Angular Analogue of Linear Momentum

Q. The gravitational field due to a disc of mass $M$ and radius $R$ at a point located $x$ distance away from the centre of disc along the axis of disc is given by

1. $\frac{2GMx}{R^2}[\frac{1}{x}-\frac{1}{\sqrt{R^2+x^2}}]$
2. $\frac{2GMx}{R^2}\left[\frac{1}{\sqrt{R^2+x^2}}\right]$
3. $\frac{GMx}{R^2}\left[\frac{1}{\sqrt{R^2+x^2}}\right]$
4. $\frac{GMx}{R^2}[\frac{1}{x}-\frac{1}{\sqrt{R^2+x^2}}]$

Q.

Four identical solid spheres each of mass $m$ and radius $a$ are placed with their centres on the four corners of a square of side $b$. The moment of inertia of the system about one side of square where the axis of rotation is parallel to the plane of the square is:

1. $\frac{4}{3}m{a}^2$

2. $\frac{8}{5}m{a}^2+m{b}^2$

3. $\frac{4}{5}m{a}^2+2m{b}^2$

4. $\frac{8}{5}m{a}^2+2m{b}^2$

Q. A bob of mass m attached to an inextensible string of length l is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed $\omega rad/S$ about the vertical. About the point of suspension,

1. angular momentum changes in direction but not in magnitude.
2. angular momentumchanges both in direction and in magnitude.
3. angular momentum is conserved.
4. angular momentum changes in magnitude but not in direction.

Q. Paragraph
A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity $\omega$ is an example of a non-inertial frame of reference. The relationship between the force ${\overrightarrow{F}}_{\text{rot}}$ experience by a particle of mass $m$ moving on the rotating disc and the force ${\overrightarrow{F}}_{\text{in}}$ experienced by the particle in an inertial frame of reference is
${\overrightarrow{F}}_{\text{rot}}={\overrightarrow{F}}_{\text{in}}+2m({\overrightarrow{v}}_{\text{rot}}\times \overrightarrow{\omega })+m(\overrightarrow{\omega }\times \overrightarrow{r})\times \overrightarrow{\omega },$

where ${\overrightarrow{v}}_{\text{rot}}$ is the velocity of the particle in the rotating frame of reference and $\overrightarrow{r}$ is the position vector of the particle with respect to the centre of the disc. Now consider a smooth slot along a diameter of a disc of radius $R$ rotating counterclockwise with a constant angular speed $\omega$ about its vertical axis through its center. We assign a coordinate system with the origin at the center of the disc, the $x-$axis along the slot, the $y-$ axis perpendicular to the slot and the $z-$ axis along the rotation axis $\overrightarrow{\omega =\omega \overrightarrow{k}}$. A small block of mass $m$ is gently placed in the slot at $\overrightarrow{r}=\left(R/2\right)\overrightarrow{i}$ at $t=0$ and is constrained to move only along the slot.


The distance $r$ of the block at time $t$ is

1. $\frac{R}{2}\cos 2\omega t$
2. $\frac{R}{2}\cos \omega t$
3. $\frac{R}{4}(e^{\omega t}+e^{-\omega t})$
4. $\frac{R}{4}(e^{2\omega t}+e^{-2\omega t})$

Q.

A circular race track of radius 300 m is banked at an angle of 15 degrees. If the coefficient of friction between the wheels of a race car and the road is 0.2, what is the optimum speed of the race car to avoid wear and tear on its tires ?

Q.

Explain circular motion of car on a level road for 5 marks

Q.

The moment of inertia is constant, the time period is doubled, and what happens to the angular momentum of the body?

Q. A mass $m$ is moving with a constant velocity $v$ along a line parallel to $X-$ axis, away from origin. Then, its angular momentum with respect to the origin

1. is zero
2. remains constant
3. goes on increasing
4. goes on decreasing

Q. What is the direction of angular acceleration?

Q.

A body is moving in a low circular orbit about a planet of mass $M$ and radius $R$. The radius of the orbit can be taken to be $R$ itself. Then the ratio of the speed of this body in the orbit to the escape velocity from the planet is:

1. $2$

2. $\sqrt{2}$

3. $1$

4. $\frac{1}{\sqrt{2}}$

Q. If the time of flight of a bullet over a horizontal range R is T, then the inclination of the direction of projection with the horizontal is

1. $ta{n}^{-1}\left[\frac{g{T}^2}{2R}\right]$
   
2. $ta{n}^{-1}\left[\frac{2R^2}{gT}\right]$
   
3. $ta{n}^{-1}\left[\frac{2R}{g^2\pi }\right]$
   
   
4. $ta{n}^{-1}\left[\frac{2R}{gT}\right]$

Q. A particle of mass $m$ is projected with velocity $v$ at an angle $\theta$ with the horizontal. Find its angular momentum about the point of projection when it is at the highest point of its trajectory.

1. $\frac{m{v}^2{\sin }^2\theta \cos \theta }{2g}$
2. $\frac{m{v}^3\sin \theta \cos \theta }{g}$
3. $\frac{m{v}^3{\sin }^2\theta \cos \theta }{2g}$
4. $\frac{m{v}^3{\sin }^2\theta \cos \theta }{g}$

Q. A mass $m$ is placed at point $\text{P}$ at a distance $h$ along the normal through the centre $\text{O}$ of a thin circular ring of mass $M$ and radius $r$ as shown in figure. If the mass is moved further away such that, $\text{OP}$ becomes $2h$, by what factor, the force of gravitation will decrease, if $h=r$

1. $\frac{3\sqrt{2}}{4\sqrt{3}}$
2. $\frac{5\sqrt{2}}{\sqrt{3}}$
3. $\frac{4\sqrt{3}}{5}$
4. $\frac{4\sqrt{2}}{5\sqrt{5}}$

Q. The position of a particle is given by $\overrightarrow{r}=(\hat{i}+2\hat{j}-\hat{k})$ and momentum $\overrightarrow{P}=(3\hat{i}+4\hat{j}-2\hat{k})$. The direction of angular momentum is perpendicular to

1. X-axis
2. Y-axis
3. Z-axis
4. Line at equal angles to all the three axes

Q. The time dependence of the position of a particle of mass $2kg$ is given by $\overrightarrow{r}\left(t\right)=2t\hat{i}-3t^2\hat{j}\text{m}$. Its angular momentum with respect to origin at time $t=2s$ is given by $-N\hat{k}{\text{kg.m}}^2\text{/s}$, the value of $N$ is

Q.

A flywheel of moment of inertia $5.0kg-m^2$ is rotated at a speed of 60rad/s. Because of the friction at the axle, it comes to rest in 5.0 minutes.

Find

(a) the average torque of the friction,

(b) the total work done by the friction and

(c) the angular momentum of the wheel 1 minute before it stops rotating.

Q. A particle is moving along a straight line.Find the magnitude of angular momentum about origin when particle is at minimum distance from origin.

1. $12{\text{kg m}}^2\text{/s}$
2. $6{\text{kg m}}^2\text{/s}$
3. $8{\text{kg m}}^2\text{/s}$
4. $10{\text{kg m}}^2\text{/s}$

Q.

A particle is projected at time t =0 from a point P with a speed $v_0$ at an angle of 45${}^{\circ }$ to the horizontal. Find the magnitude and the direction of the angular momentum of the particle about the point P at time t =$\frac{v_0}{g}$.

1. 

2. 

3. 

4. 

Q. Two bodies have their moments of inertia $I$ and $2I$ respectively about their axis of rotation. If their kinetic energies of rotation are equal, their angular momenta will be in the ratio:

1. $2:1$
2. $1:2$
3. $1:\sqrt{2}$
4. $\sqrt{2}:1$

Q.

A wheel starting from rest is uniformly accelerated at 4 rad/s${}^2$ for 10 seconds . It is allowed to rotate uniformly for the next 10 seconds and is finally brought to rest in the next 10 seconds. Find the total angle rotated by the wheel.

Q.

Two particles of mass m each are attached to a light rod of length d, one at its centre and the other at a free end. The rod is fixed at the other end and is rotated in a plane at an angular speed $\omega$. Calculate the angular momentum of the particle at the end with respect to the particle at the centre.

1. 

2. 

3. 

4. 

Q.

A particle of mass m moves along line PC with velocity v as shown. What is the angular momentum of the particle about P?

1. $mvL$

2. $mvI$

3. $mvr$

4. $Zero$

Q. A thin rod of length $L$ is bent to form a circle. Its mass is $M$. What force will act on the mass $m$ placed at the centre of the circle?

1. $\frac{4{\pi }^{2}GMm}{L^2}$
2. $\frac{GMm}{4{\pi }^2{L}^2}$
3. $\frac{2\pi GMm}{L^2}$
4. zero

Q. The point masses having mass $m$ and $2m$ are placed at distance $d$ from each other. The point on the line joining the point masses, where gravitational field intensity is zero will be at distance

1. $\frac{2d}{\sqrt{3}+1}$ from point mass $2m$.
2. $\frac{2d}{\sqrt{3}-1}$ from point mass $2m$.
3. $\frac{d}{1+\sqrt{2}}$from point mass $m$.
4. $\frac{d}{1-\sqrt{2}}$from point mass $m$.

Q. The masses of the blocks A and B are $0.5\text{kg}$ and $1\text{kg}$ respectively. These are arranged as shown in the figure and are connected by a massless string. The coefficient of friction between all contact surfaces is $0.4$. The force, necessary to move the block B with constant velocity, will be $(g=10{\text{m/s}}^2)$

1. $5\text{N}$
2. $10\text{N}$
3. $15\text{N}$
4. $20\text{N}$

Q. Statement - $\text{I}$: $\overrightarrow{L}=I\overrightarrow{\omega }$ is always true for bodies of all shapes.
Statement- $\text{II}$: $\overrightarrow{\tau }=\frac{d\overrightarrow{L}}{dt}$ is always true in inertial frames.

1. Statement $\text{I}$ is true, statement $\text{II}$ is true and statement$\text{II}$ is correct explanation for statement $\text{I}$
2. Statement $\text{I}$ is true, statement $\text{II}$ is true but statement $\text{II}$ is not correct explanation for statement $\text{I}$
3. Statement $\text{I}$ is false but statement $\text{II}$ is true.
4. Statement $\text{I}$ is true but statement $\text{II}$ is false.

Q. Four balls $A,B,C$ and $D$ are placed on a smooth horizontal surface. Ball $B$ collides with ball $C$ first with an initial velocity $V.$ if all collisions are perfectly elastic then total number of collisions among balls are:-

1. $10$
2. $4$
3. $3$
4. Cannot be determined

Q. A uniform solid disc, is performing pure rolling on a fixed horizontal surface as shown in figure.

O, A and B are three points lying on y-axis where as C is top most point and D is bottom most point of the disc at the instant shown. Choose the correct option(s).

1. Angular momentum of disc with respect to C and D are same
2. Magnitude of angular momentum of disc with respect to A (among O, A, B, C and D) is minimum
3. Magnitude of angular momentum of disc with respect to B (among O, A, B, C and D) is minimum
4. Angular momentum with respect to O and C are same

Q. A uniform disc of mass $m$ radius $R$ translating with a speed $v$ on a smooth horizontal table is hinged at one point on the circumference all of a sudden. The angular velocity of the disc is

1. $\frac{2v}{R}$
2. $\frac{v}{3R}$
3. $\frac{2v}{3R}$
4. 0

Q. . The is a non-conducting rod of length L and negligible mass with two small balls each ofmass m and electric charge Q attached to its ends.The rod can rotate in the horizontal plane about fixed vertical axis crossing it a distancefromone of its ends. A uniform horizontal electric field E(along +x axis) is established. At first the rod is in unstable equilibrium. If it is disturbed slightly from this position, the maximum velocity attained by the ball which is closer to the axis in subsequent motion is