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.
A small block of mass m slides without friction down a wedge-shaped block of mass M and opening angle \alpha as depicted below.
The wedge-shaped block itself slides along the horizontal frictionless floor along the +\text{ve x-direction}. Find the horizontal acceleration \ddot{\text{X}} = \text{d}^2\text{X}/\text{dt}^2 of the wedge-shaped block by the Lagrangian method and using the displacements \text{X} and \text{S} of the blocks as generalized coordinates.
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 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.
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.
A light rod of length 100\,\mathrm{cm} is suspended from the ceiling, horizontally by means of two vertical wires of equal length tied to its ends. One of the wires is made of steel and its cross-section is 0.05\,\mathrm{sq.\ cm} and the other is of brass of cross-section 0.1\,\mathrm{sq.\ cm}. Find the position along the rod at which a weight may be hung to produce
(i) Equal stresses in both the wires,
(ii) Equal strain in both the wires. Young's modulus of elasticity of brass and steel are 1.0 \times 10^{11}\,\mathrm{N/m^2} and 2.0 \times 10^{11}\,\mathrm{N/m^2} respectively.
An observer detects two explosions, one that occurs near him at a certain time and another that occurs 2\,\mathrm{ms} later 100\,\mathrm{km} away. Another observer finds that the two explosions occur at the same place. What time interval separates the explosions to the second observer?
A capillary tube having 1.0\,\mathrm{mm} diameter, 20\,\mathrm{cm} in length is fitted horizontally to a vessel in which alcohol is kept fully up to the neck. Density of alcohol is 8 \times 10^{2}\,\mathrm{kg/m^3}. The depth of the centre of the capillary tube below the surface of alcohol is 40\,\mathrm{cm}. Find the amount of alcohol that will flow out of the capillary tube in 10 minutes. Coefficient of viscosity of alcohol is 0.0012\,\mathrm{N\,s/m^2}.
(i) Calculate the mass and momentum of a proton of rest mass 1.67\times10^{-27}\,\mathrm{kg} moving with a velocity of 0.8c, where c is the velocity of light. If it collides and sticks to a stationary nucleus of mass 5\cdot0\times10^{-26}\,\mathrm{kg}, find the velocity of the resultant particle.
(ii) Calculate the mass of the particle whose kinetic energy is half of its total energy. Find the velocity with which the particle is travelling.
A body of mass m at rest splits into two masses m_1 and m_2 by an explosion. After the split the bodies move with a total kinetic energy T in opposite direction. Show that their relative speed is \sqrt{\frac{2Tm}{m_1m_2}}.
The radius of the Earth is 6.4 \times 10^{6}\ \mathrm{m}, its mean density is 5.5 \times 10^{3}\ \mathrm{kg/m^{3}} and the universal gravitational constant is 6.66 \times 10^{-1}\ \mathrm{Nm^{2}/kg^{2}}. Calculate the gravitational potential on the surface of the Earth.
A particle moving in a central force field describes the path r=ke^{\alpha\theta}, where k and \alpha are constants. If the mass of the particle is m, find the law of force.
An electron is moving under the influence of a point nucleus of atomic number Z. Show that the orbit of the electron is an ellipse.
On a space-time diagram after the Lorentz transformation with given v we get new tilted axis ct' and x'. For a point on the ct' axis which is at the (geometrical) distance \delta on the diagram measured from the origin, find the values of t' and t.
A comet of mass m moves towards the Sun with initial velocity v_0. The mass of the sun is M and its radius is R. Find the total cross-section \sigma for striking the Sun. Take the Sun to be at rest and ignore all other bodies.
A block of mass m is sliding on a wedge of mass M as shown in the figure below. The wedge can slide on the horizontal table. Find the equation of motion.
(i) Define the angular velocity \omega of a rigid body rotating about some axis.
(ii) Then derive the relation \vec{v} = \vec{\omega} \times \vec{r} for the velocity of a point in the body with position vector \vec{r} relative to an origin on the axis. Draw a diagram explaining all the vectors.
During take-off, an aircraft accelerates horizontally in a straight line at a rate A. A small bob of mass m is suspended on a string attached to the roof of the cabin, and a hydrogen balloon (total mass m) is tethered to the floor by a string. For each, determine the tension in the string and the equilibrium angle \theta between string and vertical. Draw a neat diagram to explain your answer.
A rigid body is rotating under the influence of an external torque (N) acting on it. If T is the kinetic energy and \omega is the angular velocity, show that \dfrac{dT}{dt} = N \cdot \omega in the principal axes system.
A homogeneous right triangular pyramid with the base side a and height \frac{3a}{2} is shown below. Obtain the moment of inertia tensor of the pyramid:
