Rolling Motion

A wheel starts from rest has an angular acceleration that is given by $\alpha =6t^2$. The angle it turns in time $t$ is given as:

Angular momentum and axial velocity of a body of mass $\mathrm{m}$ are related as 

When the torque acting upon a system is zero, which of the following will be constant?

A rod is rotating about one end. If a force ${\mathrm{F}}_1$ acting on the other end produces torque $\mathrm{T}$ and supplies power ${\mathrm{P}}_1$ find the value of force ${\mathrm{F}}_2$ that will produce the same amount of power when it acts at the midpoint of the rod. Assume that all the forces are ${\perp }^{\mathrm{r}}$ to the axis of rotation & axis of rod. The rod starts from rest.

A ring is rotating about a diameter. The radius$=5\mathrm{cm}$ & mass of ring$=1\mathrm{kg}$. A force is applied on the ring such that it is ${\perp }^{\mathrm{r}}$ to the axis and vector $\mathrm{AB}$. The magnitude of force is $10\mathrm{N}\sqrt{2}$ and the work done by the torque, when the ring rates by ${90}^{\mathrm{o}}$

A ring of radius $7 \mathrm{cm}$ is rotating about the central axis ${\perp }^{\mathrm{r}}$ to its plane. A force acts on it tangentially, such that it does a work of $10\mathrm{J}$ in a complete rotation. Find the value of that force.

When the pitching of a ship is upward, the effect of gyroscopic couple acting on it will be

A disc is spinning with an angular velocity $? \raisebox{1ex}{$\mathrm{rad}$}\!\left/ \!\raisebox{-1ex}{$\sec $}\right.$ about the axis of spin. The couple applied to the disc causing precession will be

In an automobile, if the vehicle makes a left turn, the gyroscopic torque

The rotor of a ship rotates in clockwise direction when viewed from the stern and the ship takes a left turn. The effect of the gyroscopic couple acting on it will be 

A disc spinning on its axis at $20 \raisebox{1ex}{$\mathrm{rad}$}\!\left/ \!\raisebox{-1ex}{$\sec $}\right.$ will undergo precession when a torque $100 \mathrm{N}-\mathrm{m}$ is applied about an axis normal to it at an angular speed, if mass moment of inertia of the disc is the $1 \mathrm{kg}-{\mathrm{m}}^2$-

The velocities are in ground frame and the cylinder is performing pure rolling on the plank, velocity of point 'A' would be

A body rolls down an inclined plane. If its kinetic energy of rotation is $40\%$ of its kinetic energy of translation, then the body is

A uniform rod of length $L$ is free to rotate in a vertical plane about a fixed smooth hinge $\left(P\right)$ at one end. The rod is released from horizontal position as shown in the figure. When it has turned through angle $\theta ,$ its angular velocity $\omega$ is

A thin uniform rod of length $l$ and mass $m$ suspended vertically is free to rotate about a smooth horizontal axis which passes through its end A. A point mass $m$ moving horizontally with velocity $v_0$ hits the lower end $B$ of the rod and sticks to it. After collision the rod just reaches the horizontal position. Which of the following statements is WRONG?

Moment of inertia of uniform horizontal solid cylinder of mass $M$ about an axis passing through its edge and perpendicular to the axis of the cylinder when its length is $6$ times its radius $R$ is

A uniform rod $AB$ of length $l$ and mass $m$ is free to rotate about point  $A$. The rod is released from rest in the horizontal position. Given that the moment of inertia of the rod about $A$ is $\frac{m{l}^2}{3},$ the initial angular acceleration of the rod will be :-

A sphere of radius $R$ is moving in $+x$-axis with a velocity $v_C$ and rotates with angular speed $\omega$ as shown below. Assume $v_c>\omega R$ then where will be the instantaneous axis of rotation ?

A uniform rod rotates in a vertical plane about the shown axis. Angular velocity at the moment is $\omega$ . Then, force on the rod by the axis at the moment

A rod of length $L$ is hinged from one end. It is brought to a horizontal position and released. The angular velocity of the road, when it is in vertical position is