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A spherical shell of \( 1 \, \text{kg} \) mass and radius \( R \) is rolling with angular speed \( \omega \) on horizontal plane (as shown in figure). The magnitude of angular momentum of the shell about the origin \( O \) is \( \frac{a}{3} R^{2} \omega \). The value of \( a \) will be:

A metre scale is balanced on a knife edge at its centre. When two coins, each of mass \( 10 \, \text{g} \) are put one on the top of the other at the \( 10.0 \, \text{cm} \) mark the scale is found to be balanced at \( 40.0 \, \text{cm} \) mark. The mass of the metre scale is found to be \( x \times 10^{-2} \, \text{kg} \). The value of \( x \) is __________.

A thin circular ring of mass \( M \) and radius \( R \) is rotating with a constant angular velocity \( 2 \,\text{rad s}^{-1} \) in a horizontal plane about an axis vertical to its plane and passing through the center of the ring. If two objects each of mass \( m \) be attached gently to the opposite ends of a diameter of ring, the ring will then rotate with an angular velocity (in \( \text{rad s}^{-1} \)).

A solid cylinder length is suspended symmetrically through two massless strings, as shown in the figure. The distance from the initial rest position, the cylinder should be unbinding the strings to achieve a speed of \( 4 \, \text{ms}^{-1} \), is ________ \( \text{cm} \). (Take \( g = 10 \, \text{ms}^{-2} \)).

A rolling wheel of \( 12 \, \text{kg} \) is on an inclined plane at position \( P \) and connected to a mass of \( 3 \, \text{kg} \) through a string of fixed length and pulley as shown in figure. Consider \( PR \) as friction free surface. The velocity of centre of a mass of the wheel when it reaches at the bottom \( Q \) of the inclined plane \( PQ \) will be \( \frac{1}{2} \sqrt{x g h} \, \text{m} / \text{s} \). The value of \( x \) is ________.

Moment of Inertia (M.I.) of four bodies having same mass '\( M \)' and radius '\( 2R \)' are as follows:
\( I_{1} \) = M.I. of solid sphere about its diameter
\( I_{2} \) = M.I. of solid cylinder about its axis
\( I_{3} \) = M.I. of solid circular disc about its diameter
\( I_{4} \) = M.I. of thin circular ring about its diameter
If \( 2 ( I_{2} + I_{3} ) + I_{4} = x \cdot I_{1} \) then the value of \( x \) will be __________.

Match List-I with List-II

Choose the correct answer from the options given below:

A uniform disc with mass \( M = 4\,\text{kg} \) and radius \( R = 10\,\text{cm} \) is mounted on a fixed horizontal axle as shown in figure. A block with mass \( m = 2\,\text{kg} \) hangs from a massless cord that is wrapped around the rim of the disc. During the fall of the block, the cord does not slip and there is no friction at the axle. The tension in the cord is _______ \( \text{N} \). (Take \( g = 10\,\text{ms}^{-2} \)).

A solid spherical ball is rolling on a frictionless horizontal plane surface about its axis of symmetry. The ratio of rotational kinetic energy of the ball to its total kinetic energy is:

A \( \sqrt{34} \, \text{m} \) long ladder weighing \( 10 \, \text{kg} \) leans on a frictionless wall. Its feet rest on the floor \( 3 \, \text{m} \) away from the wall as shown in the figure. If \( F_f \) and \( F_w \) are the reaction forces of the floor and the wall, then ratio of \( \frac{F_w}{F_f} \) will be: (Use \( g = 10 \, \text{m} / \text{s}^2 \).)

The torque of a force \( 5\hat{i} + 3\hat{j} - 7\hat{k} \) about the origin is \( \tau \). If the force acts on a particle whose position vector \( 2\hat{i} + 2\hat{j} + \hat{k} \), then the value of \( \tau \) will be

A pulley of radius \( 1.5 \, \text{m} \) is rotated about its axis by a force \( F = (12t - 3t^{2}) \, \text{N} \) applied tangentially (while \( t \) is measured in seconds). If moment of inertia of the pulley about its axis of rotation is \( 4.5 \, \text{kg m}^{2} \), the number of rotations made by the pulley before its direction of motion is reversed, will be \( {K}/{\pi} \). The value of \( K \) is _______.

A disc of mass \( 1 \, \text{kg} \) and radius \( R \) is free to rotate about a horizontal axis passing through its centre and perpendicular to the plane of disc. A body of same mass as that of disc is fixed at the highest point of the disc. Now the system is released, when the body comes to the lowest position, its angular speed will be \( 4\sqrt{\dfrac{x}{3R}} \, \text{rad} \, \text{s}^{-1} \) where \( x = \) ________.
\( (g = 10 \, \text{m} \, \text{s}^{-2}) \)

The radius of gyration of a cylindrical rod about an axis of rotation perpendicular to its length and passing through the center will be ________ \( \text{m} \). (Given, the length of the rod is \( 10\sqrt{3} \, \text{m} \).)

Four identical discs each of mass 'M' and diameter 'a' are arranged in a small plane as shown in figure. If the moment of inertia of the system about \( OO' \) is \( \dfrac{x}{4} \, Ma^{2} \). Then, the value of \( x \) will be __________.

The moment of inertia of a uniform thin rod about a perpendicular axis passing through one end is \( I_{1} \). The same rod is bent into a ring and its moment of inertia about a diameter is \( I_{2} \). If \( \frac{I_{1}}{I_{2}} \) is \( \frac{x \pi^{2}}{3} \), then the value of \( x \) will be ______.

A solid cylinder and a solid sphere, having same mass \( M \) and radius \( R \), roll down the same inclined plane from top without slipping. They start from rest. The ratio of velocity of the solid cylinder to that of the solid sphere, with which they reach the ground, will be:

The position vector of \( 1 \, \text{kg} \) object is \( \vec{r} = (3\hat{i} - \hat{j}) \, \text{m} \) and its velocity \( \vec{v} = (3\hat{j} + \hat{k}) \, \text{m}\,\text{s}^{-1} \). The magnitude of its angular momentum is \( \sqrt{x} \, \text{N}\,\text{m} \) where \( x \) is _______.

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JEE Main 2022 29th July Evening shift

The torque of a force \( 5\hat{i} + 3\hat{j} - 7\hat{k} \) about the origin is \( \tau \). If the force acts on a particle whose position vector \( 2\hat{i} + 2\hat{j} + \hat{k} \), then the value of \( \tau \) will be