Kamal A.A. 1000 Solved Problems in Classical Physics: An Exercise Book
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Problem Index 781
6.32 Length of a simple pendulum that has the same T as a swinging rod.
6.33 g by a physical pendulum.
6.34 Frequency of oscillations of a semicircular disc pivoted freely.
6.35 Ratio of time periods of two modes of oscillations of a ring suspended on a
nail.
6.36 Torsional oscillator.
6.37 Small oscillations of a pulley-spring-mass system.
6.38 Small oscillation of a cylinder attached to two springs.
6.39 Period of small oscillations of a particle in a one-dimensional potential field.
6.40 Effective spring constant of two springs in series.
6.41 Effective spring constant of two springs in parallel.
6.42 T for a block plus two spring system.
6.43 Oscillations of wire-spring-mass system.
6.44 Natural frequency of oscillation of a system consisting of a mass attached to
one end of a rod which is connected to the centre of a cylinder.
6.45 Natural frequency of an oscillating semi-circular disc.
6.46 Eigen frequencies and normal modes of coupled pendula.
6.47 Equations of motion and energy in normal coordinates. Characteristics of nor-
mal coordinates.
6.48 To find frequency components and beat frequency from the resultant
displacement.
6.49 Frequency of vibration of HCl molecule.
6.50 Resultant of three vibrations in the same straight line.
6.51 Logarithmic decrement for the vibrations of the mass-spring system.
6.52 Underdamped, overdamped and critically damped motion for the given
equation.
6.53 Natural period, damping constant and logarithmic energy decrement.
6.54 Solution of equation of motion for a damped oscillator.
6.55 Position of a weight attached to a vertical spring and nature of oscillations.
6.56 Resonance frequency when periodic force is applied.
6.57 To find amplitude, phase lag, Q-factor and power dissipation from the equa-
tion for forced oscillations.
6.58 Q-factor for electric bell given frequency and time constant.
6.59 An oscillator has T = 3 s. Its amplitude decreases by 5% each cycle. To find
energy decrease, time constant and Q-factor.
6.60 A damped oscillator loses 3% of energy in each cycle. Number of cycles
required for half of energy to be dissipated and Q-factor.
6.61 Decrease of amplitude in each cycle when ω
= 9ω
0
/10.
6.62 For small damping ω
(1 −r
2
/8 mk)ω
0
.
6.63 Time elapsed between successive maximum displacements of a damped
oscillator.
6.64 Period of oscillation from logarithmic decrement.
6.65 ω
0
and frequency of driving force from the given equation.
6.66 To show that t
1/2
= t
c
ln 2.
6.67 Fraction of energy decrease in each cycle and Q-factor.
782 Problem Index
Chapter 7
Lagrangian and Hamiltonian Mechanics
7.1 Equations of motion for a particle under force μm/r
2
.
7.2 Lagrangian for simple pendulum and proof for SHM.
7.3 Equations of motion for masses of Atwood machine by Lagrangian method.
7.4 Double Atwood machine.
7.5 Lagrangian and its equation of motion from given T and V .
7.6 Lagrangian for an SHO and time period.
7.7 Acceleration of a block on a fixed incline.
7.8 Acceleration of a block and inclined plane resting on smooth horizontal table.
7.9 Sliding of a bead on a straight wire which is constrained to rotate.
7.10 Spherical pendulum.
7.11 Equation of motion for a system of two blocks connected by spring on smooth
horizontal table.
7.12 A double pendulum.
7.13 Hamilton’s equations for spherical pendulum.
7.14 Hamilton’s equation and solution from H for one-dimensional SHO.
7.15 Planetary motion using Hamilton’s equations.
7.16 Natural frequencies of two coupled blocks.
7.17 Equations of motion of a simple pendulum pivoted to a block which slides on
a smooth horizontal plane.
7.18 Equations of motion of an insect on a rod turning about one fixed end.
7.19 A rod attached at one end by a cord to a fixed end. To find inclination of string
and rod when the system revolved about vertical through pivot.
7.20 Lagrangian in (r, θ ) coordinates for a central potential and corresponding p
r
and p
θ
and H and conservation of E and J.
7.21 Coupled linear differential equations from Lagrangian equations and normal
modes for a system of two masses plus three springs.
7.22 Lagrangian eigenfrequencies and normal modes for a system of two identical
beads connected by two springs to a fixed wall.
7.23 Lagrangian and eigenfrequencies of a system of two beads of different masses
connected to a wall by two springs.
7.24 Normal modes of oscillation for a system of three particles connected by
springs.
7.25 Derivation of H for a single particle under conservative force.
7.26 Motion of a pendulum mounted on a block which can freely move on a hori-
zontal surface.
7.27 Sliding of a particle down a smooth spherical bowl when it is (a) fixed and (b)
freetomove.
7.28 Hamilton and Hamiltonian equations from the given Lagrangian.
7.29 Lagrangian and equation of motion as in the given diagram.
7.30 Lagrangian and Lagrangian equation and equilibrium positions of a particle
moving on an elliptic orbit in vertical plane.
Problem Index 783
7.31 Double pendulum with equal masses and lengths.
7.32 Coupled pendulums by Lagrangian method.
7.33 A bead sliding freely on a circular wire which rotates on a horizontal plane.
7.34 Equations of motion of a block-wedge spring system.
7.35 Rolling of a ball down a wedge which itself can slide on a horizontal table.
Chapter 8
Waves
8.1 Solution of a one-dimensional wave equation.
8.2 y = 2A sin(nπ x/L) cos 2π ft for standing wave is a solution of wave equa-
tion.
8.3 String plucked at the centre.
8.4 Superposition of the waves y
1
= A sin(kx − ωt) and y
2
= 3A sin(kx −ωt).
8.5 A sinusoidal wave has v = 8 m/s and λ = 2 m. To find K , f , ω and wave
equation.
8.6 Equation of wave given k, ω and A.
8.7 When a standing wave is formed each point undergoes SHM transversely.
8.8 Frequencies of the first three harmonics for a plucked string.
8.9 Ratio of linear mass density of two strings.
8.10 A, f , v and λ of a transverse wave.
8.11 A and v of component waves of the given vibration. Distance between the
nodes and the transverse velocity at a given point at a given time.
8.12 Phase velocity and phase difference.
8.13 Amplitude of resultant motion.
8.14 Suitable functions for one-dimensional wave equation.
8.15 Displacement of a cord when plucked at x = L/3.
8.16 Equation of a wave in the negative x-direction.
8.17 Superposition of the waves y
1
= A sin(kx − ωt) and y
2
= A sin(kx + ωt) is
a standing wave.
8.18 Superposition of a harmonic wave with another wave travelling in the same
direction but differing by phase difference δ of the same amplitude.
8.19 Average rate of energy transmission of a travelling wave.
8.20 Frequency of the fork by beat frequency with a monochord.
8.21 Velocity of a moving pulse.
8.22 Mass density of piano string frequencies of the first two harmonics. Length of
a flute pipe.
8.23 (a) Sketch of first and second harmonic waves on a stretched string and (b) v
and distance between nodes for given standing wave.
8.24 Wave function for the progressive wave from given data on A, ω and k.
8.25 Units for (F/μ)
1/2
.
8.26 For a sinusoidal wave on a string slope ∂y/∂x is equal to ∂y/∂t divided by v.
8.27 Wave equation for transverse waves.
784 Problem Index
8.28 Reflection of wave at the joint of two wires.
8.29 From the sketch to find f and λ if v is given and to find equation f or the
wave.
8.30 Given the linear density of the string to find energy sent down the string per
second in prob. (8.29).
8.31 Energy of the string in t he nth mode.
8.32 (a) Fundamental frequency of a steel bar for longitudinal vibrations and (b)
comparison of frequencies for (i) free at both ends; (ii) damped at midpoint of
bar (i); and (iii) clamped at both ends.
8.33 Fundamental frequency of vertical oscillations of a mass attached to a wire as
(a) a simple oscillator and (b) system of vibrator fixed at one end and mass
loaded at the other.
8.34 For a mass loaded bar the frequency condition kL tan kL = M/m reduces to
that of SHO for kL < 0.2.
8.35 Velocity of long waves compared with those in deep liquid and canal waves.
8.36 Maximum depth of liquid for which the formula v
2
= gh represents velocity
of waves of length λ within 1%.
8.37 Surface tension of water by Ripple method.
8.38 Minimum velocities of surfaces waves for mercury and water.
8.39 v
p
and v
g
from dispersion relation for a piano wire.
8.40 v
p
and v
g
from dispersion relation for water waves of very short wavelength
in deep water.
8.41 Given general dispersion relation for water wave to show that v
p
= v
g
=
√
gh
and for deep water v
p
= (g/k + Sk/ρ)
1/2
and to find v
g
.
8.42 v
p
and v
g
in deep water for small ripples.
8.43 v
p
v
g
= c
2
for a relativistic particle.
8.44 Wavelength of surface waves on water.
8.45 v
2
p
= g/k for deep water waves. To show that v
g
= v
p
/2.
8.46 v
p
and v
g
from dispersion relation of sound in air.
8.47 Given v
2
p
= (g/k + Sk/ρ) for deep water waves, to show v
p
is minimum for
λ = 2π(S/ρg)
1/2
.
8.48 Relation between pressure amplitude and displacement amplitude and that
they are out of phase by 90
◦
.
8.49 Pressure amplitude corresponding to the threshold of hearing intensity.
8.50 Intensity of wave from intensity level of ordinary conversation.
8.51 A point source of sound radiates energy of 4 W. To find I and I.L at 25 m from
source.
8.52 Maximum displacement from maximum pressure variation, f , ρ and v.
8.53 Ratio of intensities of two sound waves one in air and the other in water having
equal pressure amplitude.
8.54 Pressure amplitude, f , ρ and v from the equation. P = 2.4sinπ(x − 330t)
for progressive wave.
8.55 Amplitude of air vibrations by a note of given frequency and intensity.
8.56 To show a plane wave of effective acoustic pressure of a microbar has I.L of
74 dB.
Problem Index 785
8.57 Energy density and effective pressure of a plane wave in air of 70 dB I.L.
8.58 Pressure amplitude for I = 1W/m
2
at pain threshold.
8.59 Theoretical speed of sound in H
2
at 0
◦
C.
8.60 Given speed of sound in H
2
at 0
◦
C, to calculate speed in O
2
.
8.61 Two sound waves have I
1
= 0.4 and 10 W/m
2
. How many decibels is one
louder than the other?
8.62 Ratio of intensities if one sound is 6.0 dB higher than the other.
8.63 Distance to which sound is audible if the source radiates at the rate of 0.009 W.
8.64 Calculation of displacement amplitude from the pressure amplitude.
8.65 Two sound waves of equal pressure amplitude and frequencies traverse in liq-
uids with ρ
1
/ρ
2
= 3/4 with v
1
/v
2
= 3/2. To compare displacement ampli-
tudes, intensities and energy densities.
8.66 One sound wave travels in air and the other in water, their intensities and
frequencies being equal. To find ratio of wavelengths, pressure amplitude and
particle amplitudes.
8.67 Characteristic impedance.
8.68 Intensity of a beam of plane waves, pressure amplitude displacement ampli-
tude, acoustic particle velocity amplitude and condensation amplitude.
8.69 Laplace formula for sound velocity in a gas.
8.70 An empirical formula for sound velocity as a function of temperature.
8.71 The sound of whistle is reflected from the wall of the rock as the engine
approaches a tunnel. To find the ratio of frequencies of reflected and direct
sounds heard by the engine driver given the speed of the train.
8.72 Doppler effect when two train moves towards each other.
8.73 Doppler effect when two trains move away from each other.
8.74 Maximum and minimum frequencies heard by a listener from a rotating
whistle.
8.75 The Kundt’s tube experiment.
8.76 Wavelength of sound from a motion train when (i) train at rest; (ii) moving
towards you; and (iii) moving away from you.
8.77 Shock wave, Mach number and angle of Mach cone.
8.78 Reverberation time for a room, Sabine’s formula.
8.79 Echo of drum beating from a mountain.
8.80 Echo of rifle shot fired in the valley formed between two parallel mountains.
8.81 Beat frequency heard by a man who walks in line between two whistles sound-
ing neighbouring frequencies.
8.82 Frequency of the unknown frequency by beat frequency in transferring load
from one fork to another.
8.83 Beats produced by a fork sounding with an open organ pipe of an appropriate
length.
8.84 Falling plate experiment in sound.
8.85 Effective length of a resonance tube closed at one end.
8.86 When an open pipe is suddenly closed so that f
3
of the closed pipe is higher
by 100 Hz than f
2
of original pipe. To find f
1
of original pipe.
786 Problem Index
Chapter 9
Fluid Dynamics
9.1 Velocity of water in the narrower portion of pipe.
9.2 Verification of continuity equation from velocity components.
9.3 Work done in forcing water through pipe.
9.4 Lift on the wing of aeroplane.
9.5 Rate of flow of water using a venturi meter.
9.6 Speed of a plane using a Pitot tube.
9.7 Velocity of the spray in a sprinkler.
9.8 Test of steady incompressible flow.
9.9 Speed of flow past the lower and upper surface of the wing of an aeroplane.
9.10 Pressure drop and velocity in the throat of a venturimeter.
9.11 Application of Reynolds number to steady and turbulent flow.
9.12 A tube open at one end and closed at the other with small orifice is filled with
liquid. To find the efflux velocity when rotated in a horizontal plane, about an
axis through the open end.
9.13 Application of pitot tube.
9.14 Rate of flow of water in a horizontal pipe of varying cross-sections.
9.15 A cylinder with a small hole at the bottom is filled with water and fitted with
a piston. To find the work done by a constant force when the system is rotated
horizontally to squeeze all water from the cylinder.
9.16 A cylindrical vessel with water is r otated about its vertical axis. To find pres-
sure distribution and shape of free space of water.
9.17 Speed of water flowing from a water tap.
9.18 Application of Torricelli’s theorem.
9.19 Application of Torricelli’s theorem.
9.20 Application of Torricelli’s theorem.
9.21 Water leaks through a hole in the bottom of a tank. To find time for water level
to decrease from h
1
and h
2
.
9.22 Velocity of efflux through a hole in a tank filled with water.
9.23 Application of Torricelli’s theorem to two tanks filled with a liquid and car-
rying hole of different areas and at different depths, the volume of flux being
identical.
9.24 Efflux velocity of water in a bottom orifice in a tank filled with water +
kerosene.
9.25 Two identical holes are punched on opposite sides at different height in a ves-
sel filled with water. To calculate resultant force of reaction of water flow.
9.26 Pressure required to maintain water flow through a tube.
9.27 Pressure difference across a composite tube in Poiseuille’s experiment.
9.28 Terminal velocity of rain drops.
9.29 Water flow through horizontal tubes connected in parallel.
9.30 Water flow through horizontal tubes connected in series.
Problem Index 787
Chapter 10
Heat and Matter
10.1 Mean free path of molecule given collision frequency and mean molecular
speed.
10.2 rms of a molecule, mean free path and collision frequency.
10.3 Mean free path of gas molecules.
10.4 v
mp
and v
av
and T for two graphs for Maxwell–Boltzmann distribution.
10.5 Mort probable speed assuming Maxwell–Boltzmann distribution.
10.6 < E >, v
rms
, v
mp
and v
av
for CO
2
gas.
10.7 Volume of a helium-filled weather balloon when it rises to high altitude.
10.8 Using van der Waal’s equation to estimate the molar density of H
2
Sgas.
10.9 Thermal expansion of a bimetal bar.
10.10 Buckled rail due to thermal expansion.
10.11 Length of a steel and copper rod such that the difference is constant at any
temperature.
10.12 Volume of mercury in a flask such that the volume of air inside the flask is
constant at any temperature.
10.13 Tension in a wire with the decrease in temperature.
10.14 Specific gravity bottle experiment.
10.15 Determination of the gas constant.
10.16 Rising air bubble in a lake; Boyle’s law.
10.17 Load carried by a balloon to a given height.
10.18 Two glass bulbs in communication kept at different temperatures. To find the
pressure.
10.19 Conduction of heat through slabs in series.
10.20 Conduction of heat through slabs in parallel.
10.21 Temperature of interface of copper and iron bars i n heat conduction.
10.22 Melting of ice kept at one end of a copper bar while the other end is heated.
10.23 Formula for heat conduction in a metal at low temperature given that thermal
conductivity is proportional to absolute temperature.
10.24 Radial flow of heat between concentric spheres.
10.25 Radial flow of heat in a material across a coaxial cylinders.
10.26 Rate of addition of ice at the bottom of a layer of ice in a pond when air
temperature drops to −10
◦
C.
10.27 Application of Newton’s law of cooling.
10.28 Water equivalent of calorimeter from Newton’s law of cooling.
10.29 Rate of cooling of steel balls of different radii.
10.30 Application of resistance thermometer.
10.31 Solar constant from the given data.
10.32 Temperature of sun’s surface from given data.
10.33 Rate of heat loss by radiation at a given temperature.
10.34 Application of Wien’s displacement law.
788 Problem Index
10.35 Latent heat of fusion determination.
10.36 Mean specific heat and specific heat at midpoint.
10.37 Equilibrium temperature from the mixture of liquid (A, B), (B, C) and
(A, C).
10.38 Rise of temperature of the bullet in its collision with a block of wood (a)
fixed and (b) free to move.
10.39 Rise in temperature due to water fall.
10.40 Rise in temperature due to fall of a lead piece from a given height on a non-
conducting slab.
10.41 Work done on a gas during an adiabatic compression.
10.42 A thermodynamic cycle.
10.43 In an adiabatic process of a monatomic ideal gas PV
5/3
= const.
10.44 Work done on a gas during an isothermal compression.
10.45 P − V diagram for a sequence of thermodynamic processes.
10.46 Number of degrees of freedom for gas molecules.
10.47 Efficiency of a heat engine.
10.48 Temperatures of the source and sink of a heat engine.
10.49 Air pressure at a given attitude h, number density of gas molecules and p at
h/2.
10.50 P − V diagram for the Carnot cycle and the Sterling cycle.
10.51 Entropy change for isobaric and isochoric processes.
10.52 Change of values of thermodynamic parameters in reversible isothermal
expansion.
10.53 Internal energy, heat, enthalpy, work and Gibbs function.
10.54 Shear modulus of a material.
10.55 Stress, strain and Young’s modulus of a wire.
10.56 Isothermal elasticity and adiabatic elasticity.
10.57 Poisson’s ratio from the ratio of Young’s modulus and the rigidity modulus.
10.58 Young’s modulus from the depression of a wire caused by attaching known
load at midpoint of a horizontal wire.
10.59 Speed of an object released from a catapult.
10.60 Maximum angular speed of an object attached to the end of a wire whirled
in a horizontal plane.
10.61 Maximum length of a wire which will not break under its own weight when
it hangs freely.
10.62 Capillary rise.
10.63 Volume of air bubble at depth 100 m – gas equation.
10.64 Energy released in coalescing of droplets.
10.65 Work done in blowing a soap bubble to a larger size.
10.66 A hollow vessel with a small hole of radius r is immersed to a known depth
when water just penetrates into vessel, to find r.
10.67 Depth of water column at which an air bubble is in equilibrium.
10.68 Inadequate capillary tube length.
10.69 Effect of charge on soap bubble.
10.70 Radius of bigger bubble when two bubbles coalesce.
Problem Index 789
Chapter 11
Electrostatics
11.1 (a) Force between two charges. Position of neutral point (b) E and F of
proton at the position of electron in H atom.
11.2 (a) Tension in the thread when a charged ball hangs in an electric field and
(b) V
b
− V
a
positive or negative.
11.3 Electric potential along the axis of a charged circular loop.
11.4 Potential energy of four charges at the corners of a square and their stability.
11.5 V at the surface of a charged liquid drop, when two such drops coalesce, V
at the surface of new drop.
11.6 A charged pendulum bob is in equilibrium in a horizontal electric field. Ten-
sion and angle of the thread with the vertical.
11.7 An infinite number of charges, each equal to q are placed at x = 1, 2, 4, 8,...
units. V and E at x = 0.
11.8 In prob. (11.7) what will be V and E if consecutive charges have opposite
sign.
11.9 A charged particle is released from rest on the axis of a fixed oppositely
charged ring. To show that the particle has SHM and to find time period of
oscillation.
11.10 Three charges each of value q are at the corners of an equilateral triangle and
the fourth one at the centre. It Q =−q, charges move toward or away from
the centre. Value of Q for the charge to be stationary.
11.11 Two identically charged spheres are suspended by strings of equal length.
The strings make an angle of 30
◦
with each other. When immersed in a
liquid the angle remains the same. To find the dielectric constant of the
liquid.
11.12 One charge is placed at one corner of a square and another at the center. Work
done to move the charge from the centre to an empty corner.
11.13 A charged pith ball suspended by a thread is deflected by a known distance
by a horizontal electric field. To find E.
11.14 Suspension of a charged oil drop under electric field and gravity.
11.15 Equal amount of charge on the earth and the moon to nullify gravitational
attraction.
11.16 An energy output of 10
−5
J results from a spark between insulated surfaces
at P.D 5 × 10
6
V. To find q transferred and number of electrons flowed.
11.17 E at a given distance along the axis of a charged rod.
11.18 E along the axis of a charged disc.
11.19 Electronic charge by Millikan’s oil drop method.
11.20 Total charge on a circular wire which has cos
2
θ charge density dependence.
11.21 Strength of electric force compared to gravitation force in H atom.
11.22 Charges on two spheres given their combined charge and force of repulsion
at the given distance.
790 Problem Index
11.23 Four charges are placed at the four corners of a square as in the diagram. To
find E and its direction.
11.24 E on the perpendicular bisector of a thin charged rod.
11.25 A thin non-conducting charged rod is bent to form an arc of circle and sub-
tends an angle θ
o
at the centre of circle. To find E at the centre of circle.
11.26 In prob. (11.3) to find the distance at which E is maximum.
11.27 E and V at the centre of a charged non-conducting hemispherical cup.
11.28 E varies as 1/r
4
for an electric quadrupole.
11.29 Electric and gravitational forces between two bodies each of mass m and
charge q will be equal if q/m = 8.6 × 10
−10
/kg.
11.30 Two equally charged spheres are suspended from the same point by silk
thread of the same length. To find the rate at which charge leaks out given
the relative velocity of approach.
11.31 A long charged thread is placed on the axis of a charged ring with one end
coinciding with the centre of the ring. To find force of interaction.
11.32 A very long wire with charge density λ is placed on the x-axis with one end
coinciding with the origin. To calculate E from the y-axis.
11.33 E from the potential φ = cxy.
11.34 To show that the locus of zero potential points for two fixed charges is a
circle.
11.35 Two identical rings charged to Q
1
and Q
2
coaxially placed at a fixed dis-
tance. To find the work done in moving a charge q from the centre of one
ring to that of the other.
11.36 To find x if the potentials at (0, 2a) and P
2
(x, 0) are equal and to find the
potential.
11.37 Value of Q if the interaction energy of three charges +q
1
, +q
2
and Q placed
at the vertices of a right-angled isosceles triangle is zero.
11.38 Work done in assembling the charges at the four corners of a square as in the
diagram.
11.39 Total potential energy of a charged sphere.
11.40 A linear quadrupole.
11.41 Charge that can be placed on a sphere for the given field strength and the
corresponding V .
11.42 Speed of electron when it approaches a positive charge.
11.43 For the linear quadrupole of prob. (11.40)forr >> d, E(r) ∝ 1/r
3
.
11.44 Force and acceleration of electron when it passes through a hole in a con-
denser plate.
11.45 Potential on the axis of a charged disc and the limiting case of x >> R.
11.46 Kepler’s third l aw of motion is applicable to electron in H atom.
11.47 Equal charges are placed on four corners of a square. To calculate F on one
of them due to other three.
11.48 E on the perpendicular bisector of a dipole. For x >> d/2, E ∝ 1/r
3
.
11.49 Application of Gauss’ law to an infinite sheet of charge. σ of the sheet by
deflection of a charged mass hanging by a silk thread.