Rotational Mechanics — JEE Main PYQs

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 $\frac{X}{5}$. The value of $x$ 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:

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.

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 $\mathrm{M}/2$ is placed gently on th …

A circular table is rotating with an angular velocity of $\omega$ 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 m on the groove. All the surface are smooth. If the radius of th …

A string is wrapped around the rim of a wheel of moment of inertia $0.40{\mathrm{kgm}}^2$ and radius 10 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 N. The angular velocity of the wheel after 10 s is …

A circular disc reaches from top to bottom of an inclined plane of length $l$. When it slips down the plane, if takes t s . When it rolls down the plane then it takes ${\left(\frac{\alpha }{2}\right)}^{1/2}\mathrm{ts}$, where $\alpha$ 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 $\frac{7}{x}$ where $x$ is ____.

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

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

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

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

The dimensional formula of angular impulse is :

Two discs of moment of inertia ${\mathrm{I}}_1=4\mathrm{kg}{\mathrm{m}}^2$ and ${\mathrm{I}}_2=2\mathrm{kg}{\mathrm{m}}^2$ about their central axes & normal to their planes, rotating with angular speeds $10\mathrm{rad}/\mathrm{s}$ & …

Consider a Disc of mass 5 kg, radius 2 m, rotating with angular velocity of 10 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 …

A body of mass 'm' is projected with a speed 'u' making an angle of $45^{\circ }$ with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as $\frac{\sqrt{2}{\mathrm{mu}}^3}{\mathrm{Xg}}$. The value of 'X' is ______.

Two identical spheres each of mass 2 kg and radius 50 cm are fixed at the ends of a light rod so that the separation between the centers is 150 cm. Then, moment of inertia of the system about an axis perpendicular to the rod and passing through its middle point is …

The orbital angular momentum of a satellite is L, when it is revolving in a circular orbit at height h from earth surface. If the distance of satellite from the earth centre is increased by eight times to its initial value, then the new angular momentum will be-

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

A ring and a solid sphere rotating about an axis passing through 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 $\sqrt{\frac{2}{x}}$. The value of $x$ is ______

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

If the earth suddenly shrinks to $\frac{1}{64}$ th of its original volume with its mass remaining the same, the period of rotation of earth becomes $\frac{24}{x}h$. The value of $x$ is ____ .

A solid sphere of mass 500 g and radius 5 cm is rotated about one of its diameter with angular speed of $10{\mathrm{rad}\mathrm{s}}^{-1}$. 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 $x$ wi …

For a rolling spherical shell, the ratio of rotational kinetic energy and total kinetic energy is $\frac{\mathrm{x}}{5}$. The value of $x$ is ____ .

A disc is rolling without slipping on a surface. The radius of the disc is R . At $\mathrm{t}=0$, 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 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 ______ $\mathrm{rad}/\mathrm{s}$.

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

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 ($k_{\text{sph}}:k_{\text{cyl}}$) is $2:\sqrt{x}$, then value of $x$ is ______.

Two identical solid spheres each of mass 2 kg and radii 10 cm are fixed at the ends of a light rod. The separation between the centres of the spheres is 40 cm. The moment of inertia of the system about an axis perpendicular to the rod passing through its middle point is …

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 moment of inertia of t …

A solid cylinder is released from rest from the top of an inclined plane of inclination $30^{\circ }$ and length 60 cm. If the cylinder rolls without slipping, its speed upon reaching the bottom of the inclined plane is ______ ${\mathrm{ms}}^{-1}$. (Given …

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

A uniform solid cylinder with radius R and length L has moment of inertia ${\mathrm{I}}_1$, about the axis of cylinder. A concentric solid cylinder of radius ${\mathrm{R}}^{\prime }=\frac{\mathrm{R}}{2}$ and length ${\mathrm{L}}^{\prime }=\frac{\mathrm{L}}{2}$ is carved out of th …

An object of mass 8 kg is hanging from one end of a uniform rod CD of mass 2 kg and length 1 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 …

$I_{CM}$ is 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 its moment of inertia about an axis AB perpendicular to plane and parallel to axis CM at a distance $\frac{2}{3}R$ from center. Wher …

If a solid sphere of mass 5 kg and a disc of mass 4 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 $\frac{x}{7}$. The value of $x$ is ____.

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

A particle of mass 100 g is projected at time $t=0$ with a speed $20{\text{ms}}^{-1}$ 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 …

A uniform disc of mass 0.5 kg and radius r is projected with velocity 18 m/s at t=0 s on a rough horizontal surface. It starts off with a purely sliding motion at t=0 s. After 2 s it acquires a purely rolling motion (see figure). The total kinetic energy of the disc after 2 s wil …

A thin uniform rod of length 2 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 …

A solid sphere of mass 1 kg rolls without slipping on a plane surface. Its kinetic energy is $7\times 10^{-3}\mathrm{J}$. The speed of the centre of mass of the sphere is $\_\_\_\_$ $\mathrm{cm}{\mathrm{s}}^{-1}$.

Two discs of same mass and different radii are made of different materials such that their thicknesses are 1 cm and 0.5 cm respectively. The densities of materials are in the ratio 3:5. The moment of inertia of these discs respectively about their diameters will be in the ratio o …

Moment of inertia of a disc of mass M and radius 'R' about any of its diameter is $\frac{M{R}^2}{4}$. The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, $\frac{x}{2}M{R}^2$. 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 is $2\hat{i}+2\hat{j}+\hat{k}$, then the value of $\tau$ will be :

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${}^{\prime }$ is $\frac{x}{4}M{a}^2$. Then, the value of $x$ will be _____.

A pulley of radius 1.5 m is rotated about its axis by a force $F=\left(12t-3t^2\right)N$ applied tangentially (while $t$ is measured in seconds). If moment of inertia of the pulley about its axis of rotation is $4.5\mathrm{kg}{\mathrm{m}}^2$, the number of rotations made …

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 radius of gyration of a cylindrical rod about an axis of rotation perpendicular to its length and passing through the center will be ______ m.

Given, the length of the rod is $10\sqrt{3}$ m.

A disc of mass 1 kg and radius R is free of 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 lowe …

A uniform disc with mass $\mathrm{M}=4\mathrm{kg}$ and radius $\mathrm{R}=10\mathrm{cm}$ is mounted on a fixed horizontal axle as shown in figure. A block with mass $\mathrm{m}=2\mathrm{kg}$ hangs from a massless cord that is wrapped around the rim of the disc. During …