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A force of \( -P \hat{k} \) acts on the origin of the coordinate system. The torque about the point \( (2,-3) \) is \( P(a\hat{i}+b\hat{j}) \). The ratio of \( a/b \) is \( x/2 \). The value of \( x \) is ______.

Moment of inertia of a disc of mass \( M \) and radius ‘\( R \)’ about any of its diameter is \( \frac{MR^{2}}{4} \). The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, \( \frac{x}{2} MR^{2} \). The value of \( x \) is _______.

A disc is rolling without slipping on a surface. The radius of the disc is \( R \). At \( t = 0 \), the top most point on the disc is \( A \) as shown in figure. When the disc completes half of its rotation, the displacement of point \( A \) from its initial position is:

The moment of inertia of a semicircular ring about an axis, passing through the center and perpendicular to the plane of ring, is \( \dfrac{1}{x} MR^{2} \), where \( R \) is the radius and \( M \) is the mass of the semicircular ring. The value of \( x \) will be ________.

A solid sphere of mass \( 1 \, \text{kg} \) rolls without slipping on a plane surface. Its kinetic energy is \( 7 \times 10^{-3} \, \text{J} \), the speed of the centre of mass of the sphere is _____ \( \text{cm s}^{-1} \).

A ring and a solid sphere rotating about an axis passing though their centers have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of sphere is \( \sqrt{\dfrac{2}{x}} \). The value of \( x \) is _________.

A solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about axis of rotation of the sphere to the total energy of moving sphere is \( \pi : 22 \) then, the value of its angular speed will be ______ rad/s.

As per the given figure, a small ball \( P \) slides sown the quadrant of a circle and hits the other ball \( Q \) of equal mass which is initially at rest. Neglecting the effect of friction and assume the collision to be elastic, the velocity of ball \( Q \) after collision will be: \( (g = 10 \, \text{m/s}^2) \).

A solid sphere of mass \( 500 \, \text{g} \) and radius \( 5 \, \text{cm} \) is rotated about one of its diameter with angular speed of \( 10 \, \text{rad s}^{-1} \). If the moment of inertia of the sphere about its tangent is \( x \times 10^{-2} \) times its angular momentum about the diameter. Then the value of \( x \) will be ______.

If the earth suddenly shrinks to \( \dfrac{1}{64} \) th of its original volume with its mass remaining the same, the period of rotation of earth becomes \( \dfrac{24}{x} \, \text{h} \). The value of \( x \) is ________.

A thin uniform rod of length \( 2 \, \text{m} \), cross sectional area \( A \) and density \( d \) is rotated about an axis passing through the centre and perpendicular to its length with angular velocity \( \omega \). If value of \( \omega \) in terms of its rotational kinetic energy \( E \) is \( \sqrt{\frac{\alpha E}{A d}} \) then value of \( \alpha \) is _______.

A solid sphere of mass \( 2 \, \text{kg} \) is making pure rolling on a horizontal surface with kinetic energy \( 2240 \, \text{J} \). The velocity of centre of mass of the sphere will be ________ \( \text{ms}^{-1} \).

A solid cylinder is released from rest from the top of an inclined plane of inclination \( 30^\circ \) and length \( 60 \, \text{cm} \). If the cylinder rolls without slipping, its speed upon reaching the bottom of the inclined plane is \( \_\_\_\_\_\_ \) \( \text{ms}^{-1} \). (Given \( g = 10 \, \text{ms}^{-2} \))

Two identical solid spheres each of mass \( 2 \, \text{kg} \) and radii \( 10 \, \text{cm} \) are fixed at the ends of a light rod. The separation between the centres of the spheres is \( 40 \, \text{cm} \). The moment of inertia of the system about an axis perpendicular to the rod passing through its middle point is ______ \( \times 10^{-3} \, \text{kg-m}^2 \).

A solid sphere and a solid cylinder of same mass and radius are rolling on a horizontal surface without slipping. The ratio of their radius of gyrations respectively \( (k_{sph} : k_{cyl}) \) is \( 2 : \sqrt{x} \). The value of \( x \) is __________.

A ball of mass \( 200 \, \text{g} \) rests on a vertical post of height \( 20 \, \text{m} \). A bullet of mass \( 10 \, \text{g} \), travelling in horizontal direction, hits the centre of the ball. After collision both travels independently. The ball hits the ground at a distance \( 30 \, \text{m} \) and the bullet at a distance of \( 120 \, \text{m} \) from the foot of the post. The value of initial velocity of the bullet will be (if \( g = 10 \, \text{m/s}^{2} \)).

A spherical body of mass \( 2 \, \text{kg} \) starting from rest acquires a kinetic energy of \( 10000 \, \text{J} \) at the end of \( 5^{\text{th}} \) second. The force acted on the body is \( \text{N} \).

If a solid sphere of mass \( 5 \, \text{kg} \) and a disc of mass \( 4 \, \text{kg} \) have the same radius. Then the ratio of moment of inertia of the disc about a tangent in its plane to the moment of inertia of the sphere about its tangent will be \( \dfrac{x}{7} \). The value of \( x \) is _______.

For a rolling spherical shell, the ratio of rotational kinetic energy and total kinetic energy is \( \dfrac{x}{5} \). The value of \( x \) is _____.

A hollow spherical ball of uniform density rolls up a curved surface with an initial velocity \( 3 \, \text{m/s} \) (as shown in figure). Maximum height with respect to the initial position covered by it will be ______ \( \text{cm} \) (take, \( g = 10 \, \text{m/s}^2 \)).

An object of mass \( 8 \, \text{kg} \) is hanging from one end of a uniform rod CD of mass \( 2 \, \text{kg} \) and length \( 1 \, \text{m} \) pivoted at its end C on a vertical wall as shown in figure. It is supported by a cable AB such that the system is in equilibrium. The tension in the cable is: (Take \( g = 10 \, \text{m}/\text{s}^2 \))

A circular plate is rotating in horizontal plane, about an axis passing through its center and perpendicular to the plate, with an angular velocity \( \omega \). A person sits at the center having two dumbbells in his hands. When he stretches out his hands, the moments of inertia of the system becomes triple. If \( E \) be the initial kinetic energy of the system, then final kinetic energy will be \( \dfrac{E}{x} \). The value of \( x \) is ________.

A light rope is wound around a hollow cylinder of mass \( 5 \, \text{kg} \) and radius \( 70 \, \text{cm} \). The rope is pulled with a force of \( 52.5 \, \text{N} \). The angular acceleration of the cylinder will be __________ \( \text{rad s}^{-2} \).

Given below are two statements: one is labeled as Assertion A and the other is labeled as Reason R.
Assertion A: An electric fan continues to rotate for some time after the current is switched off.
Reason R: Fan continues to rotate due to inertia of motion.
In the light of above statements, choose the most appropriate answer from the options given below.

\( I_{CM} \) is the moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc. \( I_{AB} \) is it’s moment of inertia about an axis AB perpendicular to plane and parallel to axis CM at a distance \( \frac{2}{3} R \) from center. Where \( R \) is the radius of the disc. The ratio of \( I_{AB} \) and \( I_{CM} \) is \( x:9 \). The value of \( x \) is ___________.

A uniform solid cylinder with radius \( R \) and length \( L \) has moment of inertia \( I_1 \), about the axis of the cylinder. A concentric solid cylinder of radius \( R' = \frac{R}{2} \) and length \( L' = \frac{L}{2} \) is carved out of the original cylinder. If \( I_2 \) is the moment of inertia of the carved out portion of the cylinder then \( \frac{I_1}{I_2} = \) ________ . (Both \( I_1 \) and \( I_2 \) are about the axis of the cylinder)

In the equation \( \left[ X + \dfrac{a}{y^2} \right]\left[ Y - b \right] = RT \), \( X \) is pressure, \( Y \) is volume, \( R \) is universal gas constant and \( T \) is temperature. The physical quantity equivalent to the ratio \( \dfrac{a}{b} \) is :

Solid sphere \( A \) is rotating about an axis \( PQ \). If the radius of the sphere is \( 5 \, \mathrm{cm} \) then its radius of gyration about \( PQ \) will be \( \sqrt{x} \, \mathrm{cm} \). The value of \( x \) is ______.

A uniform disc of mass \( 0.5 \, \text{kg} \) and radius \( r \) is projected with velocity \( 18 \, \text{m/s} \) at \( t = 0 \, \text{s} \) on a rough horizontal surface. It starts off with a purely sliding motion at \( t = 0 \, \text{s} \). After \( 2 \, \text{s} \) it acquires a purely rolling motion (see figure). The total kinetic energy of the disc after \( 2 \, \text{s} \) will be _____ \( \text{J} \) (Given, coefficient of friction is \( 0.3 \) and \( g = 10 \, \text{m/s}^{2} \).)

A particle of mass \( 100 \, \text{g} \) is projected at time \( t = 0 \) with a speed \( 20 \, \text{m} \, \text{s}^{-1} \) at an angle \( 45^\circ \) to the horizontal as given in the figure. The magnitude of the angular momentum of the particle about the starting point at time \( t = 2 \, \text{s} \) is found to be \( \sqrt{K} \, \text{kg} \, \text{m}^2 / \text{s} \). The value of \( K \) is ________. (Take \( g = 10 \, \text{m} \, \text{s}^{-2} \)).

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JEE Main 2023 24th Jan Morning Shift

Solid sphere \( A \) is rotating about an axis \( PQ \). If the radius of the sphere is \( 5 \, \mathrm{cm} \) then its radius of gyration about \( PQ \) will be \( \sqrt{x} \, \mathrm{cm} \). The value of \( x \) is ______.