Angular Momentum and its Conservation

A planet of mass $m$ is in an elliptical orbit about the sun $\left(m<<M_{\text{sun}}\right)$ with an orbital period $T$. If $A$ be the area of orbit, then its angular momentum would be

On the rotating disc $A$ of mass$M$, another disc of same dimension but of mass$\frac{ M}{4}$is placed gently with same axis. New angular velocity of system.

A particle of mass $1 \mathrm{kg}$ is moving along the line $y = x + 3$ ($x$ and$y$ are in meter) with speed $3 \mathrm{m} {\mathrm{s}}^{-1}$. The magnitude of angular momentum of the particle about origin is

 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

A bar of mass $M=1.00 \mathrm{kg}$ and length $L=0.20 \mathrm{m}$ is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. A small mass $m=0.10 \mathrm{kg}$ is moving on the same horizontal surface with $5.00 \mathrm{m} {\mathrm{s}}^{–1}$ speed on a path perpendicular to the bar. It hits the bar at a distance $\frac{L}{2}$ from the pivoted end and returns back on the same path with speed $v$. After this elastic collision, the bar rotates with an angular velocity $\omega$. Which of the following statement is correct?

If the earth suddenly shrinks to $\frac{1}{64}\mathrm{th}$ 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 rolls on a horizontal plane. The ratio of angular momentum about COM to total energy is $\frac{\pi }{22}$. The angular speed of the sphere is $x \mathrm{rad} {\mathrm{s}}^{-1}$. Find the value of $x$

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 sphere is released on a smooth inclined plane from the top. When it moves down its angular momentum is:
 

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 $x$.

The angular momentum of a particle with respect to the origin will not be zero, if

Two persons stand at the edges of a rotating circular platform at diametrically opposite points. If they start moving towards each other at uniform velocity, then its

A uniform solid cylinder of mass $1 \mathrm{kg}$ and radius $20 \mathrm{cm}$ rotates about a fixed vertical frictionless axle with an angular speed of $50 \mathrm{rad}/\mathrm{s}$. A particle of mass $0.5 \mathrm{kg}$ now sticks on cylinder at a distance of $15 \mathrm{cm}$ from the axle. Magnitude of change in angular velocity (in $\mathrm{rad}/\mathrm{s}$) of the cylinder is _____ .

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 $L$.

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

A uniform solid cylinder of mass $1 \mathrm{kg}$ and radius $20 \mathrm{cm}$ rotates about a fixed vertical frictionless axle with an angular speed of $50\mathrm{rad} {\mathrm{s}}^{-1}$. A particle of mass $0.5 \mathrm{kg}$ now sticks on cylinder at a distance of $15 \mathrm{cm}$ from the axle. Magnitude of change in angular velocity (in $\mathrm{rad} {\mathrm{s}}^{-1}$) of the cylinder is