Kamal A.A. 1000 Solved Problems in Classical Physics: An Exercise Book
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x Contents
3 Rotational Kinematics ..........................................103
3.1 Basic Concepts and Formulae ...............................103
3.2 Problems .................................................107
3.2.1 Motion in a Horizontal Plane ........................107
3.2.2 Motion in a Vertical Plane ...........................112
3.2.3 Loop-the-Loop ....................................112
3.3 Solutions .................................................114
3.3.1 Motion in a Horizontal Plane ........................114
3.3.2 Motion in a Vertical Plane ...........................123
3.3.3 Loop-the-Loop ....................................129
4 Rotational Dynamics ...........................................135
4.1 Basic Concepts and Formulae ...............................135
4.2 Problems .................................................138
4.2.1 Moment of Inertia..................................138
4.2.2 Rotational Motion .................................139
4.2.3 Coriolis Acceleration ...............................149
4.3 Solutions .................................................151
4.3.1 Moment of Inertia..................................151
4.3.2 Rotational Motion .................................157
4.3.3 Coriolis Acceleration ...............................184
5 Gravitation ....................................................189
5.1 Basic Concepts and Formulae ...............................189
5.2 Problems .................................................193
5.2.1 Field and Potential .................................193
5.2.2 Rockets and Satellites ..............................196
5.3 Solutions .................................................201
5.3.1 Field and Potential .................................201
5.3.2 Rockets and Satellites ..............................213
6 Oscillations ....................................................235
6.1 Basic Concepts and Formulae ...............................235
6.2 Problems .................................................245
6.2.1 Simple Harmonic Motion (SHM) .....................245
6.2.2 Physical Pendulums ................................248
6.2.3 Coupled Systems of Masses and Springs ...............251
6.2.4 Damped Vibrations.................................253
6.3 Solutions .................................................254
6.3.1 Simple Harmonic Motion (SHM) .....................254
6.3.2 Physical Pendulums ................................267
6.3.3 Coupled Systems of Masses and Springs ...............273
6.3.4 Damped Vibrations.................................279
Contents xi
7 Lagrangian and Hamiltonian Mechanics ..........................287
7.1 Basic Concepts and Formulae ...............................287
7.2 Problems .................................................288
7.3 Solutions .................................................296
8Waves.........................................................339
8.1 Basic Concepts and Formulae ...............................339
8.2 Problems .................................................345
8.2.1 Vibrating Strings...................................345
8.2.2 Waves in Solids....................................350
8.2.3 Waves in Liquids ..................................350
8.2.4 Sound Waves......................................352
8.2.5 Doppler Effect ....................................354
8.2.6 Shock Wave.......................................355
8.2.7 Reverberation .....................................355
8.2.8 Echo .............................................355
8.2.9 Beat Frequency ....................................355
8.2.10 Waves in Pipes ....................................356
8.3 Solutions .................................................356
8.3.1 Vibrating Strings...................................356
8.3.2 Waves in Solids....................................371
8.3.3 Waves in Liquids ..................................373
8.3.4 Sound Waves......................................378
8.3.5 Doppler Effect ....................................384
8.3.6 Shock Wave.......................................386
8.3.7 Reverberation .....................................386
8.3.8 Echo .............................................387
8.3.9 Beat Frequency ....................................388
8.3.10 Waves in Pipes ....................................389
9 Fluid Dynamics ................................................391
9.1 Basic Concepts and Formulae ...............................391
9.2 Problems .................................................394
9.2.1 Bernoulli’s Equation ...............................394
9.2.2 Torricelli’s Theorem ................................396
9.2.3 Viscosity .........................................397
9.3 Solutions .................................................398
9.3.1 Bernoulli’s Equation ...............................398
9.3.2 Torricelli’s Theorem ................................403
9.3.3 Viscosity .........................................406
10 Heat and Matter ...............................................409
10.1 Basic Concepts and Formulae ...............................409
10.2 Problems .................................................414
xii Contents
10.2.1 Kinetic Theory of Gases ............................414
10.2.2 Thermal Expansion ................................416
10.2.3 Heat Transfer .....................................418
10.2.4 Specific Heat and Latent Heat ........................420
10.2.5 Thermodynamics ..................................420
10.2.6 Elasticity .........................................423
10.2.7 Surface Tension ...................................425
10.3 Solutions .................................................425
10.3.1 Kinetic Theory of Gases ............................425
10.3.2 Thermal Expansion ................................430
10.3.3 Heat Transfer .....................................433
10.3.4 Specific Heat and Latent Heat ........................439
10.3.5 Thermodynamics ..................................441
10.3.6 Elasticity .........................................452
10.3.7 Surface Tension ...................................455
11 Electrostatics ..................................................459
11.1 Basic Concepts and Formulae ...............................459
11.2 Problems .................................................465
11.2.1 Electric Field and Potential ..........................465
11.2.2 Gauss’ Law .......................................473
11.2.3 Capacitors ........................................476
11.3 Solutions .................................................482
11.3.1 Electric Field and Potential ..........................482
11.3.2 Gauss’ Law .......................................506
11.3.3 Capacitors ........................................517
12 Electric Circuits ...............................................535
12.1 Basic Concepts and Formulae ...............................535
12.2 Problems .................................................538
12.2.1 Resistance, EMF, Current, Power .....................538
12.2.2 Cells .............................................544
12.2.3 Instruments .......................................544
12.2.4 Kirchhoff’s Laws ..................................547
12.3 Solutions .................................................552
12.3.1 Resistance, EMF, Current, Power .....................552
12.3.2 Cells .............................................562
12.3.3 Instruments .......................................564
12.3.4 Kirchhoff’s Laws ..................................569
13 Electromagnetism I ............................................579
13.1 Basic Concepts and Formulae ...............................579
13.2 Problems .................................................583
Contents xiii
13.2.1 Motion of Charged Particles in Electric
and Magnetic Fields ............................... 583
13.2.2 Magnetic Induction ................................587
13.2.3 Magnetic Force ....................................593
13.2.4 Magnetic Energy, Magnetic Dipole Moment............595
13.2.5 Faraday’s Law.....................................596
13.2.6 Hall Effect ........................................599
13.3 Solutions .................................................599
13.3.1 Motion of Charged Particles in Electric
and Magnetic Fields ............................... 599
13.3.2 Magnetic Induction ................................606
13.3.3 Magnetic Force ....................................619
13.3.4 Magnetic Energy, Magnetic Dipole Moment............622
13.3.5 Faraday’s Law.....................................625
13.3.6 Hall Effect ........................................629
14 Electromagnetism II ............................................631
14.1 Basic Concepts and Formulae ...............................631
14.2 Problems .................................................637
14.2.1 The RLC Circuits ..................................637
14.2.2 Maxwell’s Equations, Electromagnetic Waves,
Poynting Vector ................................... 642
14.2.3 Phase Velocity and Group Velocity ...................649
14.2.4 Waveguides .......................................650
14.3 Solutions .................................................651
14.3.1 The RLC Circuits ..................................651
14.3.2 Maxwell’s Equations and Electromagnetic Waves,
Poynting Vector ................................... 665
14.3.3 Phase Velocity and Group Velocity ...................692
14.3.4 Waveguides .......................................696
15 Optics ........................................................703
15.1 Basic Concepts and Formulae ...............................703
15.2 Problems .................................................713
15.2.1 Geometrical Optics.................................713
15.2.2 Prisms and Lenses .................................715
15.2.3 Matrix Methods ...................................717
15.2.4 Interference .......................................717
15.2.5 Diffraction ........................................721
15.2.6 Polarization .......................................724
15.3 Solutions .................................................725
15.3.1 Geometrical Optics.................................725
15.3.2 Prisms and Lenses .................................728
15.3.3 Matrix Methods ...................................737
xiv Contents
15.3.4 Interference .......................................740
15.3.5 Diffraction ........................................751
15.3.6 Polarization .......................................760
Subject Index ......................................................765
Problem Index .....................................................769
Chapter 1
Kinematics and Statics
Abstract Chapter 1 is devoted to problems based on one and two dimensions.
The use of various kinematical formulae and the sign convention are pointed out.
Problems in statics involve force and torque, centre of mass of various systems and
equilibrium.
1.1 Basic Concepts and Formulae
Motion in One Dimension
The notation used is as f ollows: u = initial velocity, v = final velocity, a = accele-
ration, s = displacement, t = time (Table 1.1).
Table 1.1 Kinematical equations
UV ASt
(i) v = u + at X
(ii) s = ut + 1/2at
2
X
(iii) v
2
= u
2
+ 2as X
(iv) s =
1
2
(u + v)t X
In each of the equations u is present. Out of the remaining four quantities only
three are required. The initial direction of motion is taken as positive. Along this
direction u and s and a are taken as positive, t is always positive, v can be positive
or negative. As an example, an object is dropped from a rising balloon. Here, the
parameters for the object will be as follows:
u = initial velocity of the balloon (as seen from the ground)
u =+ve, a =−g. t =+ve, v =+ve or −ve depending on the value of t, s =+ve
or −ve, if s =−ve, then the object is found below the point it was released.
Note that (ii) and (iii) are quadratic. Depending on the value of u, both the
roots may be real or only one may be real or both may be imaginary and therefore
unphysical.
1
2 1 Kinematics and Statics
v–t and a–t Graphs
The area under the v–t graph gives the displacement (see prob. 1.11) and the area
under the a–t graph gives the velocity.
Motion in Two Dimensions – Projectile Motion
Equation: y = x tan α −
1
2
gx
2
u
2
cos
2
α
(1.1)
Fig. 1.1 Projectile Motion
Time of flight: T =
2u sin α
g
(1.2)
Range: R =
u
2
sin 2α
g
(1.3)
Maximum height: H =
u
2
sin
2
α
2g
(1.4)
Velocity: v =
g
2
t
2
− 2ug sin α.t +u
2
(1.5)
Angle: tan θ =
u sin α − gt
u cos α
(1.6)
Relative Velocity
If v
A
is the velocity of A and v
B
that of B, then the relative velocity of A with respect
to B will be
v
AB
= v
A
− v
B
(1.7)
Motion in Resisting Medium
In the absence of air the initial speed of a particle thrown upward is equal to that
of final speed, and the time of ascent is equal to that of descent. However, in the
presence of air resistance the final speed is less than the initial speed and the time of
descent is greater than that of ascent (see prob. 1.21).
1.2 Problems 3
Equation of motion of a body in air whose resistance varies as the velocity of the
body (see prob. 1.22).
Centre of mass is defined as
r
cm
=
m
i
r
i
m
i
=
1
M
m
i
r
i
(1.8)
Centre of mass velocity is defined as
V
c
=
1
M
m
i
˙r
i
(1.9)
The centre of mass moves as if the mass of various particles is concentrated at
the location of the centre of mass.
Equilibrium
A system will be in translational equilibrium if F = 0. In terms of potential
∂V
∂x
= 0, where V is the potential. The equilibrium will be stable if
∂
2
V
∂x
2
< 0.
A system will be in rotational equilibrium if the sum of the external torques is zero,
i.e. τ
i
= 0
1.2 Problems
1.2.1 Motion in One Dimension
1.1 A car starts from rest at constant acceleration of 2.0 m/s
2
. At the same instant
a truck travelling with a constant speed of 10 m/s overtakes and passes the car.
(a) How far beyond the starting point will the car overtake the truck?
(b) After what time will this happen?
(c) At that instant what will be the speed of the car?
1.2 From an elevated point A, a stone is projected vertically upward. When the
stone reaches a distance h below A, its velocity is double of what it was at a
height h above A. Show that the greatest height obtained by the stone above A
is 5h/3.
[Adelaide University]
1.3 A stone is dropped from a height of 19.6 m, above the ground while a second
stone is simultaneously projected from the ground with sufficient velocity to
enable it to ascend 19.6 m. When and where the stones would meet.
1.4 A particle moves according to the law x = A sin πt, where x is the displace-
ment and t is time. Find the distance traversed by the particle in 3.0 s.
4 1 Kinematics and Statics
1.5 A man of height 1.8 m walks away from a lamp at a height of 6 m. If the man’s
speed is 7 m/s, find the speed in m/s at which the tip of the shadow moves.
1.6 The relation 3t =
√
3x + 6 describes the displacement of a particle in one
direction, where x is in metres and t in seconds. Find the displacement when
the velocity is zero.
1.7 A particle projected up passes the same height h at 2 and 10 s. Find h if g =
9.8m/s
2
.
1.8 Cars A and B are travelling in adjacent lanes along a straight road (Fig. 1.2).
At time, t = 0 their positions and speeds are as shown in the diagram. If car A
has a constant acceleration of 0.6m/s
2
and car B has a constant deceleration of
0.46 m/s
2
, determine when A will overtake B.
[University of Manchester 2007]
Fig. 1.2
1.9 A boy stands at A in a field at a distance 600 m from the road BC. In the field
he can walk at 1 m/s while on the road at 2 m/s. He can walk in the field along
AD and on the road along DC so as to reach the destination C (Fig. 1.3). What
should be his route so that he can reach the destination in the least time and
determine the time.
Fig. 1.3
1.10 Water drips from the nozzle of a shower onto the floor 2.45 m below. The drops
fall at regular interval of time, the first drop striking the floor at the instant the
third drop begins to fall. Locate the second drop when the first drop strikes the
floor.
1.11 The velocity–time graph for the vertical component of the velocity of an object
thrown upward from the ground which reaches the roof of a building and
returns to the ground is shown in Fig. 1.4. Calculate the height of the building.
1.2 Problems 5
Fig. 1.4
1.12 A ball is dropped into a lake from a diving board 4.9 m above the water. It
hits the water with velocity v and then sinks to the bottom with the constant
velocity v. It reaches the bottom of the lake 5.0 s after it is dropped. Find
(a) the average velocity of the ball and
(b) the depth of the lake.
1.13 A stone is dropped into the water from a tower 44.1 m above the ground.
Another stone is thrown vertically down 1.0 s after the first one is dropped.
Both the stones strike the ground at the same time. What was the initial veloc-
ity of the second stone?
1.14 A boy observes a cricket ball move up and down past a window 2 m high. If
the total time the ball is in sight is 1.0 s, find the height above the window that
the ball rises.
1.15 In the last second of a free fall, a body covered three-fourth of its total path:
(a) For what time did the body fall?
(b) From what height did the body fall?
1.16 A man travelling west at 4 km/h finds that the wind appears to blow from
the south. On doubling his speed he finds that it appears to blow from the
southwest. Find the magnitude and direction of the wind’s velocity.
1.17 An elevator of height h ascends with constant acceleration a. When it crosses
a platform, it has acquired a velocity u. At this instant a bolt drops from the
top of the elevator. Find the time for the bolt to hit the floor of the elevator.
1.18 A car and a truck are both travelling with a constant speed of 20 m/s. The
car is 10 m behind the truck. The truck driver suddenly applies his brakes,
causing the truck to decelerate at t he constant rate of 2 m/s
2
. Two seconds later
the driver of the car applies his brakes and just manages to avoid a rear-end
collision. Determine the constant rate at which the car decelerated.
1.19 Ship A is 10 km due west of ship B. Ship A is heading directly north at a speed
of 30 km/h, while ship B is heading in a direction 60
◦
west of north at a speed
of 20 km/h.