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A solid sphere of mass \( 5 \, \text{kg} \) and radius \( 10 \, \text{cm} \) is kept in contact with another solid sphere of mass \( 10 \, \text{kg} \) and radius \( 20 \, \text{cm} \). The moment of inertia of this pair of spheres about the tangent passing through the point of contact is ______ \( \mathrm{kg \, m^2} \).

A solid sphere of radius \( 10 \, \text{cm} \) is rotating about an axis which is at a distance \( 15 \, \text{cm} \) from its centre. The radius of gyration about this axis is \( \sqrt{n} \, \text{cm} \). The value of \( n \) is __________.

Two small balls with masses \( m \) and \( 2m \) are attached to both ends of a rigid rod of length \( d \) and negligible mass. If angular momentum of this system is \( L \) about an axis \( (A) \) passing through its centre of mass and perpendicular to the rod then angular velocity of the system about \( A \) is:

A thin uniform rod (X) of mass \( M \) and length \( L \) is pivoted at a height \( \left( \dfrac{L}{3} \right) \) as shown in the figure. The rod is allowed to fall from a vertical position and lie horizontally on the table. The angular velocity of this rod when it hits the table top, is ____________ . \( (g = \text{gravitational acceleration}) \).

Moment of inertia about an axis \( AB \) for a rod of mass \( 40\ \text{kg} \) and length \( 3\ \text{m} \) is same as that of a solid sphere of mass \( 10\ \text{kg} \) and radius \( R \) about an axis parallel to \( AB \) axis with separation of \( 3\ \text{m} \) as shown in figure below. The value of \( R \) is given as \( \sqrt{\frac{\alpha}{2}} \). The value of \( \alpha \) is __________.

Match the LIST-I with LIST-II.

List-I	List-II
A. Planck's constant	I. \( ML^2T^{-2} \)
B. Stopping potential	II. \( T^{-1} \)
C. Work function	III. \( ML^2T^{-1} \)
D. Threshold frequency	IV. \( ML^2T^{-3}A^{-1} \)

Choose the correct answer from the options given below:

A solid sphere of radius \( 4 \, \text{cm} \) and mass \( 5 \, \text{kg} \) is rotating (rotation axis is passing through the centre of the sphere) with an angular velocity of \( 1200 \, \text{rpm} \). It is brought to rest in \( 10 \, \text{s} \) by applying a constant torque. The torque applied and the number of rotations it made before it comes to rest are ______ and ______ respectively.

Two masses \( 400 \, \text{g} \) and \( 350 \, \text{g} \) are suspended from the ends of a light string passing over a heavy pulley of radius \( 2 \, \text{cm} \). When released from rest the heavier mass is observed to fall \( 81 \, \text{cm} \) in \( 9 \, \text{s} \). The rotational inertia of the pulley is ______ \( \text{kg m}^2 \). \( (g = 9.8 \, \text{m}^2) \)

A circular disc has radius \( R_1 \) and thickness \( T_1 \). Another circular disc made of the same material has radius \( R_2 \), and thickness \( T_2 \). If the moment of inertia of both discs are same and \( \frac{R_1}{R_2} = 2 \) then \( \frac{T_1}{T_2} = \frac{1}{\alpha} \). The value of \( \alpha \) is __________.

10.

Two identical thin rods of mass \( M \) \( \text{kg} \) and length \( L \) \( \text{m} \) are connected as shown in figure. Moment of inertia of the combined rod system about an axis passing through point \( P \) and perpendicular to the plane of the rods is \( \frac{x}{12} \, M L^2 \, \text{kg m}^2 \). The value of \( x \) is _________________.

11.

Two cars \( A \) and \( B \) each of mass \( 10^{3} \, \text{kg} \) are moving on parallel tracks separated by a distance of \( 10 \, \text{m} \), in same direction with speeds \( 72 \, \text{km/h} \) and \( 36 \, \text{km/h} \). The magnitude of angular momentum of car \( A \) with respect to car \( B \) is _______ \( \text{J·s} \).

12.

A solid sphere (A) of mass \( 5m \) and a spherical shell (B) of mass \( m \), both having same radius, are placed on a rough surface. When a force of same magnitude is applied tangentially at the highest points of A and B, they start rolling without slipping with an acceleration of \( a_{A} \) and \( a_{B} \), respectively. The ratio of \( a_{A} \) and \( a_{B} \) is:

13.

A solid cylinder having radius \( R \) and length \( L \) is slipping on a rough horizontal plane. At time \( t = 0 \) the cylinder has a translational velocity \( v_0 = 49 \, \text{m} \, \text{s}^{-1} \), perpendicular to its axis and a rotational velocity \( \dfrac{v_0}{4R} \) about the centre. The time taken by the cylinder to start rolling is ______ seconds. [Coefficient of kinetic friction \( \mu_K = 0.25 \) and \( g = 9.8 \, \text{m} \, \text{s}^{-2} \)]

14.

A uniform solid cylinder of length \( L \) and radius \( R \) has moment of inertia about its axis equal to \( I_1 \). A small co-centric cylinder of length \( L / 3 \) and radius \( R / 3 \) carved from this cylinder has moment of inertia about its axis equals to \( I_2 \). The ratio \( {I_1}/{I_2} \) is _____________.

15.

Two masses \( m \) and \( 2m \) are connected by a light string going over a pulley (disc) of mass \( 30m \) with radius \( r = 0.1 \, \text{m} \). The pulley is mounted in a vertical plane and it is free to rotate about its axis. The \( 2m \) mass is released from rest and its speed when it has descended through a height of \( 3.6 \, \text{m} \) is ______ \( \text{m/s} \). (Assume string does not slip and \( g = 10 \, \text{m/s}^{2} \))

16.

Two circular discs of radius each \( 10 \, \text{cm} \) are joined at their centres by a rod of length \( 30 \, \text{cm} \) and mass \( 600 \, \text{gm} \) as shown in figure. If the mass of each disc is \( 600 \, \text{gm} \) and applied torque between two discs is \( 43 \times 10^{5} \, \text{dyne}\!\cdot\!\text{cm} \), the angular acceleration of the discs about the given axis \( AB \) is _________ \( \text{rad}/ \text{s}^{2} \).

17.

A solid sphere of mass \( M \) and radius \( R \) is divided into two unequal parts. The smaller part having mass \( \frac{M}{8} \) is converted into a sphere of radius \( r \) and the larger part is converted into a circular disc of thickness \( t \) and radius \( 2R \). If \( I_1 \) is moment of inertia of a sphere having radius \( r \) about an axis through its centre and \( I_2 \) is the moment of inertia of a disc about its diameter, the ratio of their moment of inertia

18.

The pulley shown in figure is made using a thin rim and two rods of length equal to diameter of the rim. The rim and each rod have a mass of \( M \). Two blocks of mass of \( M \) and \( m \) are attached to two ends of a light string passing over the pulley, which is hinged to rotate freely in vertical plane about its center. The magnitudes of the acceleration experienced by the blocks is __________. (assume no slipping of string on pulley).

19.

Suppose there is a uniform circular disc of mass \( M \; \mathrm{kg} \) and radius \( r \; \mathrm{m} \) shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis \( A \) of the disc is given by \( \dfrac{x}{256} M r^{2} \). The value of \( x \) is ________.

20.

A uniform rod of mass \( m \) and length \( l \) suspended by means of two identical inextensible light strings as shown in figure. Tension in one string immediately after the other string is cut, is __________. \( (g \) acceleration due to gravity\() \).

21.

The moment of inertia of a square loop made of four uniform solid cylinders, each having radius \( R \) and length \( L \) \( (R < L) \) about an axis passing through the mid points of opposite sides, is: [Take the mass of the entire loop as \( M \)]

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