(i) Define cyclic coordinates and find their connections with the symmetries of the system.
(ii) A system with two degrees of freedom is described by the Lagrangian \text{L} = \frac{1}{2}\text{m}_1 \dot{\text{q}}_1^2 + \frac{1}{2}\text{m}_2 \text{q}_1^2 \dot{\text{q}}_2^2 - \frac{\alpha}{\text{q}_1}, \alpha is constant. Find the cyclic coordinates, conserved quantities and symmetries for this system, if any.
Consider two inertial frames \text{S} and \text{S}'. \text{S}' is moving along \text{y}-direction with a constant speed \text{v}_0. Find how the first order partial derivatives with respect to \text{x}, \text{y}, \text{z} and \text{ct} are connected in these two inertial frames. Show that the D'Alembertian, \nabla^2 - \frac{1}{\text{c}^2}\frac{\partial^2}{\partial \text{t}^2} is invariant under the Lorentz transformation.
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A particle of mass \text{m} moves under the attractive central force \text{F} = -\frac{\text{C}}{\text{r}^{\text{n}+1}}. Find the condition for which the particle will have stable circular orbit. (\text{C} > 0, a constant).
Write down the Euler equations for torque-free motion of a rigid body. Solve these equations to find precessing motion for a symmetric top.
Two particles each of rest mass 1\text{ gm} collide head-on with the speed 0\cdot 8\text{c} to stick together at rest. Find the mass of the final composite.
Construct the Lagrangian of a simple pendulum whose string is replaced by a spring with spring constant k and rest length l_0 oscillating in a vertical x - y plane. Find the equations of motion.
The Lagrangian of a system is given by L (\vec{r}, \vec{v}) = \frac{1}{2} m (\vec{v} + \vec{a})^2 + \vec{b} \cdot \vec{v} - \vec{c} \cdot \vec{r} where, \vec{a}, \vec{b} and \vec{c} are constant vectors. Construct the Hamiltonian of the system and find the canonical equations of motion.
Define Coriolis force. A particle is dropped from a height h vertically at the northern hemisphere at latitude \theta. Find its deflection from the vertical line due to Coriolis force.
Find the equation of orbit for a particle of mass m, moving in the influence of central force F(r) in terms of u \left(\equiv \frac{1}{r}\right).
Consider two inertial frames S and S'. S is at rest and S' is moving along x - x' direction with constant speed v. (i) Find how different components of momentum four-vector in S and S' frames are related. (ii) Show that p^\mu p_\mu is Lorentz invariant.
The moment of inertia tensor of a cube of mass M and side a is given by the matrix I = \frac{M a^2}{12} \begin{bmatrix} 8 & -3 & -3 \\ -3 & 8 & -3 \\ -3 & -3 & 8 \end{bmatrix}. Calculate the principal moments of inertia of the cube.
Define generalized coordinates. How many generalized coordinates are required to describe the dynamics of (i) a free rigid body in 3 dimension ? (ii) a solid cylinder rolling in an inclined plane without slipping ?
Write down the expression for kinetic energy of a relativistic particle with rest mass m_0, moving with a speed v. (i) Show that the kinetic energy reduces to \frac{1}{2} m_0 v^2 in the non-relativistic limit. (ii) Find the first relativistic correction to the non-relativistic kinetic energy.
What are the coordinates of the centre of mass of the system of masses shown in the figure?
A particle of mass m moves under the action of a central force whose potential is V(r) = Kmr^4 (K > 0). Calculate the kinetic energy for which the orbit will be circle of radius R, about the origin.
Why can Electron-Positron pair production through a high energy photon not take place in vacuum?
A particle of mass m moves under the action of a central force whose potential is V(r) = Kmr^4 (K > 0). Calculate the kinetic energy for which the orbit will be circle of radius R, about the origin.
Calculate the inertia tensor for a rigid body consisting of three particles of masses 3\text{ gm}, 1\text{ gm}, 2\text{ gm} located at (1, -1, 2)\text{ cm}, (1, 0, 2)\text{ cm}, (-2, 1, 0)\text{ cm} respectively.
A hoop is rolling down on an inclined plane without slipping. Find its velocity at the bottom of the inclined plane.