(i) Derive the expressions for gravitational potentials at a point (I) outside the spherical shell, (II) inside the spherical shell. (ii) Calculate the escape velocity of a body of mass 10\text{ kg} from the surface of Moon \left( g_{\text{Moon}} = \frac{1}{6} g_{\text{Earth}} \right). Mass of Moon = 7\cdot3 \times 10^{22}\text{ kg} Radius of Moon = 1\cdot7 \times 10^6\text{ m}
Calculate the gravitational self-energy of the Earth. Given : Mass of Earth M_e = 6 \times 10^{24}\text{ kg} and the Radius of Earth R_e = 6\cdot4 \times 10^6\text{ m}
A hoop is rolling down on an inclined plane without slipping. Find its velocity at the bottom of the inclined plane.
(i) The quantities of rotatory motion are analogous to those of translatory motion. Write the corresponding equations of translatory and rotatory motion. (ii) Describe the theorems of perpendicular and parallel axes in case of a plane lamina.
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.
In a reference frame S, an event 1 occurs at the origin at t = 0 and another event 2 occurs at x = 4000\text{ m} and time t = 5 \times 10^{-6}\text{ s}. Find the time interval between the events as registered by clock in a frame S' moving with speed v = 0 \cdot 6 \, c relative to S along the common X-X' axis, the origin coinciding at t = t' = 0\,.
Why can Electron-Positron pair production through a high energy photon not take place in vacuum?
What are the coordinates of the centre of mass of the system of masses shown in the figure?
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.
Define streamline flow of a fluid. Using the equation of continuity for an isotropic fluid, find different components of total energy per unit volume.
(i) Prove that the separation of two colliding particles is same, when observed in centre of mass and laboratory systems. (ii) Determine the kinetic energy of a thin disc of mass 0\cdot5\text{ kg} and radius 0\cdot2\text{ m} rotating with 100 rotations per second around the axis passing through its centre and perpendicular to its plane.
Calculate the velocity of a particle having kinetic energy four times the rest mass energy.
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.
Consider a particle of mass m moving in a potential field of the form \text{V}(\vec{\text{r}}) = \text{V}_0(\text{x}^2 + \text{y}^2), where \text{V}_0 is a constant. What are the conserved physical quantities for the particle ?
Show that for very small velocity, the equation for kinetic energy, K=\Delta mc^2 becomes K=\dfrac{1}{2}m_0v^2, where notations have their usual meanings.
A particle P of mass m_1 collides with another particle Q of mass m_2 at rest. The particles P and Q travel at angles \theta and \phi, respectively, with respect to the initial direction of P. Derive the expression for the maximum value of \theta.
Define moment of inertia and radius of gyration of a body of mass M rotating about an axis. State and prove Parallel Axis theorem on moment of inertia.
Two spaceships are approaching each other as depicted below.
Spaceship 1 is moving with a velocity \vec{\text{u}}_1 = 0\cdot 7\text{c }\hat{\text{x}}, while spaceship 2 is moving with a velocity \vec{\text{u}}_2 = -\,0\cdot 8\text{c }\hat{\text{x}} as measured by the stationary observer on Earth. Here, c is the velocity of light in vacuum and \hat{\text{x}} is the unit vector along the x-axis.
(i) What, according to Galilean addition of velocity, is the speed of spaceship 2 with respect to spaceship 1 (in terms of c) ?
(ii) What, according to relativistic addition of velocities, is the speed of spaceship 2 with respect to spaceship 1 (in terms of c) ?
(iii) Spaceship 1 emits a light signal towards spaceship 2. The observer on spaceship 2 measures the frequency of the light signal to be 10^{16}\text{ Hz}. What was the original frequency of the light signal ?
A particle of rest mass \text{m}_0 is moving in a stationary frame K with a velocity \vec{\text{u}} making an angle \theta with the x-axis as shown in the figure.
Find the corresponding angle \theta' made by the velocity vector of the particle with the \text{x}' axis in the frame \text{K}' that is moving with respect to K along the x-direction at a constant speed V.
Two spaceships approach each other, both moving with same speed as measured by a stationary observer on the Earth. Their relative speed is 0.7c. Determine the velocity of each spaceship as measured by the stationary observer on the Earth.
