Four solid spheres A, B, C, and D each of mass m and radius a, are placed with their centres on the four corners of square of side b as shown in the figure below:
Calculate the moment of inertia of the system about one side of the square, Also, calculate the moment of inertia of the system about a diagonal of the square.
Show that the Young's modulus Y, modulus of rigidity n and Poisson's ratio \sigma are related by the equation Y = 2n(1 + \sigma).
A horizontal pipe of non-uniform bore has water flowing through it such that the velocity of flow is 40\ \mathrm{cm/s} at a point where the pressure is 2\ \mathrm{cm} of mercury column. What is the pressure at a point where the velocity of flow is 60\ \mathrm{cm/s}? (Take, g = 980\ \mathrm{cm/s^2} and density of water = 1\ \mathrm{g/c.c.})
A force field is given by \vec{F} = (2xy + z^3)\hat{i} + x^2\hat{j} + 3xz^2\hat{k} Is it a conservative field ? If so, what is the scalar potential ?
(i) The distance between the centres of the carbon and oxygen atoms in the carbon monoxide (CO) gas molecule is 1.130\times10^{-10}\ \mathrm{m}. Locate the centre of mass of the molecule relative to the carbon atom.
(ii) Find the centre of mass of a homogeneous semicircular plate of radius a.
A student is working in a physics laboratory, which is at temperature 27^{\circ}\mathrm{C}, on a sonometer to study formation of stationary waves. The cross-sectional area of the sonometer wire is 0.85\times10^{-6}\ \mathrm{m^2} and a tension of 20\ \mathrm{N} is applied on it. If the rigid supports are 1.2\ \mathrm{m} apart and the temperature of the wire drops by 7^{\circ}\mathrm{C}, calculate the (i) final tension and (ii) fundamental frequency of vibration of the wire. Take, coefficient of linear expansion and isothermal Young's modulus as 1.5 \times 10^{-5}\ \mathrm{K}^{-1} and 2.0 \times 10^{11}\ \mathrm{N\,m}^{-2} respectively.
A body moving in an inverse square attractive field traverses on elliptical orbit with eccentricity e and period \gamma. Find the time taken by the body to traverse the half of the orbit that is nearer the centre of force. Explain briefly why a comet spends only 18\% of its time on the half of its orbit that is nearer the sun.
What are cyclic coordinates for a system of particles ? Discuss the significance of the Hamilton's principle. Show how Lagrange's equations can be derived from Hamilton's principle.
Derive the expression for Coriolis force and show that this force is perpendicular to the velocity and to the axis of rotation. What is the nature of this force?
Using Euler's equations for a force free motion of a rigid body, show that the Kinetic energy remains constant throughout the motion of the rigid body.
Write down precisely the conservation theorems for energy, linear momentum, and angular momentum of a particle with their mathematical forms.
Show that the rest mass energy of an electron is 0.51\,\mathrm{MeV}. (use the standard values of the physical parameters).
Prove mathematically that the addition of any velocity of a particle to the velocity of light in free space merely reproduces the velocity of light in free space only.
How does one obtain the angular velocity of the Earth about the North Pole with respect to a fixed star as 7.292 \times 10^{-5}\,\mathrm{sec}^{-1}? Explain your method of calculating the above value.
Show that the moment of inertia of a circular disc of mass M and radius R about an axis passing through its centre and perpendicular to its plane is \frac{1}{2}MR^2.
A particle of mass m is moving with the velocity \vec{v} on the surface of the earth. Discuss the nature of its motion in both the hemispheres due to the Coriolis force \vec{F} = -2\,m\,\vec{\omega} \times \vec{v}.
Show that the differential scattering cross-section can be expressed as \sigma(\theta)=\frac{s}{\sin\theta}\left|\frac{ds}{d\theta}\right|, where s is the impact parameter and \theta is the scattering angle.
Using Poiseuille's formula, show that the volume of a liquid of viscosity coefficient \eta passing per second through a series of two capillary tubes of lengths l_1 and l_2 having radii r_1 and r_2 is obtained as Q=\frac{\pi p}{8\eta\left[\frac{l_1}{r_1^4}+\frac{l_2}{r_2^4}\right]}. where p is the effective pressure difference across the series.
Discuss that the motion of a particle in a central field gets restricted to a plane provided its angular momentum is nonzero. What will happen if the angular momentum is zero?
Solve Euler's equation of motion for a rigid body to show that the torque-free motion of a spherical top is a uniform rotation about an axis fixed in space.
