PYQ Bank

Consider a particle of mass m in two dimensions experiencing a central force \vec{F} = -k\vec{r}, where k is a positive constant and \vec{r} is the radius vector of the particle relative to the force center.

(i) What is the angular momentum \vec{J} of the particle relative to the force center? Show that \vec{J} is conserved.

(ii) Write down the system of equations of motion in two dimensions in polar coordinates. Reduce this system to a one-equation problem and find the equation for the effective potential energy U_{\text{eff}}.

A cardiologist reports to her patient that the radius of the left anterior descending artery of the heart has narrowed by 10\%. What percent increase in the blood pressure is required to maintain the normal blood flow through this artery? Assume that the viscosity of the blood and the length of the artery remain unchanged.

A uniform rod of length L and mass m stands vertically upright on a rough floor and then tips over. What is the rod's angular velocity when it hits the floor?

Consider a symmetric top of mass M, with its tip held fixed, rotating in a gravitational field. Assuming that the origins of the fixed and body coordinate systems coincide, determine the Lagrangian of the top. Are there any angular momenta which are conserved? If yes, find their expressions.

A pion at rest decays into a muon and a neutrino (see the figure) :

Find the velocity of the muon.

The energy of a photon is expressed as E = h\nu, where h is the Planck's constant and \nu is the frequency of the photon. The momentum of the photon is \frac{h\nu}{c}, where c is the speed of light. Show that if a photon scatters from a free electron (of mass m_e), the scattered photon has energy E' = E \left[ 1 + \frac{E}{m_e c^2} (1 - \cos\theta) \right]^{-1} where \theta is the angle through which the photon scatters. Also, show that the electron acquires a kinetic energy T = \frac{E^2}{m_e c^2} \left[ \frac{1 - \cos\theta}{1 + \frac{E}{m_e c^2}(1 - \cos\theta)} \right]

A cube of mass M and side ‘a’ is rotating with angular velocity \omega around one of its edges, which is, say, along the x-axis. Obtain the expressions for its angular momentum and kinetic energy. (Given that the I_{XX} = \frac{2}{3} Ma^2, I_{YX} = -\frac{1}{4} Ma^2 and I_{ZX} = -\frac{1}{4} Ma^2)

Derive the expression for the gravitational self-energy of a uniform solid sphere of mass M and radius R.

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 rest mass 1\text{ kg} and velocity of magnitude 0\cdot 9c collides with a particle of mass 2\text{ kg} at rest. After collision the two particles coalesce and form a single particle of mass M and velocity V. Determine M and V.

Write down the Euler equations for torque-free motion of a rigid body. Solve these equations to find precessing motion for a symmetric top.

A solid shaft of mass M, length l and radius r is to be replaced by a lighter hollow shaft of the same length l and having the same ratings of \tau/\theta, where \tau is the couple and \theta is the angle of twist. Estimate the percentage reduction in mass of the hollow shaft if the outer radius of the shaft is twice the inner radius. Assume the material of the new shaft is same as that of the replaced shaft.

A body moves about a point ‘O’ under no force, the principal moments of inertia at ‘O’ being 3A, 5A and 6A. The components of the initial angular velocity about the principal axes are \omega_1 = n, \omega_2 = 0 and \omega_3 = n. Find the components \omega_1, \omega_2 and \omega_3 for large values of time t.

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).

(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 a large stationary cylinder of inner radius R. A smaller solid cylinder of radius r rolls without slipping inside the larger cylinder. Determine the equation of motion of the smaller cylinder.

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.

A particle of mass m\text{ kg} having an initial velocity V_0 is subjected to a retarding force proportional to its instantaneous velocity. Obtain the expression for the velocity and position of the particle as a function of time.

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).