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A circular table is rotating with an angular velocity of \( \omega \, \text{rad/s} \) about its axis (see figure). There is a smooth groove along a radial direction on the table. A steel ball is gently placed at a distance of \( 1 \, \text{m} \) on the groove. All the surfaces are smooth. If the radius of the table is \( 3 \, \text{m} \), the radial velocity of the ball w.r.t., the table at the time ball leaves the table is \( x \sqrt{2} \, \omega \, \text{m/s} \), where the value of \( x \) is __________.

Three balls of masses \( 2\,\text{kg} \), \( 4\,\text{kg} \) and \( 6\,\text{kg} \) respectively are arranged at centre of the edges of an equilateral triangle of side \( 2\,\text{m} \). The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of triangle, will be ______ \( \text{kg m}^{2} \).

A heavy iron bar of weight \(12\,\text{kg}\) is having its one end on the ground and the other on the shoulder of a man. The rod makes an angle \(60^{\circ}\) with the horizontal, the weight experienced by the man is:

Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for moment of Inertia about their diameter axis \( AB \) as shown in figure is \( \sqrt{\frac{8}{x}} \). The value of \( x \) is:

A hollow sphere is rolling on a plane surface about its axis of symmetry. The ratio of rotational kinetic energy to its total kinetic energy is \( \dfrac{x}{5} \). The value of \( x \) is _________.

A body of mass \( 5 \, \text{kg} \) moving with a uniform speed \( 3\sqrt{2} \, \text{m s}^{-1} \) in \( X\text{-}Y \) plane along the line \( y = x + 4 \). The angular momentum of the particle about the origin will be _____ \( \text{kg m}^2 \text{ s}^{-1} \).

A uniform rod AB of mass \( 2 \, \text{kg} \) and length \( 30 \, \text{cm} \) at rest on a smooth horizontal surface. An impulse of force \( 0.2 \, \text{Ns} \) is applied to end B. The time taken by the rod to turn through at right angles will be \( \dfrac{\pi}{x} \, \text{s} \), where \( x = \) __________.

A thin circular disc of mass \( M \) and radius \( R \) is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity \( \omega \). If another disc of same dimensions but of mass \(\displaystyle \frac{M}{2} \) is placed gently on the first disc co-axially, then the new angular velocity of the system is:

Consider a Disc of mass \( 5 \, \text{kg} \), radius \( 2 \, \text{m} \), rotating with angular velocity of \( 10 \, \text{rad/s} \) about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate together without slipping is _____ \( \text{J} \).

Two blocks of mass \( 2 \, \text{kg} \) and \( 4 \, \text{kg} \) are connected by a metal wire going over a smooth pulley as shown in figure. The radius of wire is \( 4.0 \times 10^{-5} \, \text{m} \) and Young’s modulus of the metal is \( 2.0 \times 10^{11} \, \text{N} / \text{m}^{2} \). The longitudinal strain developed in the wire is \( \dfrac{1}{\alpha \pi} \). The value of \( \alpha \) is __________. [Use \( g = 10 \, \text{m} / \text{s}^{2} \)]

If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be _________ hours \( 30 \) minutes.

Two identical spheres each of mass \( 2 \, \text{kg} \) and radius \( 50 \, \text{cm} \) are fixed at the ends of a light rod so that the separation between the centres is \( 150 \, \text{cm} \). then, moment of inertia of the system about an axis perpendicular to the rod and passing through its middle point is \( \dfrac{x}{20} \, \text{kg m}^2 \) , where the value of \( x \) is ________.

A cylinder is rolling down on an inclined plane of inclination \( 60^\circ \). It’s acceleration during rolling down will be \( \frac{x}{\sqrt{3}} \, \text{m /s}^{2} \), where \( x = \) ________________ (use \( g = 10 \, \text{m /s}^{2} \)).

A solid circular disc of mass \( 50 \, \text{kg} \) rolls along a horizontal floor so that its center of mass has a speed of \( 0.4 \, \text{m/s} \). The absolute value of work done on the disc to stop it is:

A ring and a solid sphere roll down the same inclined plane without slipping. They start from rest. The radii of both bodies are identical and the ratio of their kinetic energies is \( \dfrac{7}{x} \), where \( x \) is ____________ .

Two discs of moment of inertia \( I_1 = 4 \,\mathrm{kg\,m^2} \) and \( I_2 = 2 \,\mathrm{kg\,m^2} \), about their central axes & normal to their planes, rotating with angular speeds \( 10 \,\mathrm{rad/s} \) & \( 4 \,\mathrm{rad/s} \) respectively are brought into contact face to face with their axes of rotation coincident. The loss in kinetic energy of the system in the process is ______ J.

A circular disc reaches from top to bottom of an inclined plane of length \( l\). When it slips down the plane, it takes \( t \, \text{s} \). When it rolls down the plane then it takes \( \left( \dfrac{\alpha}{2} \right)^{1/2} t \, \text{s} \), where \( \alpha \) is ........

Four particles each of mass \( 1 \, \text{kg} \) are placed at four corners of a square of side \( 2 \, \text{m} \). Moment of inertia of system about an axis perpendicular to its plane and passing through one of its vertex is __________ \( \text{kg} \, \text{m}^2 \).

A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed \( v \). The sphere and the cylinder reaches upto maximum heights \( h_{1} \) and \( h_{2} \), respectively, above the initial level. The ratio \( h_{1} : h_{2} \) is \( \frac{n}{10} \). The value of \( n \) is __________.

A string is wrapped around the rim of a wheel of moment of inertia \( 0.40 \, \text{kg m}^{2} \) and radius \( 10 \, \text{cm} \). The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of \( 40 \, \text{N} \). The angular velocity of the wheel after \( 10 \, \text{s} \) is \( x \, \text{rad/s} \), where \( x \) is _______.

A heavy iron bar, of weight \( W \) is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle \( \theta \) with the horizontal. The weight experienced by the person is:

Two cars are travelling towards each other at speed of \( 20 \, \text{m s}^{-1} \) each. When the cars are \( 300 \, \text{m} \) apart, both the drivers apply brakes and the cars retard at the rate of \( 2 \, \text{m s}^{-2} \). The distance between them when they come to rest is:

A particle of mass $m$ is projected with a velocity '$u$' making an angle of $30^\circ$ with the4 horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height $h$ is:

A disc of radius \( R \) and mass \( M \) is rolling horizontally without slipping with speed \( v \). It then moves up an inclined smooth surface as shown in figure. The maximum height that the disc can go up the incline is:

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JEE Main 2024 5th April Morning Shift

Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for moment of Inertia about their diameter axis \( AB \) as shown in figure is \( \sqrt{\frac{8}{x}} \). The value of \( x \) is: