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JEE Mains | PYQ

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 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 __________.

The torque due to the force \( (2\hat{i} + \hat{j} + 2\hat{k}) \) about the origin, acting on a particle whose position vector is \( (\hat{i} + \hat{j} + \hat{k}) \) would be:

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 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 ______.

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 wheel is rolling on a plane surface. The speed of a particle on the highest point of the rim is \( 8\,\text{m/s} \). The speed of the particle on the rim of the wheel at the same level as the centre of wheel, will be:

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:

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:

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 cord of negligible mass is wound around the rim of a wheel supported by spokes with negligible mass. The mass of wheel is \( 10 \, \text{kg} \) and radius is \( 10 \, \text{cm} \) and it can freely rotate without any friction. Initially the wheel is at rest. If a steady pull of \( 20 \, \text{N} \) is applied on the cord, the angular velocity of the wheel, after the cord is unwound by \( 1 \, \text{m} \), would be:

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 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 solid disc of radius \( 20 \, \text{cm} \) and mass \( 10 \, \text{kg} \) is rotating with an angular velocity of \( 600 \, \text{rpm} \), about an axis normal to its circular plane and passing through its centre of mass. The retarding torque required to bring the disc at rest in \( 10 \, \text{s} \) is __________ \( \pi \times 10^{-1} \, \text{Nm} \).

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 ________.

Three equal masses \( m \) are kept at vertices \( (A, B, C) \) of an equilateral triangle of side \( a \) in free space.
At \( t = 0 \), they are given an initial velocity \( \overrightarrow{{v}_{A}} = v_{0} \, \overrightarrow{AC} \), \( \overrightarrow{{v}_{B}} = v_{0} \, \overrightarrow{BA} \) and \( \overrightarrow{{v}_{C}} = v_{0} \, \overrightarrow{CB} \). Here, \( \overrightarrow{AC}, \overrightarrow{CB} \) and \( \overrightarrow{BA} \) are unit vectors along the edges of the triangle. If the three masses interact gravitationally, then the magnitude of the net angular momentum of the system at the point of collision is:

Moment of inertia of a square plate of side \( l \) about the axis passing through one of the corner and perpendicular to the plane of square plate is given by:

A thin solid disk of \( 1 \, \text{kg} \) is rotating along its diameter axis at the speed of \( 1800 \, \text{rpm} \). By applying an external torque of \( 25\pi \, \text{Nm} \) for \( 40 \, \text{s} \), the speed increases to \( 2100 \, \text{rpm} \). The diameter of the disk is ______ m.

A force \( \vec{F} = 4\hat{i} + 3\hat{j} + 4\hat{k} \) is applied on a intersection point of \( x = 2 \) plane and x-axis. The magnitude of torque of this force about a point \( (2, 3, 4) \) is __________. (Round off to the Nearest Integer)

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 particle of mass '\( m \)' is moving in time '\( t \)' on a trajectory given by
\( \vec r = 10 \alpha t^{2}\,\hat{i} + 5\beta (t-5)\,\hat{j} \)
Where \( \alpha \) and \( \beta \) are dimensional constants.
The angular momentum of the particle becomes the same as it was for \( t = 0 \) at time \( t = \) __________ seconds.

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 __________.

Consider a uniform wire of mass \( M \) and length \( L \). It is bent into a semicircle. Its moment of inertia about a line perpendicular to the plane of the wire passing through the centre is:

A solid cylinder of mass \( m \) is wrapped with an inextensible light string and, is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is:

[The coefficient of static friction, \( \mu_s \) is \( 0.4 \).]

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 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.

An object of mass \( 'm' \) is projected from origin in a vertical \( xy \) plane at an angle \( 45^\circ \) with the \( x \)-axis with an initial velocity \( v_0 \). The magnitude and direction of the angular momentum of the object with respect to origin, when it reaches at the maximum height, will be [\( g \) is acceleration due to gravity]

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: \( (g = 10 \, \text{m/s}^2) \).

Match List-I with List-II

Choose the correct answer from the options given below:

Consider a badminton racket with length scales as shown in the figure.

If the mass of the linear and circular portions of the badminton racket are same \( (M) \) and the mass of the threads are negligible, the moment of inertia of the racket about an axis perpendicular to the handle and in the plane of the ring at, \( \dfrac{r}{2} \) distance from the end \( A \) of the handle will be ________ \( M r^2 \).

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 ____________ . \( (g = \text{gravitational acceleration}) \).

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 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 ______.

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 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 _____ \( \text{kg m}^2 \text{ s}^{-1} \).

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 square Lamina \( OABC \) of length \( 10 \, \text{cm} \) is pivoted at \( 'O' \). Forces act at Lamina as shown in figure. If Lamina remains stationary, then the magnitude of \( F \) is:

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 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 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 \( x = \) __________.

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 __________.

A uniform solid cylinder of mass \( 'm' \) and radius \( 'r' \) rolls along an inclined rough plane of inclination \( 45^\circ \). If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder axis will be:

A circular disc reaches from top to bottom of an inclined plane of length '\( L \)'. When it slips down the plane, it takes time '\( t_1 \)'. When it rolls down the plane, it takes time \( t_2 \). The value of \( \dfrac{t_2}{t_1} \) is \( \sqrt{\dfrac{3}{x}} \). The value of \( x \) will be _______.

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 sphere is rolling without slipping on a horizontal plane. The ratio of the linear kinetic energy of the centre of mass of the sphere and rotational kinetic energy is:

The moment of inertia of a solid disc rotating along its diameter is \( 2.5 \) times higher than the moment of inertia of a ring rotating in similar way. The moment of inertia of a solid sphere which has same radius as the disc and rotating in similar way, is \( n \) times higher than the moment of inertia of the given ring. Here, \( n = \) ______. Consider all the bodies have equal masses.

Two iron solid discs of negligible thickness have radii \( R_{1} \) and \( R_{2} \) and moment of inertia \( I_{1} \) and \( I_{2} \), respectively. For \( R_{2} = 2 R_{1} \), the ratio of \( I_{1} \) and \( I_{2} \) would be \( \frac{1}{x} \), 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:

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 \( g = 10 \, \text{ms}^{-2} \))

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 _____ \( \text{J} \).

A thin circular ring of mass \( M \) and radius \( r \) is rotating about its axis with an angular speed \( \omega \). Two particles having mass \( m \) each are now attached at diametrically opposite points. The angular speed of the ring will become:

A solid sphere with uniform density and radius \( R \) is rotating initially with constant angular velocity \( (\omega_{1}) \) about its diameter. After some time during the rotation its starts loosing mass at a uniform rate, with no change in its shape. The angular velocity of the sphere when its radius become \( R/2 \) is \( x \omega_{1} \). The value of \( x \) is ________.

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 \( g = 10 \, \text{m} / \text{s}^{2} \)]

Match the LIST-I with LIST-II.

Choose the correct answer from the options given below:

The moment of inertia of a circular ring of mass \( M \) and diameter \( r \) about a tangential axis lying in the plane of the ring is:

A uniform rod of mass \( 250 \, g \) having length \( 100 \, cm \) is balanced on a sharp edge of \( 40 \, cm \) mark. A mass of \( 400 \, g \) is suspended at \( 10 \, cm \) mark. To maintain the balance of the rod, the mass to be suspended at \( 90 \, cm \) mark, is:

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 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.

The coordinates of a particle with respect to origin in a given reference frame is $(1, 1, 1)$ meters. If a force of $\vec{F} = \hat{i} - \hat{j} + \hat{k}$ acts on the particle, then the magnitude of torque (with respect to origin) in z-direction is _____________.

The solid cylinder of length \( 80 \, \text{cm} \) and mass \( M \) has a radius of \( 20 \, \text{cm} \). Calculate the density of the material used if the moment of inertia of the cylinder about an axis \( CD \) parallel to \( AB \) as shown in figure is \( 2.7 \, \text{kg m}^{2} \).
 

Which of the following are correct expression for torque acting on a body?
A. \( \vec{\tau} = \vec{r} \times \vec{L} \)
B. \( \vec{\tau} = \dfrac{d}{dt}\left( \vec{r} \times \vec{p} \right) \)
C. \( \vec{\tau} = \vec{r} \times \dfrac{d\vec{p}}{dt} \)
D. \( \vec{\tau} = I \vec{\alpha} \)
E. \( \vec{\tau} = \vec{r} \times \vec{F} \)
( \( \vec{r} \) = position vector; \( \vec{p} \) = linear momentum; \( \vec{L} \) = angular momentum; \( \vec{\alpha} \) = angular acceleration; \( I \) = moment of inertia; \( \vec{F} \) = force; \( t \) = time )
Choose the correct answer from the options given below:

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 ________.

Angular momentum of a single particle moving with constant speed along circular path:

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 \( g = 10 \, \text{m/s}^{2} \)).

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 __________.

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 \( \text{N} \).

A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is \( \sqrt{\dfrac{x}{5}} \) where \( x = \)

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 force of \( 49 \, \text{N} \) acts tangentially at the highest point of a sphere (solid) of mass \( 20 \, \text{kg} \), kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is:

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 _______.

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 _________________.

A sphere of mass \( 2\,\text{kg} \) and radius \( 0.5\,\text{m} \) is rolling with an initial speed of \( 1\,\text{m}\,\text{s}^{-1} \) goes up an inclined plane which makes an angle of \( 30^\circ \) with the horizontal plane, without slipping. How long will the sphere take to return to the starting point \( A \)?

A triangular plate is shown. A force \( \vec{F} = 4\hat{\imath} - 3\hat{\jmath} \) is applied at point \( P \). The torque at point \( P \) with respect to point \( 'O' \) and \( 'Q' \) are:

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 system consists of two identical spheres each of mass \( 1.5 \, \text{kg} \) and radius \( 50 \, \text{cm} \) at the ends of a light rod. The distance between the centres of the two spheres is \( 5 \, \text{m} \). What will be the moment of inertia of the system about an axis perpendicular to the rod passing through its midpoint ?

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 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, \( g = 10 \, \text{m/s}^2 \)).

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 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 _______ \( \text{J·s} \).

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 \( g = 10 \, \text{m} / \text{s}^2 \).)

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 ........

A solid disc of radius '\( a \)' and mass '\( m \)' rolls down without slipping on an inclined plane making an angle \( \theta \) with the horizontal. The acceleration of the disc will be \( \dfrac{2}{b} g \sin \theta \) where \( b \) is __________ . (Round off to the Nearest Integer)
(\( g \) = acceleration due to gravity \( \theta =\)  angle as shown in figure) 

A solid sphere of mass \( 'm' \) and radius \(' r '\) is allowed to roll without slipping from the highest point of an inclined plane of length \( 'L' \) and makes an angle \( 30^\circ \) with the horizontal. The speed of the particle at the bottom of the plane is \( v_1 \). If the angle of inclination is increased to \( 45^\circ \) while keeping \( L \) constant. Then the new speed of the sphere at the bottom of the plane is \( v_2 \). The ratio \( v_1^2 : v_2^2 \) is:

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:

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 \( g = 10 \, \text{m}/\text{s}^2 \))

Two discs have moment of inertia \( I_1 \) and \( I_2 \) about their respective axes perpendicular to the plane and passing through the centre. They are rotating with angular speeds, \( \omega_1 \) and \( \omega_2 \) respectively and are brought into contact face to face with their axes of rotation coaxial. The loss in kinetic energy of the system in the process is given by:

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 \( g = 9.8 \, \text{m} \, \text{s}^{-2} \)]

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 mass \( M \) hangs on a massless rod of length \( l \) which rotates at a constant angular frequency. The mass \( M \) moves with steady speed in a circular path of constant radius. Assume that the system is in steady circular motion with constant angular velocity \( \omega \). The angular momentum of \( M \) about point \( A \) is \( L_A \) which lies in the positive \( z \) direction and the angular momentum of \( M \) about point \( B \) is \( L_B \). The correct statement for this system is:

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 _____________.

A uniform circular disc of radius \( 'R' \) and mass \(' M' \) is rotating about an axis perpendicular to its plane and passing through its centre. A small circular part of radius \( \dfrac{R}{2} \) is removed from the original disc as shown in the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above. 

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 \( g = 10 \, \text{m/s}^{2} \))

Two bodies, a ring and a solid cylinder of same material are rolling down without slipping an inclined plane. The radii of the bodies are same. The ratio of velocity of the centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is \( \dfrac{\sqrt{x}}{2} \). Then, the value of \( x \) is __________.

The following bodies,
(\( 1 \)) a ring (\( 2 \)) a disc (\( 3 \)) a solid cylinder (\( 4 \)) a solid sphere of same mass ‘\( m \)’ and radius ‘\( R \)’ are allowed to roll down without slipping simultaneously from the top of the inclined plane. The body which will reach first at the bottom of inclined plane is _______.
[Mark the body as per their respective numbering given in the question]

A, B and C are disc, solid sphere and spherical shell respectively with same radii and masses. These masses are placed as shown in figure.

The moment of inertia of the given system about PQ axis is \( \dfrac{x}{15} I \), where \( I \) is the moment.of inertia of the disc about its diameter. The value of x is ____________.
 

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

In the given figure, two wheels \( P \) and \( Q \) are connected by a belt \( B \). The radius of \( P \) is three times as that of \( Q \). In case of same rotational kinetic energy, the ratio of rotational inertias \( \left( \dfrac{I_{1}}{I_{2}} \right) \) will be \( x : 1 \). The value of \( x \) will be ________.

A \( 2 \, \text{kg} \) steel rod of length \( 0.6 \, \text{m} \) is clamped on a table vertically at its lower end and is free to rotate in vertical plane. The upper end is pushed so that the rod falls under gravity. Ignoring the friction due to clamping at its lower end, the speed of the free end of rod when it passes through its lowest position is _____ \( \text{ms}^{-1} \). (Take \( g = 10 \, \text{ms}^{-2} \))

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 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 __________ \( \text{rad s}^{-2} \).

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 __________.

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 ___________.

Consider a situation in which a ring, a solid cylinder and a solid sphere roll down on the same inclined plane without slipping. Assume that they start rolling from rest and having identical diameter.
The correct statement for this situation 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 \( 10\sqrt{3} \, \text{m} \).)

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 _________ \( \text{rad}/ \text{s}^{2} \).

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 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)

Choose the correct answer from the options given below:

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

A body rolls down an inclined plane without slipping. The kinetic energy of rotation is \( 50\% \) of its translational kinetic energy. The body is:

A solid sphere and a hollow sphere of the same mass and of same radius are rolled on an inclined plane. Let the time taken to reach the bottom by the solid sphere and the hollow sphere be \( t_{1} \) and \( t_{2} \) respectively, then:

The figure shows two solid discs with radius \( R \) and \( r \) respectively. If mass per unit area is same for the both, what is the ratio of MI of bigger disc around \( AB \) (which is \( \perp \) to the plane of the disc and passing through its centre) to MI of smaller disc around one of its diameters lying on its plane? Given 'M' is the mass of the larger disc. (MI stands for moment of inertia)

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:

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 ______.

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: Moment of inertia of a circular disc of mass '\( M \)' and radius '\( R \)' about \( X \), \( Y \) axes (passing through its plane) and \( Z \)-axis which is perpendicular to its plane were found to be \( I_x \), \( I_y \) and \( I_z \) respectively. The respective radii of gyration about all the three axes will be the same.
Reason R: A rigid body making rotational motion has fixed mass and shape.
In the light of the above statements, choose the most appropriate answer from the options given below :

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 uniform thin bar of mass \( 6 \, \text{kg} \) and length \( 2.4 \, \text{meter} \) is bent to make an equilateral hexagon. The moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of hexagon is ______ \( \times 10^{-1} \, \text{kg m}^{2} \).

A rod of linear mass density ‘\( \lambda \)’ and length ‘\( L \)’ is bent to form a ring of radius ‘\( R \)’. Moment of inertia of ring about any of its diameter 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:

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 ______.

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).

A wheel of radius \( 0.2 \, \text{m} \) rotates freely about its center when a string that is wrapped over its rim is pulled by force of \( 10 \, \text{N} \) as shown in figure. The established torque produces an angular acceleration of \( 2 \, \text{rad}/\text{s}^{2} \). Moment of inertia of the wheel is __________ \( \text{kg}\,\text{m}^{2} \). (Acceleration due to gravity \( = 10 \, \text{m}/\text{s}^{2} \))

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 ________.

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 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:

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:

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\() \).

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 _______.

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 \( M \)]

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 \( g = 10 \, \text{m} \, \text{s}^{-2} \)).

Moment of inertia of a rod of mass \( M \) and length \( L \) about an axis passing through its center and normal to its length is '\( \alpha \)'. Now the rod is cut into two equal parts and these parts are joined symmetrically to form a cross shape. Moment of inertia of cross about an axis passing through its center and normal to plane containing cross is: