 0.1 second. The amplitude of oscillation is 2 cm.A body of mass
0.2 kg is hung from it.Calculatew the maximum P.E that will be stored in it.
11) In a vertical spring-mass system, the period of the oscillation is 0.26 second when the mass is 0.4 kg and
period becomes 0.34 second when an additional mass of 0.3 kg is added. Calculate the mass of the spring and
the force constant of the spring.
12) A particle executes S.H.M with a period of 0.001 sec and amplitude 0.5 cm.Find its acceleration when it
is at 0.2 cm away from its mean position also obtained its maximum velocity.
13) A spring is stretched by 8 cm by a force of 10 Newton. Find the force constant of the string .What will
be the period of a 4 kg mass suspend by it.
14) A body of mass 0.25 kg hangs from a spring and oscillates. Find the time period and oscillating
frequency. [Spring force constant=40 NM-1]
15) A body of mass 0.2 kg is suspended from a spring, the spring extends by 0.1m .The velocity of the body
is 0.4 m/sec.when the displacement is 0.05 m.Find the time period ,frequency and amplitude.
16) A body of mass 0.2 kg is suspended from a spring, the spring extends by 0.1 m .The velocity of the
body is 0.4 m/sec.When the displacement is0.05 m. Find the time period ,frequency and amplitude.
CHAPTER 8
DAMPED AND FORCED OSCILLATIONS
SHORT ANSWERS
1) What are damped oscillations?
2) Distinguish between free and forced oscillations?
3) Explain the phenomenon of resonance. Give examples.
4) Explain the term ‘resonance’, sharpness of resonance’ and Q-factor?
5) Define Q of damped oscillator and explain its importance.
6) Explain the phenomenon of resonance .What is q-factor.
7) Explain the terms velocity resonance and amplitude resonance.
8) What is sharpness? What is Q-factor?
9) Define Q-factor and mention its importance.
LONGANSWERSQUESTIONS:
1)What are damped oscillations? Solve the differential equation of a damped harmonic oscillator and
discuss specially the case when it is under damped.
2)Discuss the case of damped simple harmonic motion. Deduce expression for amplitude, timeperiod and
logarithmic decrement of damped oscillations.
3)Solve the differential equations of a damped harmonic oscillator. Investigate the conditions under
which the oscillations are said to be underdamped, over damped and critically damped.
4)Outline the theory of damped simple harmonic oscillations clearly analyzing the underdamped,
overdamped and critically damped cases.
5) What are damped oscillations? Deduce the equation of motion of a damped harmonic oscillator and
obtain its solution. Discuss the condition under which the oscillations are over damped.
6)What are forced oscillations? Obtain an expression for the amplitude of forced oscillations. Explain
resonance and quality factor.
7)Derive and solve the differential equations of a driven harmonic oscillator .Distinguish between
velocity resonance (energy resonance) and amplitude resonance.
PROBLEMS:
1) A point performs damped oscillations according to the law (i) the x=a0e-. btsinώt .Find (i) the oscillation
amplitude and the velocity of the Point at the moment t=0 and (ii) the moments of time at which the A point
reaches the extreme positions.
2) The oscillations of at turing fork of frequency 200 cps die away to 1/e time their amplitude in one second.
Show that the reduction in frequency due to air damping is exceeding small
3) A particle of mass 2gm is initially displaced through 2 cm and the released. The frictional force on it is
5*10-3N/ms-1.The restoring force on it is 30*10-3N/m.Is it in oscillatory motion? If so find its period
4) A damped oscillator starting from reaches first amplitude of 500mm after 100 oscillations. The periodic
time is 2.3 sec.Find the damping constant and relaxation time.
5) The Quantity factor of a sonometer wire is 2*103.On plucking it makes 240 vibrations per second.
Calculate the time in which amplitude decreases to half initial value
6) The amplitude of a second pendulum falls to half initial value in 150 sec.Calculate the Q-factor.
7) The Q value of a spring loaded with 0.3 kg is 60.It vibrates with a frequency of 2 Hz. Calculate the force
constant and mechanical resistance.
8) The frequency of a tuning fork is 300Hz.If its quality Q is 5*104 find the time after which its energy
becomes (1/10) of its initial value.
CHAPTER 9: COMPLEX VIBRATIONS
SHORTANSWERQUESTIONS
1) Explain Fourier’s theorem.
2) State and explain Fourier theorem.
3) Explain how complex vibrations can be analysed by Fourier method.
LONGANSWERQUESTIONS:
1) State Fourier’s theorm.Analyse a square wave with the help of Fourier theorem.
2) State Fourier’s theorem. Describe Fourier constants.
3) State and explain Fourier theorem. Explain how you analyse periodic functions using Fourier’s theorem.
4) State and prove Fourier theorem and use it to analyse a square wave.
5) State Fourier’s theorem. What are limitations? Evaluate the Fourier constants.
6) State and prove Fourier’s theorem. Apply the Fourier theorem to analyse a saw-tooth wave.
7) State Fourier’s theorem. Obtain Fourier series for square wave.
PROBLEMS:
1) Find the Fourier series components of a complex harmonic motion defined by
y=a when 0<t<T/2
y=0 when T/2<t<T
2) Obtain Fourier series expansion of the periodic function f (t), the period of which is T and which has the
value zero from 0 to T/2 and the value 1 from T/2 to T and zero once again and so on.
3) Find the Fourier series for the function f (t) defined by
f (t) =AsinώT 0<t<T/2
=0
T/2<t<T
CHAPTER-10
VIBRATIONS OF BARS
LONG QUESTION:
1) Derive the equation of longitudinal wave in a bar.
2) Derive the equation of transverse wave in a bar.
3) Derive the general solution for longitudinal wave in a bar. Show the modes of vibration for a) The bar fixed
at both ends and, b) A bar free at both ends and c) A bar fixed at both ends.
4) Derive the general solution for longitudinal wave in a bar. Show the modes of vibration for a) The bar free
at both ends and b) The bar free at both ends and c) A bar fixed at both ends and c) A bar fixed at both ends .
5) Write the boundary conditions for transverse vibrations of bars. Derive the equations for allowed
frequencies of the overtones for transverse waves in special cases of a) A bar fixed at one end , b) A bar free at
both ends and c) A bar fixed at both ends
6) Discuss in detail about the statement "A tuning fork can be considered as a bar free at both ends bent in to
the U shape" Discuss the working and uses of a tuning fork.
PROBLEMS:
1) Velocity of sound in copper is 3560 ms-1 .Density of copper is 8890 kg m-3 .Find the young 's modules of
copper.
2) A copper rod of length 1 m and cross sectional diameter 0.008 m is free at both ends .Find the fundamental
frequency and the frequencies of the first and second overtimes of the transverse vibrations in the bar.
3) Velocity of sound in iron is 5130 ms-1 .The density of iron is 7,800 m-3. Find the Young's modulus of iron.
4) A copper bar of length 1 m is fixed rigidly at both ends. The diameter of cross section of the bar is
0.01m.Calculate the
1) Fundamental frequencies of the longitudinal vibrations and
2) Fundamental frequencies of transverse vibrations.
5) Find the fundamental frequency of longitudinal vibration of a brass rod of length 1 m fixed at the middle (and
free at both ends). Given the velocity of sound in brass as 3400 ms-1 and density of brass is 8650 kg m-3
CHAPTER 11
VIBRATING STRINGS
LONG QUESTIONS:
1) Write laws of transverse vibrations of strings. Deduce the expression for velocity of transverse wave along a
string.
2) Derive an expression for the velocity of a transverse wave along a stretched string.
3) State the laws of transverse vibrations of stretched strings. Derive an expression for the energy of transverse
vibration of a stretched string.
4) Discuss the theory of forced vibrations of a finite string under tension.
5) Obtain an expression for the energy density of a wave traveling along a string. What are harmonics and over
tones?
SHORT QURSTIONS:
1) Explain the average power of a wave traveling on a String.
2) State the laws of vibrating string.
3) Discuss about the energy transport in the wave motion along a string.
4) Derive an expression for the velocity of wave motion propagating along a string.
4) What are transverse waves? Obtain an expression for the frequency of vibration in a stretched string. Derive
the expression for the energy of vibration.
PROBLEMS:
1) A steal wire of diameter 1 cm is kept under a tension of 5 kN .The density of steel is 7.8 gm/cm3 .Calculate
the velocity of waves.
2) The velocity of wave in a stretched string of tension 19.6 N is 500 m/s . Find the velocity of a wave in that
string with a tension of 78.4 N.
3) The speed of transverse wave on a stretched string is 500 ms-1 when it is stretched under a tension of 19.6 N
of the tension is now altered to a value of 78.4 N what will be the speed of the wave?
4) A stretched string of length 0.25 m. has a frequency of f=300 Hz in the fundamental mode.Find the velocity
of transverse in the string.
CHAPTER-12
LONG ANSWER QUESTIONS
1) a) what are ultrasonic waves?
b) Explain one method of generating ultrasonic.
c) What are the uses of ultrasonic?
2) What are ultrasonic? Derive any method of production of ultrasonics.write application of ultrasonic.
3) What is meat by piezo electric effect? Explain the production, detection and the application of ultrasonic.
4) What is piexioelectric effect? Explain how it is used to produce ultrasonic waves write short note on the
application of ultrasonic waves.
5) Describe any method of production of ultrasonic waves. Write any five applications.
6) What are ultrasonic? Give the concept of piezoelectric effect and the method of producing ultrasonic by the
piezoelectric effect method.
SHORT ANSWERS
1) What is piezoelectric effect? Mention two examples for piezoelectric effect material.
2) Write a brief note on production of ultrasonic.
3) State any four applications of ultrasonic waves.
4) What are ultrasonic? Explain how the ultrasonic are detected?
5) Write a note on the application of ultrasonic.
PEOBLEMS
1) A quartz crystal of thickness 3 mm is vibrating at its natural frequency. If the young's modulus and density of
quartz are 7.9*10 10 n/m2 and 2,650 kg/m3 respectvely.Calculatethe crystal frequency for resonant vibration.
2) Calculate the fundamental frequency of a Quartz crystal of thickness 0.001m. y=7.9*10 10 n/m2 ,