Test: Combined (CM+MP) - Physics MCQ

The Lagrangian of a particle is 

where k is a positive constant, q₁ and q₂ in generalised coordinates. The equation of motion are

Let (p.q) and (P, Q) be two pairs of canonical variables. The transformation P -q cot βp and  is canonical for β = nα. The value of n is_______    

The mass of a particle is measured to be √2 times of its rest mass. The ratio of kinetic energy and total energy is ____________ (up to two decimal places)

The percentage contraction of a rod moving with a velocity 0.8 c in a direction inclined at 60° to its own length ______% (up to one decimal place)

The Hamiltonian of a particle with mass m =1/2  units moving along x-axis inside a potential V(x) = e^x is given by H = p² + e^x Assume p> 0. The phase space trajectory of the particle will be

The Lagrangian for the case when the Hamiltonian is

In a central force field, the trajecting (in plane polar co-oridantes) of a particle is given by  where m is the mass of the particle. L is angular momentiun and e is the eccentricity of the particle's motion. Whcih one of the following conditions given rise to circular trajectory?

The minimum value of coefficient of friction for an inclined plane of angle θ = 45⁰ in order that a hoop will roll down without slipping is______(Upto one decimal place)

A distant galaxy is observed to have its hydrogen- β line shifted to a wavelength of 580 nm, away from the laboratory value of 434 mn. The approximate velocity of recession of the distant galaxy is

Two balls, of mass m and mass 2m. approach from perpendicular directions with identical speeds and collide. After the collision, the more massive ball moves with the same speed v but downward, perpendicular to its original direction. The less massive ball moves with speed U at an angle θ with respect to the horizontal. If no external forces act during the collision then the final speed of less massive ball is αv. The value of α is_______ (up to 1 decimal place)

The orthogonal trajectories to the curve x² + (y-1)² = 1 are a family of

Consider a charged sphere of radius R with charge density f(r ) = e^-r. The gradient vector to the family of equipotential surfaces of this charged surface points

The value of the integral  over a contour of unit circle centered at origin. (Taken in the anticlockwise sense) is_______(Answer should be an integer)

Given the value of H₅(0 ) is  _________ (Answer should be an integer)

The Gaimna function is defined by the integral  will be 

Consider a metric tensor   Then det (g_ij) = _____

Which one of the following combinations can be a possible set of eigenvalues for a 4 x 4 unitary matrix?

Consider a complex function f(x. y) = e^ax + i In by. If the function is analytic at (0. 1) then the possible values of (a, b) is

If δ(x) is the Dirac-Delta function. and x has the dimension of angular momentum, then δ(x) has a dimension which can be written as [L^αM^βT^y]. The value of (α,β,γ) is

The value of Trace  ______ where  is the pauli matrix  (upto 2 decimal places)

Consider a vector   The line integral  over the contour as shown is equal to _____ (upto 2 decimal places)

Given below are forces in Cartesian and spherical coordinates. Which of the following are conservative?

A particle of m moves with speed v. Ail explosion divides the particle into two half, giving each half a speed V in the centre of mass frame. Assume all motion is confined to one dimension. The increase in kinetic energy in the Lab frame is

A particle is thrown from earth's surface with speed  where M and R are mass and radius of the earth respectively. The trajectory of the particle will be a /an 

A cricular disc is rotaing about an axis, shown in figure, makes an angle θ = 30⁰ with the vertical axis. The moment of inertia of the disc about that axis is

A block of mass M with the shape of an inverted trapezoid is pushed downwards minimum force required to tip the block is

An electron moves in the lab with a speed of 0.6c. An observer moves with a velocity of 0.8 c along the direction of the motion of the electron. The kinetic energy of the electron as determined by the observer is  _____  MeV (Upto three decimal places)