PYQ Bank

Calculate the moment of inertia of a solid cone of mass M, height h, vertical half-angle \alpha and radius of its base R, about an axis passing through its vertex and parallel to its base.

Two identical relativistic particles of rest mass m and kinetic energy T collide head-on. What is the relative kinetic energy, i.e., the kinetic energy T' of one in the rest frame of the other?

A man has a mass of 100\text{ kg} on earth. When he is on the space craft an observer from the earth registers his mass as 102\text{ kg}. Determine the speed of the space craft.

A rigid body is spinning with an angular velocity of 4\ \mathrm{rad\,s^{-1}} about an axis parallel to the direction (4\hat{\jmath}-3\hat{k}) passing through the point A with \overrightarrow{OA}=2\hat{\imath}+3\hat{\jmath}-\hat{k}, where O is the origin of the coordinate system. Find the magnitude and direction of the linear velocity of the body at point P with \overrightarrow{OP}=4\hat{\imath}-2\hat{\jmath}+\hat{k}.

A particle is moving in a central force field on an orbit given by r = ke^{\alpha\theta}, where k and \alpha are positive constants, r is the radial distance and \theta is the polar angle.

(i) Find the force law for the central force field.

(ii) Find \theta(t).

(iii) Find the total energy.

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 particle of mass 'm' moves according to the equations x = a \cos \omega t, y = a \sin \omega t, z = c, where a, c, \omega are constants. Obtain the instantaneous velocity and linear momentum vectors in terms of the Cartesian components and hence the angular momentum. Find the force \vec{F} and the torque \vec{N} acting on the particle and verify that the angular momentum \vec{L} and the torque \vec{N} satisfy the relation \frac{d\vec{L}}{dt} = \vec{N}.

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

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}

On the basis of three principal moments of inertia I_A, I_B and I_C each about X, Y and Z axes respectively, how can you classify molecules?

A rigid body rotates with the angular velocity '\omega' about an axis through the origin O and having direction cosines l, m, n. Show that the moment of inertia of the rigid body about the axis is I = I_{xx} l^2 + I_{yy} m^2 + I_{zz} n^2 + 2 I_{xy} l . m + 2 I_{yz} m . n + 2 I_{zx} n . l where the symbols have their usual meanings.

The principal moments of inertia of a body at a point are given as 200, 300 and 450\text{ gm.cm}^2. Write down the equation of the ellipsoid of inertia at that point.

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.

A spaceship measures 50 m in length on ground and it measured a length of 49.7 m in space as observed from the ground. Find out the speed of the spaceship.

A particle of rest mass 'M' moving at a velocity 'u' collides with a stationary particle of rest mass 'm'. If the particles stick together, show that the speed of the composite ball is equal to v = \frac{u \gamma M}{(\gamma M + m)}, \quad \text{where } \gamma = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}}

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.

Establish the relation between the angular momentum and torque of a particle. Show that this relation leads one to the principle of conservation of angular momentum.

How can one introduce the constraints of motion through the concept of generalized coordinate systems? Write down the set of transformation equations for a system of N particles relating the generalized coordinates with the real coordinates.

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}}.