Rotational Motion

The angular velocity of a body changes from $6 \mathrm{rad} \mathrm{s}^{-1}$ to $21 \mathrm{rad} \mathrm{s}^{-1}$ in a time of 1.5 s . If the moment of inertia of the body is $\mathrm{g} \mathrm{m}^2$, then the rate of change of angular mom...

The moment of inertia of a solid cylinder of mass 2.5 kg and radius 10 cm about its axis is

A circular dise of diameter 0.8 m and mass 4 kg is rolling on a smooth horizontal plane. If 2.56 N m torque is acting on the disc, then its angular acceleration is

A solid sphere and a solid cylinder have same mass and same radius. The ratio of the moment of inertia of the solid sphere about its diameter and the moment of inertia of the solid cylinder about its axis is

As shown in the figure, two thin coplanar circular discs $A$ and $B$ each of mass $M^{\prime}$ and radius ' $r$ ' are attached to form a rigid body. The moment of inertia of this system about an axis perpendicular to the plane of disc $B$ a...

If a solid sphere is rolling without slipping on a horizontal plane, then the ratio of its rotational and total kinetic energies is

Radius of gyration of a thin uniform rod of length ' $L$ ' about an axis passing through its centre and perpendicular to its length is

A thin circular ring and a circular disc of equal mass are rolling without sliding. If their linear velocities are equal and the total kinetic energy of the disc is 6 J , then the total kinetic energy of the ring is

A circular disc of mass 20 kg and radius 1 m is rotating about an axis passing through its centre and perpendicular to its plane with an angular velocity of $2 \mathrm{rad} \mathrm{s}^{-1}$. Then, the rotational kinetic energy of the disc i...

Three thin uniform rods each of mass $M$ and length $L$ are placed along the three axes of a cartesian co-ordinate system with one end of all the rods at origin. The moment of inertia of the system of the rods about $Z$-axis is

A thin uniform circular disc rolls with a constant velocity without slipping on a horizontal surface. Its total kinetic energy is

A solid sphere of mass 4 kg and radius 28 cm is on an inclined plane. If the acceleration of the sphere when it rolls down without sliding is $3.5 \mathrm{~ms}^{-2}$, then the acceleration of the sphere when it slides down without rolling i...

If a wheel starting from rest is rotating with an angular acceleration of $\pi \mathrm{rad} \mathrm{s}^{-2}$, then the number of rotations made by the wheel in the first 6 seconds time is

If the radius of gyration of a thin circular ring about an axis passing through its centre and perpendicular to its plane is $10 \sqrt{2} \mathrm{~cm}$, then its radius of gyration about its diameter is

A uniform metal solid sphere is rotating with angular speed $\omega_0$ about diameter. If the temperature is raised by $50^{\circ} \mathrm{C}$. The angular speed will be [given $\alpha_{\text {metal }}=20 \times 10^{-5 \circ} \mathrm{C}^{-1...

One ring, one solid sphere and one solid cylinder are rolling down on same inclined plane starting from rest The radius of all the three are equal. The object reaches down with maximum velocity is

The moment of inertia of a solid sphere about its diameter is $20 \mathrm{~kg}-\mathrm{m}^2$. The moment of inertia of a thin spherical shell having the same mass and radius about its diameter is

A uniform rod of length $2 L$ is placed with one end in contact with the earth and is then inclined at an angle $\alpha$ to the horizontal and allowed to fall without slipping at contact point. When it becomes horizontal, its angular veloci...

The moment of inetia of a rod about an axis passing through its centre and perpendicular to its length is $\frac{1}{12} M L^2$, where $M$ is the mass and $L$ is the length of the rod. The rod is bent in the middle, so that the two halves ma...

A ring and a disc of same mass and same diameter are rolling without slipping. Their linear velocities are same, then the ratio of their kinetic energy is

A solid sphere of mass 50 kg and radius 20 cm is rotating about its diameter with an angular velocity d 420 rpm . The angular momentum of the sphere is

The masses of a solid cylinder and hollow cylinder ate 3.2 kg and 1.6 kg respectively. Both the solid and hollow cylinders start from rest from the top of an inclined plane and roll down without slipping. If both the cylinder have equal rad...

A solid cylinder rolls down on an inclined plane of height $h$ and inclination $\theta$. The speed of the cylinder at the bottom is

Three particles of each mass $m$ are kept at the three vertices of an equilateral triangle of side $l$. The moment of inertia of a system of the particles about any side of the triangle is

The moment of inertia of a solid sphere of mass 20 kg and diameter 20 cm about the tangent to the sphere is

A block $P$ is rotating in contact with the vertical wall of a rotor as shown in figures $A, B, C$. The relation between angular velocities $\omega_A, \omega_B$ and $\omega_C$, so that block does not slide down. ( $R_A < R_b < R_c $ radii)

A solid cylinder rolls down an inclined plane without slipping. If the translation kinetic energy of the cylinder is 140 J . The total kinetic energy of the cylinder is

The moment of inertia of a solid cylinder and a hollow cylinder of same mass and same radius about the axes of the cylinders are $I_1$ and $I_2$. The relation between $I_1$ and $I_2$ is

A force of $(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}) \mathrm{N}$ acts on a particle whose position vector with respect to the origin of the coordinate system is $(6 \hat{\mathbf{i}}+b \hat{\mathbf{j}}+12 \hat{\mathbf{k}}) ...

If $\mathbf{F}=(4 \hat{\mathbf{i}}-10 \hat{\mathbf{j}}) \mathrm{N}$ and $\mathbf{r}=(-5 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}) \mathrm{m}$, then $(\mathbf{r} \times \mathbf{F})$ is

The angular speed of a rigid body rotating about a fixed axis is $(8-2 t) \mathrm{rad} \mathrm{s}^{-1}$. The angle through which the body rotates before it comes to rest is

A solid cylinder of radius \(R\) is at rest at a height \(h\) on an inclined plane. If it rolls down then its velocity on reaching the ground is

Consider a disc of radius \(R\) and mass \(M\). A hole of radius \(\frac{R}{3}\) is created in the disc, such that the centre of the hole is \(\frac{R}{3}\) away from centre of the disc. The moment of inertia of the system along the axis pe...

A solid sphere of radius \(R\) has its outer half removed, so that its radius becomes \((R / 2)\). Then its moment of inertia about the diameter is

If an energy of 684 J is needed to increase the
speed of a flywheel from 180 rpm to
360 rpm, then find its moment of inertia.

Four spheres each of diameter \(2 a\) and mass \(m\) are placed in a way that their centers lie on the four corners of a square of side \(b\). Moment of inertia of the system about an axis along one of the sides of the square is

A sphere and a hollow cylinder without
slipping, roll down two separate inclined
planes A and B, respectively. They cover same
distance in a given duration. If the angle of
inclination of plane A is 30\(^\circ\), then the angle
of inclinati...

Two fly wheels \(A\) and \(B\) are mounted side by side with frictionless bearings on a common shaft. Their moments of inertia about the shaft are \(5.0 \mathrm{~kg}-\mathrm{m}^2\) and \(20.0 \mathrm{~kg}-\mathrm{m}^2\), respectively. Wheel...

As solid sphere of mass \(M\) and radius \(R\) spins about an axis passing through its centre making \(600 \mathrm{~rpm}\). Its kinetic energy of rotation is

Which of the following type of wheels of
same mass and radius will have largest
moment of inertia?

A girl of mass M stands on the rim of a
frictionless merry-go-round of radius R and
rotational inertia I, that is not moving. She
throws a rock of mass m horizontally in a
direction that is tangent to the outer edge of
the merry-go-round. T...

A sphere of mass m is attached to a spring of
spring constant k and is held in unstretched
position over an inclined plane as shown in
the figure. After letting the sphere go, find
the maximum length by which the spring
extends, given the s...