Imagine that a rigid body is rotating about a fixed point with angular velocity \vec{\omega}. Assuming that the coordinate axes coincide with the principal axes, if T stands for kinetic energy and G for external torque acting on the body, show that \frac{dT}{dt}=\vec{G}\cdot\vec{\omega}

Determine the number of degrees of freedom for a rigid body-

(i) moving freely in space of three dimensions;

(ii) having one point fixed;

(iii) having two points fixed.

With an appropriate diagram, deduce the velocity profile for streamline flow of a liquid through a capillary of circular cross-section. Deduce also the fraction of liquid which flows through the section up to a distance a/2 from the axis, where a is the radius of the capillary.

Prove that the time taken by the earth to travel over half of its orbit separated by the minor axis remote from the sun is two days more than half a year. Given, the period of the earth is 365 days and eccentricity of the orbit =1/60.

With an appropriate diagram, show that in the Rutherford scattering, the orbit of the particle is a hyperbola. Obtain an expression for impact parameter.

A uniform solid sphere of radius R having moment of inertia I about its diameter is melted to form a uniform disc of thickness t and radius r. The moment of inertia of the disc about an axis passing through its edge and perpendicular to the plane is also equal to I. Show that the radius r of the disc is given by r=\dfrac{2R}{\sqrt{15}}.

What are Eulerian angles? A body with rotational symmetry about an axis is rotating under gravity about a point on the axis without friction. What are the quantities remaining constant during the motion? Find them in terms of suitable Eulerian angles. Explain 'precession' and 'nutation' of such a body.

When a sphere of radius r falls down a homogeneous viscous fluid of unlimited extent with the terminal velocity v, the retarding viscous force acting on the sphere depends on the coefficient of viscosity \eta, the radius r and its velocity v. Show how Stokes' law was arrived at connecting these quantities from the dimensional considerations.

What is the significance of the null result of Michelson-Morley experiment? Does it disprove the existence of ether? Justify.

A planet revolves around the Sun in an elliptic orbit of eccentricity e. If T is the time period of the planet, find the time spent by the planet between the ends of the minor axis close to the Sun.

Show that for any rigid body consisting of at least three particles, not arranged in one straight line, number of independent degrees of freedom is six. Define Euler's angles \theta, \phi and \psi to describe the configuration of such a rigid body. Consider two frames of reference, one fixed to the body and the other to the space defined as S' = (x', y', z') and S = (x, y, z) respectively. Show that the angular momentum (\vec{L}) of the rigid body in the two frames are related by \left( \frac{d\vec{L}}{dt} \right)_S = \left( \frac{d\vec{L}}{dt} \right)_{S'} + \vec{\omega} \times \vec{L} where \vec{\omega} is the angular velocity of rotation.

Show that a four-dimensional volume element dx\,dy\,dz\,dt is invariant to Lorentz transformation.

Obtain the relativistic equation for aberration of light using velocity transformation equations.

Show that the total energy per unit mass of liquid flowing from one point to another without any friction remains constant throughout the displacement.

Obtain Poiseuille's equation for a viscous fluid flowing through a narrow tube of radius r and length l. If a spherical body of radius a is allowed to move at a speed \vec{V} through the same fluid of viscosity \eta, show that the viscous force will increase with the speed linearly.

Consider a spherical shell of mass M and radius R. Calculate the potential due to this shell at a point P when the point is (i) outside the shell and (ii) inside the shell (r < R). If the spherical shell is now replaced by a uniform solid sphere of same mass and radius, what will be its potential at the same external point?

Two bodies of masses M_1 and M_2 are placed at a distance d apart. Show that at this position where the gravitational field due to them is zero, the potential is given by V = -\frac{G}{d} (M_1 + M_2 + 2\sqrt{M_1 M_2})

A meson of rest mass $ \pi $ comes to rest and disintegrates into a muon of rest mass $ \mu $ and a neutrino of zero rest mass. Show that the kinetic energy of motion of the muon is T = \frac{(\pi - \mu)^2 c^2}{2\pi} .

Show that the Bulk modulus K, Young's modu- lus Y and Poisson's ratio $ \sigma $ are connected by the relation K = \frac{Y}{3(1 - 2\sigma)}.

A force field is given by \bar{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 ?