Test: Classical Mechanics - 2 - Physics MCQ

A pendulum consists of a mass m and a massless stick of length l. The pendulum support oscillates horizontally with a position given by x (t) = (see figure). The momentum conjugate to θ is

If the Hamiltonian of a system with two degrees of freedom is given by

then the quantity that is conserved is

The Poisson bracket  with a and b as constants is equal to

A particle of effective mass m is acted on by a potential energy given by

where, u₀, a and b are positive constants, then

The number of degrees of freedom of two particles moving on a space curve and having constant distance between them, is ______.

The average lifetime of a π -meson in its own frame of reference is 26.011s. If the π - meson moves with speed 0.95c with respect to the earth, then the average distance it travels before decaying as measured byan observer at rest on Earth is approximately close to _______________________ meters (nearest integer)

A particle of mass m is subjected to a central force given by the potential  The radius of the circular orbit for a particle with angular momentum,  The value of η is ______.

The path of a particle of mass m moving minder some potential V ( r ) is given by the r = A exp (aθ). where A and a are constants. The form of V(r) is (in the following L and E are angular momentum, and total energy, respectively)

A system with one degree of freedom has Lagrangian,  The equation of x(t) under the initial conditions t = 0, x = 0, p = mv₀, is

The dynamics of a particle moving on the real line is determined by the Hamiltonian

where μ and λ are constants. Then the value of dp/dq is

Hamiltonian of a one dimensional dynamical system is given by

The Poisson bracket {p, H} is equal to

For a charged particle of mass m and charge e in a magnetic field  the Lagrangian is 

The corresponding Hamiltonian is

Given that the linear transformation of a generalized coordinate q and corresponding momentum p, 

is canonical, the value of the constant β is ______ (Here, i = √-1)

Two events in a inertial referance frame separated by a distance 3.6 x 10⁸m and occur 2s apart. The proper time interval between the occurrence of these two events is _____ sec. (upto one decimal place)

The moment of inertia of a square frame of side length a and mass M about an axis passing through one of its
comers and perpendicular to its plane is βMa². The value of β is________(upto two decimal places)

Ail object of mass m = 100 kg and volume V= 0.18 m³ is dropped from a height h = 100m above the surface of the water. Upon entering water (β= 1000 kg/m³ ), the object experiences a resistance force given by F = -kv. where k = 10N sec in. The maximum depth to which the object will sink before starting to rise again is _____ m. (Neglect air resistance and assume the object looses negligible energy when going through the surface of the water). (Nearest integer)

A star receding from the earth at a speed of 5 x 10^-3c. The wavelength shift for the sodium D₂ line is approximately equal to ____  (nearest integer)