The Principle

Q. A thin circular ring of mass $M$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Two objects, each of mass $m$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity

1. $\frac{\omega M}{m+M}$
2. $\omega \left(\frac{M+2m}{M}\right)$
3. $\frac{\omega (M-2m)}{M+2m}$
4. $\frac{\omega M}{M+2m}$

Q. A circular platform is mounted on a frictionless vertical axle. Its radius R = 2 m and its moment of inertia about the axle is 200 kg $m^2.$ It is initially at rest. A 50 kg man stands on the edge of the platform and begins to walk along the edge at the speed of 1 $m{s}^{-1}$ relative to the ground. Time taken by the man to complete one revolution is

1. $\pi s$
2. $\frac{\pi }{2}s$
3. $\frac{3\pi }{2}s$
4. $2\pi s$

Q. A thin uniform circular disc of mass $M$ and radius $R$ is rotating with an angular velocity $\omega ,$ in a horizontal plane about an axis passing through its centre and perpendicular to its plane. Another disc of same dimensions but of mass $\frac{M}{4}$ is placed gently on the first disc co-axially. The final angular velocity of the system is

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

Q.

A particle of mass $10kg$ moves along a circle of radius $6.4cm$ with a constant tangential acceleration. What is the magnitude of this acceleration if the kinetic energy of the particle becomes equal to $8\times {10}^{-4}J$ at the end of the second revolution after the beginning of the motion?

Q. A thin horizontal circular disc is rotating about a vertical axis passing through its centre. An insect is at rest at a point near the rim of the disc. The insect now moves with constant speed $v$ along a diameter of the disc to reach it's other end. During the journey of the insect, the angular speed of the disc will

1. Remains unchanged
2. Continously decreases
3. Continously increases
4. First increase and then decrease.

Q. Two discs are rotating about their axes, normal to the plane of the discs and passing through the centre of the discs. Disc $D$ has $2\text{kg}$ mass and $0.2\text{m}$ radius and initial angular velocity of $50\text{rad/s}$. Disc $D_2$ has $4\text{kg}$ mass, $0.1\text{m}$ radius and initial angular velocity of $200\text{rad/s}$. The two discs are brought in contact face to face with their axes of rotation coincident. The final angular velocity (in $\text{rad/s}$) of the system is

1. $50\text{rad/s}$
2. $20\text{rad/s}$
3. $100\text{rad/s}$
4. $200\text{rad/s}$

Q. A force $\overrightarrow{F}=\alpha \hat{i}+3\hat{j}+6\hat{k}$ is acting at a point $\overrightarrow{r}=2\hat{i}-6\hat{j}-12\hat{k}$. Find the value of $\alpha$ for which angular momentum about origin is conserved.

1. $\text{Zero}$
2. $1$
3. $-1$
4. $2$

Q. A cubical block of side $a$ is moving with velocity $v$ on a horizontal smooth plane as shown in the figure. It hits a ridge at point $O$. The angular speed of the block after it hits $O$ is

1. $\frac{3v}{4a}$
2. $\frac{3v}{2a}$
3. $\sqrt{\frac{3}{2}}a$
4. zero

Q. A horizontal disc rotating freely about a vertical axis through its centre makes $90$ revolutions per minute. A small piece of wax of mass $m$ falls vertically on the disc and sticks to it at a distance $r$ from the axis. If the number of revolutions per minute reduces to $60$, the moment of inertia of the disc is

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

Q.

A shell at rest at the origin explodes into three fragments of masses$1kg$, $2kg$ and $mkg$. The$1kg$and$2kg$ pieces fly off with speeds of $5m{s}^{-1}$ along the $x$-axis and$6m{s}^{-1}$along the $y$-axis respectively. If the$mkg$ piece flies off with a speed of $6.5m{s}^{-1}$, the total mass of the shell must be

1. $4kg$

2. $5kg$

3. $3.5kg$

4. $4.5kg$

5. $5.5kg$

Q. The angular speed of a body changes from ${\omega }_1$ to ${\omega }_2$ due to changes in its moment of inertia without applying torque. The ratio of radius of gyration in the two cases is

1. ${\omega }_1:{\omega }_2$
2. $\sqrt{{\omega }_2}:\sqrt{{\omega }_1}$
3. $\sqrt{{\omega }_2^2}:\sqrt{{\omega }_1^2}$
4. $\sqrt{{\omega }_2^3}:\sqrt{{\omega }_1^3}$

Q. A particle moves through angular displacement $\theta$ on a circular path of radius ${}^{\prime }{\text{r}}^{\prime }$. The linear displacement will be :

1. $2\text{r}\sin \left(\frac{\theta }{2}\right)$
2. $2\text{r}\cos \left(\frac{\theta }{2}\right)$
3. $2\text{r}\tan \left(\frac{\theta }{2}\right)$
4. $2\text{r}\cot \left(\frac{\theta }{2}\right)$

Q.

Why are passengers travelling in a double decker bus allowed to stand in a lower deck, but not in the upper deck?

Q.

Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies ${\omega }_1$ and ${\omega }_2$ and have total energies $E_1$ and $E_2$, respectively. The variations of their momenta $p$ with positions $x$ are shown in figures. If $\frac{a}{b}=n^2$ and $\frac{a}{R}=n$, then the correct equation(s) is (are)

1. $E_1{\omega }_1=E_2{\omega }_2$
2. 
   $\frac{{\omega }_2}{{\omega }_1}=n^2$
3. 
   ${\omega }_1{\omega }_2=n^2$
4. 
   $\frac{E_1}{{\omega }_1}=\frac{E_2}{{\omega }_2}$

Q. If external torque acting on the system along $x-$ axis add up to zero, then

1. Angular momentum of system along $x-$ axis is always zero.
2. Angular momentum of system along $y-$ axis remains constant.
3. Angular momentum of system along $z-$ axis remains constant
4. Angular momentum of system along $x-$ axis remain constant.

Q. A turntable rotates about a fixed vertical axis, making one revolution in $10\text{s}$. The moment of inertia of the turntable about the axis of rotation is $1200{\text{kg.m}}^2$. A man of mass $80\text{kg}$, initially standing at the centre of the turntable, runs outwards along the radius. What is the approximate angular velocity of the turntable when the man is at a distance of $2\text{m}$ from the centre?

1. $2.75\text{rad/s}$
2. $1.25\text{rad/s}$
3. $0.50\text{rad/s}$
4. $0.1\text{rad/s}$
   

Q. The angular momentum of a particle about the origin is varying as $L=4\sqrt{2}t+8$ (SI units) when it moves along a straight line$y=x-4$ ($x,y$ in meters). The magnitude of force (in $\text{N}$) acting on the particle would be

Q. If two discs of moments of inertia $I_1$ and $I_2$, rotating about a collinear axis passing through their centres of mass and perpendicular to their planes with angular speeds ${\omega }_1$ and ${\omega }_2$ respectively in opposite directions are made to rotate combinedly along the same axis, then the magnitude of new angular velocity of the system is

1. $\frac{I_1{\omega }_1+I_2{\omega }_2}{I_1+I_2}$
2. $\frac{I_1{\omega }_1-I_2{\omega }_2}{I_1+I_2}$
3. $\frac{I_1{\omega }_2+I_2{\omega }_1}{I_1+I_2}$
4. $\frac{I_1{\omega }_1-I_2{\omega }_2}{I_1-I_2}$

Q. A man of mass $M=50\text{kg}$ standing on the edge of a platform (moment of inertia of the platform is $I$ and radius $R=1\text{m}$) is rotating in anticlockwise direction at an angular speed of $20\text{rad/s}$. The man starts walking along the rim with a speed $1\text{m/s}$ relative to the platform, also in the anticlockwise direction. The new angular speed of the platform in $\left(\text{rad/s}\right)$ is (take $(I=0.02{\text{kg-m}}^2)$)

Q. A dancer is rotating on a smooth horizontal floor with an angular momentum $L$. The dancer folds her hands so that her moment of inertia decreases by $25\%$. The new angular momentum is

1. $\frac{L}{4}$
2. $\frac{L}{2}$
3. $L$
4. $\frac{L}{8}$

Q. If earth suddenly shrinks and its radius becomes half, then what will be the duration of a day?

1. 3 hour
2. 6 hour
3. 9 hour
4. 12 hour

Q. Two cylinders with radii $r_1$ and $r_2$ and rotational inertia $I_1$ and $I_2$ are supported on their horizontal axles. The first one is set in rotation with angular velocity $\omega$. The axle of the other cylinder (smaller) is moved until it touches the large cylinder and is caused to rotate by the frictional forces between the two. Find the angular velocity of the two cylinders when slipping ceases finally because of the frictional forces between them.

1. ${\omega }_1=\frac{I_1{r}_2^2\omega }{I_1{r}_2^2+I_2{r}_1^2}$
2. ${\omega }_1=\frac{I_1{r}_2^2\omega }{I_1{r}_2^2-I_2{r}_1^2}$
3. ${\omega }_2=\frac{I_1\omega r_1}{I_1{r}_2-I_2{r}_1}$
4. ${\omega }_2=\frac{I_1\omega r_2}{I_1{r}_2-I_2{r}_1}$

Q. A horizontal platform is rotating with uniform angular velocity around the vertical axis passing through its centre. At some instant of time a non-viscous fluid of mass $m$ is dropped at the centre and is allowed to spread out and finally fall. The angular velocity during this period will

1. decrease continuously
2. decrease initially and will increase again
3. remains same
4. increase continuously

Q. A sphere is released from the top of a smooth inclined plane . When it moves downwards, its angular momentum will be

1. conserved about any point
2. conserved about the point of contact only
3. conserved about the centre of the sphere only
4. conserved about any point on a fixed line parallel to the inclined plane and passing through the centre of the sphere

Q. A meter stick is pivoted about its centre. A piece of wax of mass $20\text{g}$ travelling horizontally and perpendicular to it at $5\text{m/s}$ strikes and adheres to one end of the stick which starts to rotate in a horizontal circle. Given, the moment of inertia of the stick and wax about the pivot is $0.02{\text{kg m}}^2$, the angular velocity of the stick is

1. $1.58\text{rad/s}$
2. $2.50\text{rad/s}$
3. $2.24\text{rad/s}$
4. $5.00\text{rad/s}$

Q. A point object of mass $m=1\text{kg}$ moving horizontally with $10\text{m/s}$ hits the lower end of the uniform thin rod of length $2\text{m}$ and sticks to it. The rod is rested on a horizontal, frictionless surface and pivoted at the other end as shown in figure. Find out the angular velocity of the system just after collision.

1. $3.75\text{rad/s}$
2. $7.5\text{rad/s}$
3. Zero
4. $15\text{rad/s}$

Q.

The slope of kinetic energy vs displacement curve of a particle in motion is

1. equal to the acceleration of the particle

2. inversely proportional to acceleration

3. directly proportional to acceleration

4. none of these

Q. A ball is projected from ground with a velocity $v$ at an angle $\theta$ to the vertical. On its path, it makes an elastic collision with a vertical wall and returns to ground. The total time of flight of the ball is

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

Q.

A composite rod of mass 2m $\&$ length 2l consists of two identical rods joined end to end at P. The composite rod is hinged at one of its ends and is kept horizontal. If it is released from rest, ·Find its angular speed when it becomes vertical

1. 

2. 

3. 

4. 

Q. A particle of mass $m$ with speed $v$ collide inelastically with the stationary ring and gets embedded in it. Find the angular velocity with which the system rotates after the collision. Assume that the thickness of the ring is much smaller than its radius.

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