Rotation + Translation

Q. Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities ${\omega }_1$ and ${\omega }_2$. They are brought into contact face to face coinciding the axis of rotation. The expression for loss in coinciding of energy during this process is

1. $\frac{1}{8}({\omega }_1-{\omega }_2{)}^2$
2. $\frac{1}{2}I({\omega }_1+{\omega }_2{)}^2$
3. $\frac{1}{4}I({\omega }_1-{\omega }_2{)}^2$
4. $I({\omega }_1-{\omega }_2{)}^2$

Q.

What is the difference between circular and rotational motion?

Q. A hollow sphere of mass $M$ and radius $R$ is initially at rest on a horizontal rough surface. It moves under the action of a constant horizontal force $F$ as shown in figure.


The linear acceleration $\left(a\right)$ of the sphere is :

1. $a=\frac{10F}{7M}$
2. $a=\frac{7F}{5M}$
3. $a=\frac{6F}{5M}$
4. $a=\frac{F}{M}$

Q. A hollow smooth uniform sphere $A$ of mass ${}^{\prime }{m}^{\prime }$ rolls without sliding on a smooth horizontal surface. It collides elastically and head on with another stationary smooth solid sphere $B$ of the same mass $m$ and same radius. The ratio of kinetic energy of ${}^{\prime }{B}^{\prime }$ to that of ${}^{\prime }{A}^{\prime }$ just after the collision is $3:n$ where $n$ is

Q. A solid cylinder and a hollow cylinder, both of the same mass and same external diameter are released from the same height at the same time on an inclined plane. Both roll down without slipping. Which one will reach the bottom first?

1. both together
2. hollow cylinder
3. solid cylinder
4. Both together only when angle of inclination of the plane is ${45}^{\circ }$

Q. A disc of mass $M$ and radius $R$ is rolling without slipping with angular speed $\omega$ on a horizontal plane as shown in figure. The magnitude of angular momentum of the disc about the origin $O$ is:

1. $\frac{1}{2}M{R}^2\omega$
   
2. $M{R}^2\omega$
   
3. $2M{R}^2\omega$
   
4. $\frac{3}{2}M{R}^2\omega$
   

Q. A solid cylinder is released from rest from the top of an inclined plane of inclination $\theta$ and length ${}^{\prime }{l}^{\prime }$. If the cylinder rolls without slipping, then find it's speed when it reaches the bottom of inclined plane.

1. $\sqrt{\frac{4gl\sin \theta }{3}}$
2. $\sqrt{\frac{3gl\sin \theta }{2}}$
3. $\sqrt{\frac{4gl}{3\sin \theta }}$
4. $\sqrt{\frac{4g\sin \theta }{3l}}$

Q. A point $P$ is the contact point of wheel on the ground which rolls on ground without slipping. Find the value of displacement of the point $P$ when wheel completes half revolution. Radius of the wheel is $1\text{m}$.

1. $2\text{m}$
2. $\sqrt{{\pi }^2+4}\text{m}$
3. $\pi \text{m}$
4. $\sqrt{{\pi }^2+2}\text{m}$

Q. Why r cap is equal to r vector divided by r?

Q. A uniform disc is spinning about geometrical axis in free space. If its temperature is increased by $\Delta T$, the fractional change in its angular velocity is ($\alpha$ is coefficient of linear expansion and $\alpha \Delta T<<1)$

1. $2\alpha \Delta T$
2. $-\alpha \Delta T$
3. $\alpha \Delta T$
4. $-2\alpha \Delta T$

Q.

A mass m supported by a massless string wound around a uniform hollow cylinder of mass m and radius R. If the string does not slip on the cylinder, with what acceleration with the mass fall on release?

1. $\frac{2g}{3}$

2. $\frac{g}{3}$

3. $\frac{5g}{6}$

4. $\frac{g}{2}$

Q. A solid spherical ball is rolling without slipping on an inclined plane. The fraction of its total kinetic energy associated with rotation is

1. $\frac{2}{7}$
2. $\frac{2}{5}$
3. $\frac{3}{7}$
4. $\frac{3}{5}$

Q.

A cylinder of mass 10 kg and radius 15 cm is rolling perfectly on a plane of inclination ${30}^{\circ }$. The coefficient of static friction ${\mu }_s=0.25$.
(a) How much is the force of friction acting on the cylinder?
(b) What is the work done against friction during rolling?
(c) If the inclination $\theta$ of the plane is increased, at what value of $\theta$ does the cylinder begin to skid, and not roll perfectly?

Q. A hollow sphere rolls without slipping on the horizontal surface such that its translational velocity is $v$. Find that the maximum height attained by it on an inclined surface as shown in the figure.

1. $\frac{5v^2}{6g}$
2. $\frac{5v^2}{g}$
3. $\frac{v^2}{6g}$
4. $\frac{5v}{6g}$

Q. A solid sphere of mass $500\text{g}$ and radius $10\text{cm}$ rolls without slipping with a velocity of $20\text{cm/s}$. The total $K.E$. of the sphere is nearly

1. $0.001\text{J}$
2. $0.014\text{J}$
3. $0.14\text{J}$
4. $1.4\text{J}$

Q. A uniform ball of radius $r$ rolls without slipping down from the top of a sphere of radius $R$. The angular velocity of the ball when it breaks from the sphere is

1. $\sqrt{\frac{5g(R+r)}{17r^2}}$
2. $\sqrt{\frac{10g(R+r)}{17r^2}}$
3. $\sqrt{\frac{5g(R-r)}{10r^2}}$
4. $\sqrt{\frac{10g(R+r)}{17r^2}}$

Q. A solid sphere rolls down two different inclined planes of same height, but of different inclinations. Consider rolling without slipping. Then, in both cases:

1. Translational speed and time of descent will be same
   
2. Translational speed will be same, but time of descent will be different
   
3. Translational speed will be different, but time of descent will be same
   
4. Translational speed and time of descent are different
   

Q. A solid sphere of radius $r$ is gently placed on a rough horizontal surface with an initial angular speed ${\omega }_0$ but no linear velocity. If the coefficient of friction is $\mu$, then the time $t$ when the slipping will stop is

1. $\frac{2}{7}\frac{r{\omega }_0}{\mu g}$
2. $\frac{3}{7}\frac{r{\omega }_0}{\mu g}$
3. $\frac{4}{7}\frac{r{\omega }_0}{\mu g}$
4. $\frac{r{\omega }_0}{\mu g}$

Q. Four particles, each of mass $m$, are placed at the corners of a square of side length $l$. The radius of gyration of the system about an axis perpendicular to the square and passing through its center is

1. $l$
2. $\frac{l}{\sqrt{2}}$
3. $l\sqrt{2}$

4. $\frac{l}{2}$

Q. Two blocks $m_1=4kg$ and $m_2=2kg$ are connected by a weightess rod on a plane having inclination of ${37}^{\circ }$. The coefficient of dynamic friction for blocks $m_1$ and $m_2$ with the inclined plane is $\mu =0.2$. Assume
${\mu }_k={\mu }_s=0.2$ and g=10 ${ms}^{-2}$. Then the common acceleration of the two blocks and the tension in the rod will be:

1. $a=2m/s^2,T=5N$
2. $a=4.4m/s^2,T=0$
3. $a=10m/s^2,T=10N$
4. $a=15m/s^2,T=9N$

Q. Consider a non-uniform rod of length $L$, whose mass per unit length $\lambda$ varies as $\lambda =\frac{k.x^2}{L}$ where $k$ is a constant and $x$ is the distance of any point on rod from its one end. Then the centre of mass of rod is at (from the same end as $x$)

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

Q. A sphere is rolling on a frictionless surface as shown in the figure with a translational velocity $v{\text{ms}}^{-1}$. If it is to climb the inclined surface, then $v$ should be

1. $\ge \sqrt{\frac{10}{7}gh}$
2. $\ge \sqrt{2gh}$
3. $2gh$
4. $\frac{10}{7}gh$

Q. A wheel (to be considered as a ring) of mass $m$ and radius $R$ rolls without sliding on a horizontal surface with constant velocity $v$. It encounters a step of height $\frac{R}{2}$ at which it ascends without sliding. Choose the correct statement(s).

1. The angular velocity of the ring just after it comes in contact with the step is $\frac{3v}{4R}$.
2. The normal reaction due to the step on the wheel just after the impact is $\frac{mg}{2}-\frac{9m{v}^2}{6R}$.
3. The normal reaction due to the step on the wheel increases as the wheel ascends.
4. The normal reaction due to the step on the wheel stays constant as the wheel ascends.

Q.

In a children's park a heavy rod is pivoted at the centre and is made to rotate about the pivot so that the rod always remains horizontal. Two kids hold the rod near the ends and thus rotate with the rod (figure 7-E2). Let the mass of each kid be 15 kg, the distance between the points of the rod where the two kids hold it be 3.0 m and suppose that the rod rotates at the rate of 20 revolutions per minute. Find the force of friction exerted by the rod on one of the kids.

Q. A uniform ring of radius $R$ is given a back spin of angular velocity $\frac{V_0}{2R}$ and thrown on a horizontal rough surface with velocity of centre to be $V_0$. The velocity of the centre of the ring when it starts pure rolling will be

1. $\frac{V_0}{2}$
2. $\frac{V_0}{4}$
3. $\frac{3V_0}{4}$
4. $0$

Q. A uniform chain has a mass M and length L. It is placed on a frictionless table with length $L_0$ hanging over the edge. The chain begins to slide down. Then, the speed v with which the end slides down from the edge is given by

1. $v=\sqrt{\frac{g}{L}(L+L_0)}$
2. $v=\sqrt{\frac{g}{L}(L-L_0)}$
3. $v=\sqrt{\frac{g}{L}(L^2-L_0^2)}$
4. $v=\sqrt{2g(L-L_0)}$

Q. A solid sphere is given an angular velocity $\omega$ about its centre and kept on a fixed rough inclined plane. Then choose the correct statement(s).

1. If $\mu =\tan \theta$, then the sphere will be in translational equilibrium for some time and after that, pure rolling down the plane will start.
2. If $\mu =\tan \theta$, then sphere will move up the plane and frictional force acting all the time will be $2mg\sin \theta$.
3. If $\mu =\frac{\tan \theta }{2}$, there will never be pure rolling (considering inclined plane to be long enough).
4. If inclined plane is not fixed and it is on a smooth horizontal surface, then linear momentum of the system (wedge and sphere) will be conserved in horizontal direction.

Q. A uniform solid cylinder rolls without slipping on a rough horizontal floor, its centre of mass moving with a speed $v$. It makes an elastic collision with a smooth vertical wall. After impact:

1. its centre of mass will move with a speed $v$ initially
2. its motion will be rolling without slipping immediately
3. its motion will be rolling with slipping initially and its rotational motion will stop momentarily at some instant
4. its motion will be rolling without slipping only after some time.

Q. A small disc is set on rolling with speed $10\text{m/s}$ on the horizontal part of the track from right to left. There is curved part also as shown in the figure. To what height will the disc climb up on the curved path ? Assuming all track is rough and there is no slipping occur anywhere.

1. $5\text{m}$
2. $7.5\text{m}$
3. $10\text{m}$
4. $15\text{m}$

Q. A cylinder is given angular velocity ${\omega }_0$ and kept on a horizontal rough surface. The initial velocity is zero. The distance travelled by it before it starts performing pure rolling is: [Assume radius of cylinder $=R$]

1. $\frac{{\omega }_0^2{R}^2}{18\mu g}$
2. $\frac{{\omega }_0^2{R}^2}{\mu g}$
3. $\frac{{\omega }_0^2{R}^2}{2\mu g}$
4. $\frac{{\omega }_0^2{R}^2}{6\mu g}$