Pure Rolling

Q. A solid cylinder of mass $M$ and radius $R$ rolls down an inclined plane of height $h$. The angular velocity of the cylinder when it reaches the bottom of the plane will be

1. $\frac{1}{2R}\sqrt{gh}$
2. $\frac{2}{R}\sqrt{gh}$
3. $\frac{2}{R}\sqrt{\frac{gh}{2}}$
4. $\frac{2}{R}\sqrt{\frac{gh}{3}}$

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

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. 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.

Read each statement below carefully, and state, with reasons, if it is true or false;
(a) During rolling, the force of friction acts in the same direction as the direction of motion of the CM of the body.
(b) The instantaneous speed of the point of contact during rolling is zero.
(c) The instantaneous acceleration of the point of contact during rolling is zero.
(d) For perfect rolling motion, work done against friction is zero.
(e) A wheel moving down a perfectly frictionless inclined plane will undergo slipping (not rolling) motion.

Q. A solid cylinder of mass $M$ and radius $R$ rolls (without slipping) down an inclined plane of inclination $\theta$. The minimum coefficient of friction $\mu$ between the cylinder and the plane so that it rolls without slipping must be

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

Q. Derive relationship between Moment of inertia , angular velocity and torque

Q. A hoop of radius $r$ and mass $m$ rotating with an angular velocity ${\omega }_0$ is placed on a rough horizontal surface. The initial velocity of the center of the hoop is zero. What will be the velocity of the center of the hoop when it ceases to slip?

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

Q. A hollow cylinder and a solid cylinder of same mass and same external radius are rolling without slipping down an inclined plane of inclination $\theta$ from horizontal. If both of them are released at the same time, then which one reaches the bottom first?

1. Can't say anything
2. Hollow cylinder
3. Solid cylinder
4. Both reaches simultaneously

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.

Is angular momentum a scalar or vector? If it is a vector, what rule is used to determine its direction?

Q. relation between torque and angular momentum

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 drum of radius $R$ and mass $M$ rolls down without slippping along an inclined plane of inclination $\theta$. Then identify the correct statement

1. Frictional force acting on drum will dissipate energy as heat.
2. Frictional force acting on drum will oppose the rotational motion of the drum.
3. Frictional force acting on drum will oppose the translational as well as the rotational motion of drum.
4. Gravitational Potential energy of drum is $100\%$ transformed into translational and rotational $K.E$ of drum.

Q. A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping all throughout. The two climb maximum heights $h_{\text{sph}}$ and $h_{\text{cyl}}$ on the incline. The ratio $\frac{h_{\text{sph}}}{h_{\text{cyl}}}$ is given by

1. $\frac{2}{\sqrt{5}}$
2. $\frac{14}{15}$
3. $1$
4. $\frac{4}{5}$

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 $g=10{\text{m/s}}^2$]

Q. A uniform rod AB of length $l$ is rotating with an angular velocity $\omega$ while its centre moves with a linear velocity $v=\frac{\omega l}{6}$. If the end A of the rod is suddenly fixed, the angular velocity of the rod will be:

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

Q. L=Iw is not always true for all bodies (ie. of any shapes).why?

Q. A solid sphere rolls down an inclined plane and its velocity at the bottom is $v_1$. Then same sphere slides down the plane (without friction) and let its velocity at the bottom be $v_2$. Which of the following relation is correct

1. 
2. 
3. 
4. None of these

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

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

momentum measures the amount of _______ in a body.

1. inertia

2. motion

3. velocity

4. acceleration

Q. A point object of mass m is slipping down on a smooth hemispherical body of mass M and radius R. The point object is tied to a wall by an ideal string as shown. At a certain instant shown in figure, speed of the hemisphere is $v$ and its acceleration is a. Then speed $v_p$ and acceleration $a_p$ of the point object is (Assume all the surfaces in contact are frictionless).

1. $v_p=\frac{v\sqrt{3}}{2}$
2. $v_p=v$
3. $a_p=a$
4. $a_p=\sqrt{{(\frac{a\sqrt{3}}{2}+\frac{v^2}{R})}^2+{\left(\frac{a}{2}\right)}^2}$

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

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 solid sphere of uniform density and mass $M$ has radius $4\text{m}$. Its centre is at the origin of the coordinate system. Two spheres of equal radii $1\text{m}$, with their centres at $P(0,-2,0)$ and $Q(0,2,0)$ respectively, are taken out of the solid leaving behind spherical cavities as shown if figure. What is the gravitational field at the origin of the coordinate axis?

1. $\frac{31GM}{1024}$
2. $\frac{GM}{1024}$
3. $31GM$
4. Zero

Q.

A vertical capillary with inside diameter 0.50mm is submerged in to water so that the length of its part emerging outside the water surface is equal to 25mm.Find the radius of curvature of the meniscus. Surface tension of water is $73\times {10}^{-3}N/m(g=9.8m/s^2)$

1. 0.6mm

2. 0.2mm

3. 0.8mm

4. 0.4mm

Q.

A solid sphere is in pure rolling motion on an inclined surface having inclination $\theta$

1. friction will increase its angular velocity and decrease its linear velocity

2. frictional force acting on sphere is f = $\mu$ mg cos $\theta$

3. f is dissipative force

4. If $\theta$ decreases, friction will decrease

Q.

The kinetic energy of a body depends upon

1. mass of the body

2. velocity of the body

3. height of the body

4. both (a) and (b)

Q. Two metallic spheres of equal outer radii are found to have same moment of inertia about their respective diameters. Then which of the following statement(s) is/are true ?

1. Two spheres have equal mass
2. The ratio of masses is nearly 1.67 : 1
3. The spheres are made of different materials
4. Their rotational kinetic energies will be equal when rotated with equal uniform angular speed about their respective diameters