List of top Physics Questions on Classical Mechanics asked in GATE PH

Two 1 kg blocks are connected to two massless springs of spring constants \(8\text{ N.m}^{-1}\) and \(4\text{ N.m}^{-1}\). The system is kept on a frictionless horizontal floor with one end of a spring attached to a wall (see figure below). They are performing oscillatory motion along the x-axis with the normal mode frequencies \(\omega_H\) and \(\omega_L\) (\(\omega_H > \omega_L\)). The ratio \(\dfrac{\omega_H}{\omega_L}\) (rounded off to two decimal places) is ______

A 15 cm long scale is held horizontally with one of its ends on the edge of a 1 m high table and the other end resting on one's index finger. As the finger is removed (see figure below), the scale starts rotating about its end on the table. After 0.1 s, during which it has rotated by a negligibly small angle but has gained a rotational speed, it leaves the table and falls vertically towards the ground. When its centre of mass has fallen by 0.5 m, it has rotated by an angle \(\theta\). The value of \(\theta\) in degrees (rounded off to one decimal place) is ______
(\(g = 9.8\text{ m.s}^{-2}\))

A symmetric rigid body has moment of inertia \(I_1, I_2, I_3\) about its principal axes 1, 2, and 3, respectively, with \(I_1 = I_3 = I_\perp\) and \(I_2 \neq I_\perp\). It is rotating in space with no torque on it so that its angular momentum \(\vec{L}\) is constant. Let \(\omega_1, \omega_2, \omega_3\) be the components of its angular velocity along the principal axes 1, 2, and 3, respectively. Which of the following quantities is/are constant during the motion of this rigid body?

Two frames \(S\) (solid lines) and \(S'\) (dashed lines) with common origin are shown in the figure below. Frame \(S\) is inertial while \(S'\) is rotating about the common \(z\)-axis. There is a point mass fixed at \(P\) on the \(x\)-axis of the \(S\) frame. The magnitude of the centrifugal force and the Coriolis force experienced by the mass in the \(S'\) frame is \(F_{cen}\) and \(F_{cor}\), respectively. Which of the following options is correct for these forces?

The Lagrangian \(L_0 = \frac{1}{2}m\dot{q}^2 - \frac{1}{2}m\omega^2q^2\) with the generalized coordinate \(q\) is transformed to \(L = L_0 + \alpha \frac{df(q)}{dt}\). Consider the following statements:
(i) Expression for the canonical momentum does not change.
(ii) The equation of motion does not change.
Which of the following options is correct for the above statements?

A rocket of length 18.0 m is moving at speed \(0.9c\) (where \(c\) is the speed of light) parallel to its own length, relative to the earth. The length of the rocket measured in meters by an observer on earth (rounded off to two decimal places) is

On a horizontal plane, a projectile of mass \(m\) is launched from the ground with speed \(v_0\) at an angle \(\theta_0\) with the horizontal. In addition to the gravitational force (\(mg\)), it also experiences a drag force \(\vec{F}_{drag} = -\gamma \vec{v}\), where \(\vec{v}\) is its velocity and \(\gamma\) is a constant. It hits the ground at a distance \(R\) from the point of launch with its velocity making an angle \(\theta\) with the horizontal, as shown schematically in the figure below.

Then which of the following options is correct?