Angular Momentum in Case of Rotation about a Fixed Axis

 A man standing on a turn-table is rotating at a certain angular frequency with his arms outstretched. He suddenly folds his arms. If his moment of inertia with folded arms is$75\%$of that with outstretched arms, his rotational kinetic energy will

When a body is spinning on its axis in absence of any external torque, then choose the wrong statement

Earth shrinks to $\frac{1}{64}$ times of its initial volume. Time period of Earth rotation is found to be $\frac{24}{x} \mathrm{hr}$. Find the value of $x$.

Angular momentum of a system of particles changes when

Two identical solid spheres are rolling without slipping with velocity$v_0,$ over a rough horizontal surface collides elastically as shown. The final speed of centre of mass of sphere (ii) after sufficient long time will be

A ring of mass $M$and radius $R$ is rolling without slipping, the velocity of point$A$ as shown in the figure 

A small satellite of mass $m$ is revolving round a planet in circular orbit of radius $r$ and angular momentum $L$ about the centre of the planet. Kinetic energy of the satellite is given by $\frac{L^x}{xm{r}^x}$. Find $x$.

A homogeneous rod AB of length $L$ and mass $M$ is pivoted at the center O in such a way that it can rotate freely in the vertical plane. The rod is initially in the horizontal position. An insect S of the same mass $M$ falls vertically with speed $v$ on the point C mid-way between O and B. Determine the initial angular velocity $\omega$ in terms of $v$ and $L$.

 $$\begin{gathered} Find x if \\ \overrightarrow{r}= (3\hat{i}+\hat{j}) \mathrm{m} \\ v = (3\hat{j}-\hat{k}) {\mathrm{ms}}^{-1} \\ m=1 \mathrm{kg} \\ \left|\overrightarrow{\mathrm{L}}\right|=\sqrt{\mathrm{x}} \mathrm{Nm} \end{gathered}$$

Three uniform rods $AB,BC$ and $CD$ each of mass m and length $l$ are moving together with velocity $v_0$ on a smooth horizontal surface (direction of $v_0$ is perpendicular to length $ABCD$) The roads $AB$ and $CD$ are hinged at $B$ and $C$ respectively so that they can rotate freely about $B$ and $C$. At an instant during the motion, the middle road $BC$ is suddenly fixed on the horizontal surface. Find the time taken for the ends $A$ and $D$ to meet.

A particle moves in a circular path with constant speed. Find out a point about which the angular momentum of the particle is constant and another point about which it changes with time.

 A cat is at rest on a horizontal table mounted on a vertical axis. If the cat begins to walk around the perimeter, what happens to the table ?  Explain.

"It is easier to keep your balance on a moving bicycle than on a bicycle at rest ". Why ?

Explain the law of conservation of angular momentum.

Calculate the angular momentum and rotational kinetic energy of the earth about its own axis. Given mass of the earth is $6\times {10}^{24} \mathrm{kg}$ and its radius$=6.4\times {10}^3 \mathrm{km}$ [Moment of inertia of the earth about its axis of rotation is $\frac{2}{5}m{r}^2$]

A heavy circular disc is revolving in a horizontal plane about the centre which is fixed. An insect of mass $\frac{1}{n}th$  that of the disc walks from the centre along a radius and then flies away. Show that the final angular velocity is $\frac{n}{n+2}$ times the original angular velocity of the disc.

The mass of a star is $3.98\times {10}^{30} \mathrm{kg}$ and its radius is ${10}^6 \mathrm{km}$. It rotates about its axis with an angular speed of ${10}^{-6} \mathrm{rad} {\mathrm{s}}^{-1}$. Calculate the speed of the star when it collapses to a radius of ${10}^4 \mathrm{km}$?  Assume the moment of inertia of the star to be $\frac{2}{5}m{r}^2$.

The maximum and minimum distance of a comet from the sun are $1.4\times {10}^{12} \mathrm{m}$  and $7\times {10}^{10} \mathrm{m}$. If its velocity nearest to the sun is $6\times {10}^4 \mathrm{m} {\mathrm{s}}^{-1}$, what is the velocity in the farthest position?  Assume a circular path for the comet.

If earth suddenly contracts by $\frac{1}{2}$ of its present radius by how much would the day be decreased?

A small body of mass $100 \mathrm{g}$ moving with a velocity $2 \mathrm{m} {\mathrm{s}}^{-1}$ is strikes a solid cylinder of mass $2 \mathrm{kg}$ and radius $20 \mathrm{cm}$. The cylinder is initially at rest and is mounted on a fixed horizontal axle that passes through the centre of mass. The line of motion of the small body is at a perpendicular distance of $15 \mathrm{cm}$ from the centre of the cylinder. The small body strikes the cylinder and adheres to its surface. What is the angu­lar speed of the system just after the collision? Assume there is no friction.