6 questions from 2021
A pendulum consists of a bob of mass \( m = 0.1 \, \text{kg} \) and a massless inextensible string of length \( L = 1.0 \, \text{m} \). It is suspended from a fixed point at height \( H = 0.9 \, \text{m} \) above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse \( P = 0.2 \, \text{kg}\!\cdot\!\text{m/s} \) is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is \( J \, \text{kg}\!\cdot\!\text{m}^2/\text{s} \). The kinetic energy of the pendulum just after the lift off is \( K \) Joules.
The value of \( K \) is _______.
A particle of mass \( M = 0.2 \, \text{kg} \) is initially at rest in the \( xy \)-plane at a point \( (x = -l, \; y = -h) \), where \( l = 10 \, \text{m} \) and \( h = 1 \, \text{m} \). The particle is accelerated at time \( t = 0 \) with a constant acceleration \( a = 10 \, \text{m} \,/ \text{s}^{2} \) along the positive \( x \)-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by \( \vec{L} \) and \( \vec{\tau} \), respectively. \( \hat{i} \), \( \hat{j} \) and \( \hat{k} \) are unit vectors along the positive \( x \), \( y \) and \( z \)-directions, respectively. If \( \hat{k} = \hat{i} \times \hat{j} \) then which of the following statement(s) is(are) correct?
A thin rod of mass \( M \) and length \( a \) is free to rotate in horizontal plane about a fixed vertical axis passing through point \( O \). A thin circular disc of mass \( M \) and of radius \( \frac{a}{4} \) is pivoted on this rod with its center at a distance \( \frac{a}{4} \) from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary observer finds the rod rotating with an angular velocity \( \Omega \) and the disc rotating about its vertical axis with angular velocity \( 4 \Omega \). The total angular momentum of the system about the point \( O \) is \( \left( \frac{M a^{2} \Omega}{48} \right) n \). The value of \( n \) is _____ 
A horizontal force \( F \) is applied at the center of mass of a cylindrical object of mass \( m \) and radius \( R \), perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is \( \mu \). The center of mass of the object has an acceleration \( a \). The acceleration due to gravity is \( g \). Given that the object rolls without slipping, which of the following statement(s) is(are) correct? 
One end of a horizontal uniform beam of weight \( W \) and length \( L \) is hinged on a vertical wall at point \( O \) and its other end is supported by a light inextensible rope. The of the end of the rope is fixed at point \( Q \), at a height \( L \) above the hinge at pint \( O \). A block of weight \( \alpha W \) is attached at the point \( P \) of the beam, as shown in the figure (not to scale). The rope can sustain a maximum tension of \( (2\sqrt{2}\,)W \). Which of the following statement(s) is (are correct? 
A pendulum consists of a bob of mass \( m = 0.1 \, \text{kg} \) and a massless inextensible string of length \( L = 1.0 \, \text{m} \). It is suspended from a fixed point at height \( H = 0.9 \, \text{m} \) above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse \( P = 0.2 \, \text{kg}\!\cdot\!\text{m/s} \) is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is \( J \, \text{kg}\!\cdot\!\text{m}^2/\text{s} \). The kinetic energy of the pendulum just after the lift off is \( K \) Joules.
The value of \( J \) is _______.
