List of top Classical Mechanics Questions asked in GATE PH

A uniform block of mass \(M\) slides on a smooth horizontal bar. Another mass \(m\) is connected to it by an inextensible string of length \(l\) of negligible mass, and is constrained to oscillate in the X-Y plane only. Neglect the sizes of the masses. The number of degrees of freedom of the system is two and the generalized coordinates are chosen as \(x\) and \(\theta\), as shown in the figure.

If \(p_x\) and \(p_\theta\) are the generalized momenta corresponding to \(x\) and \(\theta\), respectively, then the correct option(s) is(are)

Consider a spherical galaxy of total mass \( M \) and radius \( R \), having a uniform matter distribution. In this idealized situation, the orbital speed \( v \) of a star of mass \( m \ll M \) as a function of the distance \( r \) from the galactic centre is best described by \[ (G \text{ is the universal gravitational constant}) \]

Consider a point charge +Q of mass m suspended by a massless, inextensible string of length l in free space (permittivity \( \epsilon_0 \)) as shown in the figure. It is placed at a height \( d \) (\( d > l \)) over an infinitely large, grounded conducting plane. The gravitational potential energy is assumed to be zero at the position of the conducting plane and is positive above the plane.

The time derivative of a differentiable function \( g(q_i, t) \) is added to a Lagrangian \( L(q_i, \dot{q}_i, t) \) such that \[ L' = L(q_i, \dot{q}_i, t) + \frac{d}{dt} g(q_i, t) \] where \( q_i \), \( \dot{q}_i \), \( t \) are the generalized coordinates, generalized velocities, and time, respectively. Let \( p_i \) be the generalized momentum and \( H \) the Hamiltonian associated with \( L(q_i, \dot{q}_i, t) \). If \( p_i' \) and \( H' \) are those associated with \( L' \), then the correct option(s) is(are):

If \( \vec{a} \) and \( \vec{b} \) are constant vectors, \( \vec{r} \) and \( \vec{p} \) are generalized positions and conjugate momenta, respectively, then for the transformation \( Q = \vec{a} \cdot \vec{r} \) and \( P = \vec{b} \cdot \vec{r} \) to be canonical, the value of \( \vec{a} \cdot \vec{b} \) (in integer) is \(\underline{\hspace{2cm}}\).

A hoop of mass \( M \) and radius \( R \) rolls without slipping along a straight line on a horizontal surface as shown in the figure. A point mass \( m \) slides without friction along the inner surface of the hoop, performing small oscillations about the mean position. The number of degrees of freedom of the system (in integer) is \(\underline{\hspace{2cm}}\). \includegraphics[width=0.5\linewidth]{image20.png}

Two observers \( O \) and \( O' \) observe two events \( P \) and \( Q \). The observers have a constant relative speed of 0.5c. In the units, where the speed of light, \( c \), is taken as unity, the observer \( O \) obtained the following coordinates:
\text{Event P: } x = 5, y = 3, z = 5, t = 3
\text{Event Q: } x = 5, y = 1, z = 3, t = 5
\text{The length of the space-time interval between these two events, as measured by \( O' \), is \( L \). The value of \( |L| \) (in integer) is \(\underline{\hspace{2cm}}\).}