Pure Rolling

Q. As shown in figure, a ring of radius $R$ is rolling without slipping. What will be the magnitude of velocity of point $A$ on the ring ?

1. $V_{cm}$
2. $\sqrt{2}{V}_{cm}$
3. $\frac{V_{cm}}{2}$
4. $2V_{cm}$

Q. A solid sphere of radius $R$ is resting on a smooth horizontal surface. A constant force $F$ is applied at a height $h$ from the bottom. Choose the correct alternative

1. Sphere will always slide whatever be the value of $h$
2. Sphere will roll without slipping when $h\ge 1.4R$
3. Sphere will roll without slipping if $h=1.4R$
4. None of the above

Q. A force $F$ acts tangentially at the highest point of a hollow sphere of mass $m$ kept on a rough horizontal plane. If the sphere rolls without slipping then,

1. Static friction will act on the sphere towards left.
2. Kinetic friction will act on the sphere toward right.
3. Static friction will act on the sphere toward right.
4. Kinetic friction will act on the sphere toward left.

Q. A solid sphere $A$ of mass $m$ rolls without slipping on an inclined plane of inclination ${30}^{\circ }$. Coefficient of friction between the inclined plane and the sphere is $\mu$. Then, which of the following options is satisfied when the sphere undergoes pure rolling motion?

1. $\mu \ge \frac{2\sqrt{3}}{7}$
2. $\mu \ge \frac{2}{7\sqrt{3}}$
3. $\mu <\frac{2\sqrt{3}}{7}$
4. $\mu <\frac{2}{7\sqrt{3}}$

Q. A uniform solid sphere of radius $R$ and mass $m$ rolls down an inclined plane. The coefficient of friction between the sphere and the inclined plane is $\mu$ then maximum value of $\theta$ for pure rolling is

1. ${\text{tan}}^{-1}\left(\frac{3\mu }{2}\right)$
2. ${\text{tan}}^{-1}\left(\frac{7\mu }{2}\right)$
3. ${\text{tan}}^{-1}\left(\frac{5\mu }{3}\right)$
4. ${\text{tan}}^{-1}\left(\frac{7\mu }{3}\right)$

Q. A system of uniform cylinders is shown in figure. All the cylinders are identical and there is no slipping at any contact. Velocity of lower and upper plate is $V$ and $2V$ respectively as shown in figure. Then, the ratio of angular speed of the upper cylinder to that of the lower cyllinder is

1. $3$
2. $1/3$
3. $1$
4. none of these

Q. An automobile moves on a rough road with a speed of $54{\text{kmh}}^{-1}$. The radius of its wheels is $0.45\text{m}$ and the moment of inertia of the wheel about its axis of rotation is $3{\text{kg.m}}^2$. If the wheel is brought to rest in $15\text{s}$, then find magnitude of average torque transmitted by the brakes to wheel. Assume no slipping at all points of contact.

1. $10.86{\text{kg m}}^2{\text{s}}^{-2}$
2. $2.86{\text{kg m}}^2{\text{s}}^{-2}$
3. $6.67{\text{kg m}}^2{\text{s}}^{-2}$
4. $8.58{\text{kg m}}^2{\text{s}}^{-2}$

Q. Two thin planks are moving on four identical cylinders as shown. There is no slipping at any of the contact points. Calculate the ratio of angular speed of upper cylinders to lower cylinders

Q. A sphere is pure rolling on a horizontal surface such that its centre moves with 5 m/s as shown in figure. What is the magnitude of velocity of point B on this sphere?

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

Q. As shown in figure, a ring of radius $R$ is rolling without slipping. What will be the magnitude of velocity of point $A$ on the ring ?

1. $V_{cm}$
2. $\sqrt{2}{V}_{cm}$
3. $\frac{V_{cm}}{2}$
4. $2V_{cm}$

Q.

A disc of mass M and radius R is rolling with angular speed $\omega$ on a horizontal plane as shown. The magnitude of angular momentum of the disc about the origin O is

1. $\left(\frac{1}{2}\right)M{R}_2\omega$

2. $M{R}^2\omega$

3. $\left(\frac{3}{2}\right)M{R}^2\omega$

4. $2M{R}^2\omega$

Q. Find the minimum coefficient of friction between the cylindrical shell and inclined plane as shown in the figure, so that the shell will perform pure rolling.

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

Q. A solid sphere of mass $5\text{kg}$ and radius $1\text{m}$ is kept on a rough surface. A force $F=30\text{N}$ is acting at the top most point. Friction required for pure rolling is

1. $70/9\text{N}$
2. $90/7\text{N}$
3. $90/5\text{N}$
4. $50/9\text{N}$

Q. A wheel of mass $m=1\text{kg}$ and radius $r=1\text{m}$ is under pure rolling in a straight line as shown in figure.



If $V=2\text{m/s},a=1{\text{m/s}}^2,\alpha =1{\text{rad/s}}^2$ and $\omega =2\text{rad/s}$, find the torque and $K.E$ of the wheel about the contact point.

1. ${\tau }_C=2\text{N-m},K.E=1\text{J}$
2. ${\tau }_C=1\text{N-m},K.E=2\text{J}$
3. ${\tau }_C=2\text{N-m},K.E=4\text{J}$
4. ${\tau }_C=3\text{N-m},K.E=5\text{J}$

Q. The speed of a homogeneous solid sphere after rolling down an inclined plane of vertical height $h$ from rest, without sliding is

1. $\sqrt{gh}$
2. $\sqrt{\left(6/5\right)gh}$
3. $\sqrt{\left(4/3\right)gh}$
4. $\sqrt{\left(10/7\right)gh}$

Q. A spherical ball rolls on an inclined plane without slipping. Ratio of total kinetic energy to rotational kinetic energy is

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

Q. A system of uniform cylinders is shown in figure. All the cylinders are identical and there is no slipping at any contact. Velocity of lower and upper plate is $V$ and $2V$ respectively as shown in figure. Then, the ratio of angular speed of the upper cylinder to that of the lower cyllinder is

1. $3$
2. $1/3$
3. $1$
4. none of these

Q. A plank is moving with a velocity $4\text{m/s}$. A disc of radius $1\text{m}$ rolls on it without slipping with an angular velocity of $3\text{rad/s}$ as shown in figure. Then, the velocity of the centre of the disc $\left(V_{\text{cm}}\right)$ is

1. $4\text{m/s}$
2. $1\text{m/s}$
3. $3\text{m/s}$
4. $7\text{m/s}$

Q. The ratio of the accelerations for a solid sphere (mass 'm' and radius 'R') rolling down an incline of angle ${}^{\prime }{\theta }^{\prime }$ without slipping and slipping down the incline without rolling is:

1. 7 : 5
2. 5 : 7
3. 2 : 3
4. 2 : 5

Q. Speed of the centre of a purely rolling wheel is v. A particle on the rim at the same level as the center

1. has a speed of Zero m/s
2. has a speed of v
3. has a speed of 1.414 v
4. has a speed of 2 v

Q.

A disc is rolling (without slipping) on a horizontal surface. C is its centre and Q and P are two points equidistant from C. Let $v_P,v_{Q}and{v}_C$ be the magnitude of velocities of points P, Q and C respectively, then

1. $v_Q>v_C>v_P$

2. $v_Q<v_C<v_P$

3. $v_Q=v_P,v_C=\frac{1}{2}{v}_P$

4. $v_Q<v_C>v_P$

Q. A thin spherical shell of mass $M$ and radius $R$ as shown in the figure slips on a rough horizontal plane. At the same instant, it has translational velocity $v_0$ and rotational velocity about the centre $\frac{3v_0}{R}$ . Find the translation velocity after the shell starts pure rolling.

1. $\frac{9}{5}{v}_0$
2. $2v_0$
3. $\frac{5}{9}{v}_0$
4. $\frac{v_0}{3}$

Q. A small object of uniform density rolls up a curved surface with an initial velocity $v$ (see the figure). It reaches upto a maximum height $h=\frac{3v^2}{4g}$, with respect to the initial position. The object is a:

1. ring
2. solid sphere
3. hollow sphere
4. disc

Q.

A disc is rolling (without slipping) on a horizontal surface. C is its centre and Q and P are two points equidistant from C. Let $v_P,v_{Q}and{v}_C$ be the magnitude of velocities of points P, Q and C respectively, then

1. $v_Q>v_C>v_P$

2. $v_Q<v_C<v_P$

3. $v_Q=v_P,v_C=\frac{1}{2}{v}_P$

4. $v_Q<v_C>v_P$

Q. A solid sphere $A$ of mass $m$ rolls without slipping on an inclined plane of inclination ${30}^{\circ }$. Coefficient of friction between the inclined plane and the sphere is $\mu$. Then, which of the following options is satisfied when the sphere undergoes pure rolling motion?

1. $\mu \ge \frac{2\sqrt{3}}{7}$
2. $\mu \ge \frac{2}{7\sqrt{3}}$
3. $\mu <\frac{2\sqrt{3}}{7}$
4. $\mu <\frac{2}{7\sqrt{3}}$

Q. A string is wrapped on the surface of a solid cylinder as shown in the figure. If the cylinder is released, then find the acceleration of the centre of mass of the cylinder and the tension in the string respectively. Mass of the cylinder is $M$ and radius is $r$. (No slipping occurs between any contact points)

1. $\frac{2g}{3},Mg$
2. $\frac{g}{2},\frac{2Mg}{3}$
3. $\frac{2g}{3},\frac{Mg}{3}$
4. $\frac{4g}{3},\frac{Mg}{3}$

Q. An automobile moves on a rough road with a speed of $54{\text{kmh}}^{-1}$. The radius of its wheels is $0.45\text{m}$ and the moment of inertia of the wheel about its axis of rotation is $3{\text{kg.m}}^2$. If the wheel is brought to rest in $15\text{s}$, then find magnitude of average torque transmitted by the brakes to wheel. Assume no slipping at all points of contact.

1. $10.86{\text{kg m}}^2{\text{s}}^{-2}$
2. $2.86{\text{kg m}}^2{\text{s}}^{-2}$
3. $6.67{\text{kg m}}^2{\text{s}}^{-2}$
4. $8.58{\text{kg m}}^2{\text{s}}^{-2}$

Q. A small object of uniform density rolls (without slipping) up a frictionless curved surface with an initial velocity $v$. It reaches upto a maximum height of $\frac{3v^2}{4g}$ with respect to the initial position. The object is

1. ring
2. solid sphere
3. hollow sphere
4. disc

Q. A solid sphere of mass $5\text{kg}$ and radius $1\text{m}$ is kept on a rough surface. A force $F=30\text{N}$ is acting at the top most point. Friction required for pure rolling is

1. $70/9\text{N}$
2. $90/7\text{N}$
3. $90/5\text{N}$
4. $50/9\text{N}$

Q. Two identical uniform discs roll without slipping on two different surfaces as shown in figure. If they reach points $B$ and $D$ with the same linear speed, then $v_2$ (in $\text{m/s}$) is
[Take $g=10{\text{m/s}}^2$]