Department of Physics
Indian Institute of Technology Madras
End of semester examination
PH1010 Physics I 2019 20.11.2019
Time: 9:00 AM to 12:00 Noon Max. Marks: 50
Name Roll No. Batch No. and Teacher’s name
Instructions
1. Begin by completing the information requested above. Please write your complete
name, your roll number, the name of your instructor, and your batch number.
2. This question paper cum answer sheet booklet contains 32 pages. Please check right
away that all the pages are present. You are expected to answer all the questions.
There are no negative marks.
3. Write the answers only in the allotted box. All vectors must be indicated clearly. All
expressions in your answers should be reduced to their simplest mathematical
forms and in terms of the given parameters.
4. You can use the empty pages assigned for rough work. No extra sheets will be
provided.
5. All symbols have their usual meanings unless stated otherwise. All constants are
of appropriate dimensions. Further, assume that unless otherwise stated all
constants are positive.
6. You must use only black or blue ink for writing the answers. Pencil must not be
used even in sketches.
7. Calculators, cell phones or any internet connectable device must not be in your
possession during the examination.
For use by examiners (Do not write in this space)
Q1 Q2 Q3 Q4 Total Marks
Q5 Q6 Q7 Q8 Q9
1
Useful Formulae
(a) Del operator in Cylindrical coordinate system
(b) Divergence and Curl of a vector function in spherical coordinate system
(c) Divergence and Curl of a vector function in cylindrical coordinate systems
(d) Vector Identities
2
1. Indicate whether the following statement is true or false (write True or False in the
box provided.) [1 × 9 = 9 Marks]
𝑥2
(i) The force 𝐹(𝑥) corresponding to the potential energy 𝑈(𝑥) = has two
1+𝑥 4
maxima and two minima.
(ii) If a particle of mass m moves in such a way that its position vector is given by
𝑟⃗⃗ (𝑡) = 𝑥̂ 𝑎 𝑐𝑜𝑠 𝜔𝑡 + 𝑦̂ 𝑏 𝑠𝑖𝑛 𝜔𝑡 + 𝑧̂ 𝑣0 𝑡 (where 𝑎, 𝑏, and 𝑣0 are constants, and
𝑎 ≠ 𝑏), then the resultant trajectory of the particle is an elliptical helix.
(iii) If particle of unit mass moves in a one-dimensional region with potential
𝑥2 𝑥2
energy 𝑈(𝑥) = − , then the time period of small oscillations is 𝑇 = 4𝜋 in relevant
2 3
units.
(iv) The trajectory of a particle moving under the influence of a central force with potential
𝑘
energy 𝑈(𝑟) = is circular (where 𝑘 > 0).
𝑟
⃗ represents a uniform magnetic field, then
(v) If 𝑟⃗⃗ is the position vector and 𝐵
⃗ × (𝐵
∇ ⃗ × 𝑟) = 𝐵
⃗ .
(vi) If S is the surface enclosing the volume V and 𝑛̂ is the outward unit vector normal to
𝑉
the surface S, then ∬ 𝑟 . 𝑛
̂𝑑𝑆 = .
3
⃗ . 𝑣 = 0 for an incompressible fluid flowing with a velocity 𝑣 .
(vii) ∇
(viii) The streamlines associated with the two-dimensional incompressible fluid flow with
𝑥
velocity potential 𝜙 (𝑥, 𝑦) = 𝑘 tan−1 ( ) are hyperbolic.
𝑦
(ix) The streamlines and equipotential lines in two-dimensional incompressible and
irrotational fluid flow are mutually perpendicular to each other.
3
2. Write the final answer in the box provided: Derivation is NOT required. [1 × 11 = 11
Marks]
(i) Sand drops vertically (from a negligible height) at a rate 𝜎 kg/s onto a moving
conveyor belt. A force is applied to the belt in order to keep it moving at a constant
speed 𝑣. What is the work done per unit time?
2 2 2
(ii) What is the directional derivative of 𝑓(𝑥, 𝑦, 𝑧) = 𝑓𝑜 𝑒 −[𝑥 +𝑦 +𝑧 ] at (0, 1, 1) in
1
the direction − (𝑦 ̂ + 𝑧̂ )?
4
(iii) Find the potential energy corresponding to the central force
𝑟
𝑘 𝑟 −
𝐹 (𝑟) = − (1 + 𝛽) 𝑒 𝛽 𝑟̂ , where 𝑘, 𝛽 > 0 .
𝑟2
(iv) A particle of mass m is subjected to a one-dimensional force 𝐹(𝑥) = 𝐹0 (−𝛼𝑥 +
𝛽𝑥 3 ) , where 𝛼, 𝛽 > 0. Find the time period of small oscillations around the stable
equilibrium point.
(v) A critically damped oscillator with natural frequency of oscillation 𝜔0 is found to
have position 𝑥(0) = 1 and speed 𝑣(0) = 0 at time 𝑡 = 0. The expression for its
position at 𝑡 > 0 is given by
4
(vi) A particle of mass m and angular momentum 𝐿 moves in a central force of the form
𝑘
𝐹 (𝑟) = − 𝑟̂ , where 𝑘 > 0 . Find the maximum value of the effective potential
𝑟4
energy.
(vii) A particle of mass m moves in an elliptical orbit under the influence of an inverse
square law attractive central force. If the ratio of the maximum angular speed to the
minimum angular speed of the particle in its orbit is β, then express the eccentricity of
the orbit in terms of β.
(viii) An infinitely large non-conducting plane in the x-y plane has uniform surface charge
density 𝜎. Determine the electric field 𝐸⃗ in the region 𝑧 > 0.
(ix) A current density in spherical polar coordinate system is given by 𝐽(𝑟) =
𝐽0 𝑒 −𝜆𝑟 𝑟̂ in the region 𝑟 > 0. Find the expression for the charge density 𝜌(𝑟, 𝑡) in
the given region.
(x) A two-dimensional incompressible fluid flow has velocity 𝑣 (𝑥, 𝑦) =
𝑥̂𝑦−𝑦̂𝑥
𝑘( ). Find the stream function associated with this velocity.
𝑥 2 +𝑦 2
−𝑥̂𝑦+𝑦̂𝑥
(xi) A fluid is flowing in the x-y plane with a velocity 𝑣 (𝑥, 𝑦) = 𝑘 ( ), where
√𝑥 2 +𝑦 2
k is a constant. Determine the vorticity corresponding to the given velocity.
5
3. An electron of mass 𝑚 and charge – 𝑞 is moving under the influence of a uniform
⃗ = 𝐸0 𝑦̂ and a uniform magnetic field, 𝐵
electric field, 𝐸 ⃗ = 𝐵0 𝑧̂ . The initial
position of the electron is the origin (0, 0,0) and the initial velocity is 𝑣 = 𝑣0 𝑥̂.
(i) Write the equations of motion. (ii) Determine the trajectory 𝑟 (𝑡) of the electron
at time 𝑡 > 0. [4 Marks]
6
7
8
4. Consider a particle of unit mass and total energy 𝐸 moving in a one dimensional
𝑈0
region with potential energy given by 𝑈(𝑥) = − , where U0 > 0.
𝑒 𝑥 +𝑒 −𝑥
(i) Sketch U(x) versus x (in the range −∞ < 𝑥 < ∞), suitably marking the values of
the extrema.
(ii) Determine the angular frequency ω of small oscillations about the point of stable
equilibrium.
(iii) Sketch the corresponding phase trajectories for energies, E < 0, E = 0, and E > 0.
Indicate the direction of motion with arrows. [5 Marks]
9
10
11
5. An undamped driven oscillator of mass 𝑚 and natural frequency of oscillation 𝜔0
is subjected to an external force 𝐹(𝑡) = F𝑜 cos 2 ωt , where F𝑜 and ω (ω ≠ 𝜔0 ) are
constants. The oscillator satisfies the following equation of motion
𝑚𝑥̈ + 𝑚𝜔20 𝑥 = 𝐹(𝑡)
(i) Write down the general solution 𝑥ℎ (𝑡) to the corresponding homogeneous
differential equation.
(ii) Find the particular solution 𝑥𝑝 (𝑡) to the above equation of motion.
(iii) Determine the complete solution 𝑥(t) to the equation of motion using the initial
conditions 𝑥 (t = 0) = 0, 𝑥̇ (t = 0) = 0. [5 Marks ]
12
13
14
6. A particle of mass 𝑚 moves under the influence of gravity in a circular orbit of radius
R and a time period T. At a certain instant, the angular momentum of the particle is
brought to zero. Find the time required for it to fall into the centre of the force. [4 Marks]
15
16
17
7. If S is the surface enclosing a volume V, then Gauss’s divergence
theorem states that ∭ ∇⃗ . 𝐴𝑑𝜏 = ∬ 𝐴. 𝑛̂𝑑𝑆, where 𝑛̂ is the outward unit
vector normal to the surface S.
Verify the divergence theorem by evaluating the volume and surface
integrals for the vector function
𝐴 (𝑟, 𝜃, 𝜑) = 𝑟̂ (𝑟 𝑐𝑜𝑠𝜃) + 𝜃̂ (𝑟 𝑠𝑖𝑛𝜃) + 𝜑̂ (𝑟 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜑),
using as the volume, the inverted hemispherical bowl of radius R, resting on the x-y plane
and centred at the origin as shown in the adjacent figure. [4 Marks]
18
19
20
8. (i) Using Gauss’s divergence theorem, prove that ∬ 𝑛 ⃗ 𝑑𝑆 = ∭ ∇
̂×𝐵 ⃗ ×𝐵
⃗ 𝑑𝜏.
(ii) A rigid body of volume V and surface S rotates with angular velocity 𝜔
⃗.
1
Show that 𝜔
⃗ = ⃗⃗⃗⃗ × 𝑣, where 𝑣 = 𝜔
∬ 𝑑𝑆 ⃗ × 𝑟 is the velocity of the point 𝑟 on the
2𝑉
surface S.
(Hint. Take 𝐴 = 𝐵⃗ × 𝐶 , where 𝐶 is a constant vector. Select the relevant vector identities
from those provided in page 2) [ 4 Marks]
21
22
23
𝑎𝑥−𝑏𝑦 𝑎𝑦 + 𝑏𝑥
9. A force is given by 𝐹 = 𝑥̂ ( ) + 𝑦̂ ( ), where a and b are constants.
√𝑥 2 +𝑦 2 √𝑥 2 +𝑦 2
(i) Verify whether the force is conservative or not.
(ii) Evaluate the work done by the force in moving a particle around a circle of
radius R, centred at the origin in the x-y plane.
[4 Marks]
24
25
26
Rough work
27
Rough work
28
Rough work
29
Rough work
30
Rough work
31
Rough work
32
