Rotation + Translation

Q.

The particles of a solid perform _______ motion. (translator/vibratory)

Q. Which graph describes motion of point $P$ on the disc, initially at origin?

1. 
2. 
3. 
4. $\text{None of these}$

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

Q. Two forces of magnitude $F$ are acting on a uniform disc kept on a rough horizontal surface as shown in the figure. If the frictional force by the horizontal surface on the disc is $nF$, find the value of $n$.

Q. As shown in figure, a ring is performing combined translational and rotational motion with angular velocity $2\text{rad/s}$ and the velocity of its $\text{COM}$ is $4\text{m/s}$. Find the magnitude of velocity of point $C$ on the ring.

1. $2\text{m/s}$
2. $4\text{m/s}$
3. $4\sqrt{2}\text{m/s}$
4. $2\sqrt{2}\text{m/s}$

Q.

Two cars P and Q move with velocities ${\overrightarrow{v}}_P$ and ${\overrightarrow{v}}_Q$ towards north and east respectively.

Velocity of car P w.r.t. Q (i.e., velocity of car P for an observer on cars Q) i.e., ${\overrightarrow{v}}_{PQ}$ will be in which direction?

1. North - west

2. North - east

3. South - west

4. South - east

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 $x-$axis is:

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

Q. A solid sphere of mass $5\text{kg}$ rolls without slipping on an inclined plane of inclination ${30}^{\circ }$. The linear acceleration (in $\text{m}/{\text{s}}^2$) & force of friction (in $\text{N}$) acting on the sphere are :-

1. $\frac{5g}{14},\frac{5g}{7}$
2. $\frac{5g}{7},\frac{5g}{7}$
3. $5g,\frac{2g}{3}$
4. $2g,\frac{4g}{3}$

Q. Speed of COM of a 10 Kg disc of radius 1m is 5 m/s. The angular velocity is 4 rad/s/s. Find translational KE.

1. 150 J
2. 125 J
3. 100 J
4. 175 J

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 small object of uniform density rolls up a curved surface without slipping, with an initial velocity $v\text{m/s}$. It reaches up to a maximum height of $\frac{v^2}{g}\text{m}$ with respect to the initial position. The object is

1. Ring
2. Solid sphere
3. Hollow sphere
4. Disc

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. A rotating fan only has rotatory kinetic energy.

1. False
2. True

Q. A point $P$ (on a wheel) is the contact point of the wheel and the ground. The wheel rolls on the ground without slipping. The value of displacement of the point $P$, when the wheel completes half a rotation is (if radius of 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.

Two blocks of masses 400 g and 200 g are connected through a light string going over a pulley which is free to rotate about its axis. The pulley has a moment of inertia $1.6\times {10}^{-4}kg-m^2$ and a radius 2.0 cm. Find 'x', which is the kinetic energy of the system as the 400 g block falls through 50 cm and also find 'y', the speed of the blocks at this instant.

1. x = 10.5 J, y = 2.6 m/s
2. x = 9.8 J, y = 2.6 m/s
3. x = 9.8 J, y = 1.4 m/s
4. x = 0.98 J, y = 1.4 m/s

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. $\frac{{\omega }_0^2{R}^2}{6\mu g}$

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 plank of mass $M$, whose top surface is rough with coefficient of friction $\mu$ is placed on a smooth ground. Now a disc of mass $(m=\frac{M}{2})$ and radius $R$ is placed on the plank. The disc is now given a velocity $v_0$ in the horizontal direction at $t=0$. Then, choose the correct option(s):

1. Time when disc starts rolling without slipping is $\frac{2v_0}{7\mu g}$.
2. Velocity of plank when rolling without slipping starts is $\frac{v_0}{7}$.
3. Velocity of disc when rolling without slipping starts is $\frac{5}{7}{v}_0$.
4. Angular velocity of disc when rolling without slipping starts is $\frac{4v_0}{7R}$.

Q. A solid sphere of mass $2\text{kg}$ and radius $1\text{m}$ is performing combined rotational and translational motion as shown in figure. Find the total kinetic energy of the solid sphere.

1. $14\text{J}$
2. $4\text{J}$
3. $10\text{J}$
4. $20\text{J}$

Q. A solid cylinder of mass $3\text{kg}$ is placed on a rough inclined plane of inclination ${30}^{\circ }$. The value of frictional force required, for the cylinder to roll down the inclined plane without slipping is:- (Take $g=10{\text{ms}}^{-2}$)

1. $2\text{N}$
2. $5\text{N}$
3. $15\text{N}$
4. $18\text{N}$

Q. Find the speed of uniform solid sphere after rolling down (without sliding) an inclined plane of vertical height $h=0.14\text{m}$ from rest is (Take $g=9.8{\text{m/s}}^2$)

1. $1.4\text{m/s}$
2. $1.2\text{m/s}$
3. $1\text{m/s}$
4. $1.3\text{m/s}$

Q. A plank of mass $M$, whose top surface is rough with coefficient of friction $\mu$ is placed on a smooth ground. Now a disc of mass $(m=\frac{M}{2})$ and radius $R$ is placed on the plank. The disc is now given a velocity $v_0$ in the horizontal direction at $t=0$. Then, choose the correct option(s):

1. Time when disc starts rolling without slipping is $\frac{2v_0}{7\mu g}$.
2. Velocity of plank when rolling without slipping starts is $\frac{v_0}{7}$.
3. Velocity of disc when rolling without slipping starts is $\frac{5}{7}{v}_0$.
4. Angular velocity of disc when rolling without slipping starts is $\frac{4v_0}{7R}$.

Q. A ring of radius $2\text{m}$ performs combined translational and rotational motion on a frictionless horizontal surface with an angular acceleration $4{\text{rad/s}}^2$ and the acceleration of its centre $a=4{\text{m/s}}^2$ as shown in figure. Find the acceleration of point $D$.

1. $8\hat{i}{\text{m/s}}^2$
2. $(4\hat{i}+8\hat{j}){\text{m/s}}^2$
3. $(4\hat{i}-4\hat{j}){\text{m/s}}^2$
4. $8\hat{j}{\text{m/s}}^2$

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. Speed of COM of a 20 Kg disc of radius 1m is 5 m/s. The angular velocity is 4 rad/s/s. Find rotational KE.

1. 100 J
2. 120 J
3. 80 J
4. 150 J

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

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 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.
4. None of these.

Q. A solid cylinder of mass $m$ and radius $r$ starts rolling down on inclined plane of inclination $\theta$. Friction is sufficient enough to prevent slipping. Find the speed of centre of mass of solid cylinder, when it's centre of mass has fallen through a height of $h=9\text{m}$. (Take $g=10{\text{m/s}}^2$)

1. $\sqrt{120}\text{m/s}$
2. $\sqrt{125}\text{m/s}$
3. $\sqrt{180}\text{m/s}$
4. $\sqrt{90}\text{m/s}$

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 $x-$axis is:

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

Q. A wheel of radius $r=1\text{m}$ rolls without slipping with an angular velocity of $\omega =\frac{\pi }{4}\text{rad/s}$ about its centre and velocity of its centre of mass is $v_{COM}=2\text{m/s}$. If the point $P(0,0)$ is in contact with the ground initially, find its position vector after time $t=1\text{s}$.

1. $(1-\frac{1}{\sqrt{2}})\hat{i}+(1-\frac{1}{\sqrt{2}})\hat{j}$
2. $(1-\frac{1}{2})\hat{i}+(1-\frac{1}{\sqrt{2}})\hat{j}$
3. $\hat{i}+(1-\frac{1}{\sqrt{2}})\hat{j}$
4. $(2-\frac{1}{\sqrt{2}})\hat{i}+(1-\frac{1}{\sqrt{2}})\hat{j}$