Ekksrhyky usg: jk"Vªh; ÁkS|ksfxdh laLFkku bykgkckn, bykgkckn &„ƒƒŒŒ† Hkkjr
ÒkSfrdh foÒkx
Motilal Nehru National Institute of Technology Allahabad – 211004 INDIA
Department of Physics
Tutorial Sheet (NK) 2017-18
Tutorial sheet on STR - I
1. Using Galilean transformation, show that the distance between two points in our 3-dimensional space, is independent
of the frame of reference.
2. Calculate the length and orientation of a rod of length 5 meter in a frame of reference which is moving with a velocity
0.6c in a direction making an angle of 30 ° with the rod.
3. A spaceship has a painting on its body : it comprises of a circle with a line AB across it at
45° to the vertical line PQ; (see the adjacent figure). As the spaceship shoots past another
ship in space, with a relative velocity ~v along the horizontal line MN and |~v| = 0.95c,
the observer K sitting in the second ship notices the painting. What angle does the observed
line AB make to the vertical in K’s frame?
4. Following images are observed in a frame at rest. How these images will be observed by an observer n in a frame
moving with the velocity v along x-direction (from left to right)? Discuss it in detail.
Lo Lo 2Lo
Lo Lo Lo
Lo
5. Derive and expression for Lorentz length contraction. A vector in a frame S’ is represented by 5iˆ − 8 ˆj + 9kˆ . How
can this vector be represented in other frame S while S’ is moving with velocity 0.8ciˆ with respect to S. (c=3x108
m/s)
6. A circular ring of radius ‘a’ is at rest in the x'y' plane of an inertial reference frame S' moving at constant speed v along
x-direction with respect to another inertial reference frame S
(i) show that the measurement made in frame S will indicate the ring elliptical in shape.
(ii) compute the eccentricity of this ellipse in frame S
7. A rigid rod of length L makes an angle θ with the X-axis of the system in which it is at rest in the X-Y plane. Show
that for an observer moving with respect to the rod with speed v along the positive X-direction; the apparent Length
L’ and the angle θ ’ are given by
Γ where; Γ 1
8. Why was the concept of ether introduced? Why do you think the concept was absurd? What is the significance of the
null result of Michelson-Morley experiment?
9. Why one should observe a fringe shift (theoretically) in Michelson-Morley experiment? How much be the fringe shift?
10. What are the outcomes of Michelson-Morley Experiment? Determine the volume of a cube having proper length, Lo
and moving relativistically with speed v along one of its edges.
11. Adjacent image (a circle of diameter Lo on top of parallelogram of side Lo) is observed in a frame
at rest. How this figure will be observed by an observer in a frame moving with the velocity v
along x-direction (from left to right)? Find out fundamental parameters of the image observed.
12. A circular plate moves with its plane parallel to the X-Y plane of a reference frame S at rest. Assuming its motion to be
along the axis of X, calculate the velocity at which its surface area would appear to be reduced to half to an observer in
frame S at rest.
13. A particle moves with velocity represented by a vector 3"̂
4%̂
12'( m/s in frame S’. Find the velocity of the
particle in frame Sif S’ moves with velocity 0.8c relative to S along positive X- direction.
Ekksrhyky usg: jk"Vªh; ÁkS|ksfxdh laLFkku bykgkckn, bykgkckn &„ƒƒŒŒ† Hkkjr
ÒkSfrdh foÒkx
Motilal Nehru National Institute of Technology Allahabad – 211004 INDIA
Department of Physics
Tutorial Sheet (NK) 2017-18
Tutorial sheet on KTG
1. Estimate the mean free path of an air molecule at 273K and 1 atm, assuming it to be a sphere of diameter 4× 10−10 m.
Estimate the mean time between collisions for an air molecule under these conditions, assuming the fact the air
molecules are roughly comprised of Nitrogen molecules and speed of the molecules = vrms.
2. (a) Let us consider a room of volume V , is filled with ideal gas at certain temperature, T and presure P. Let the
velocity distribution of the gas molecules obey the Maxwell-Boltzmann distribution function. If vrms, )̅ and vmp
denote the rms velocity, average velocity and most probable velocity, respectively, show that vrms : )̅ : vmp = 1.225
,,,,
:1.128 : 1. (b) For a gas obeying Maxwell-Boltzmann distribution law, show that )̅ . 4-
3. Let at temperature T0 and pressure P0, the rms speed of the molecules of certain gas is vrms. Find the speed, if
(a) the temperature is raised from 200C to 3000C;
(b) the pressure is doubled and T = T0;
(c) the molecular weight of each of the gas molecules is tripled at T = T0.
4. A beam of particles, each of mass mo and speed v, is directed along the x-axis. The beam strikes an area 1mm2 with 1×
1015 particles striking per second. Find the pressure on the area due to the beam if the particles stick to the area when
they hit. Evaluate for an electron beam in a television tube where m0 = 9.1× 10−31 kg and v = 8×107 m/s.
5. At what temperature the rms speed of gaseous hydrogen molecules (molecular weight =2) equal to that of oxygen
molecules(molecular weight = 32) at 470C?
6. Hydrogen and Oxygen are maintained under identical conditions of temperature and pressure. Calculate the ratio of
their coefficient of viscosity if diameters of these molecules are 2.5 Å and 3.5 Å.
7. The coefficient of viscosity of oxygen molecule at 150C is 196 µ-poise. Calculate the diameter of a molecule of this gas,
given R = 8.4 J mol−1 K−1 and molecular weight of oxygen = 32.
8. What are transport phenomena in gases? On the basis of kinetic theory of gases, deduce the expression for coefficient
of viscosity of a gas and discuss its dependence on temperature and pressure.
9. Show that the mean free path of the molecules in an ideal gas is
kT
λ=
2πd 2 p , where d is the diameter of a gas molecule, T and p are respectively temperature and pressure of the gas
and k is Boltzmann’s constant.
10. Discuss Maxwell’s law of distribution of velocities for gas molecules. Deduce expressions for the average speed, root
mean square speed and most probable speed of the gas molecules according
11. On the basis of kinetic theory of gases, derive the expression for coefficient of viscosity and thermal conductivity of a
gas. Hence obtain the relation between the two.
12. Discuss Maxwell’s distribution law of velocities for gas molecules and find the average speed, root mean square speed
and most probable speed on its basis
13. What is meant by mean free path and transport phenomena of a gas? Derive an expression for the coefficient of
thermal conductivity of a gas.
14. Calculate the radius of an oxygen molecule if its coefficient of thermal conductivity, K=2.4×10-2 J/m-s-K at 0°C and
Cv=2.09×104J/kgmole-K. Given Boltzmann’s constant kB =1.38×10-23 JK-1, Avogadro’s number N=6.023×1026 kg-
mole-1 and mass of oxygen molecule m = 5.31×10-26 kg.
15. Starting from the Maxwell’s velocity distribution function, obtain an expression for the energy distribution function for
translational kinetic energy E.
16. A gas of N molecules has the hypothetical speed distribution shown in figure [Note that N(v)=0 for v>vo]
i. Express a in terms of N and vo
ii. What fraction of the molecules has a speed between 1.5 vo and 2.0 vo?
N(v)
iii. Express the average speed of the molecules in terms of vo. a
iv. Find vrms.
0 vo 2vo
v
