Pure Rolling

A flat surface of a thin uniform disk $A$ of radius $R$ is glued to a horizontal table. Another thin uniform disk $B$ of mass $M$ and with the same radius $R$ rolls without slipping on the circumference of $A$, as shown in the figure. A flat surface of $B$ also lies on the plane of the table. The center of mass of $B$ has fixed angular speed $\omega$ about the vertical axis passing through the center of $A$. The angular momentum of $B$ is $nM\omega R^2$ with respect to the center of $A$. Which of the following is the value of $n$?

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A disc of radius $0.1\mathrm{ }\mathrm{m}$ rolls without sliding on a horizontal surface with a velocity of $6\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$ . It then ascends a smooth continuous track as shown in figure. The height upto which it will ascend is (in $\mathrm{cm}$) : ( $g=10\mathrm{ }{\mathrm{m} \mathrm{s}}^{-2}$ )

A ring of radius $2 \mathrm{m}$ weights $100 \mathrm{kg}.$ It rolls (pure rolling) along a horizontal floor so that its centre of mass has a speed of $20 \mathrm{cm} {\mathrm{s}}^{-1}$. If work done to stop it is $x \mathrm{J}$. Then $x$ will be

A cylinder of mass $M_{\mathrm{C}}$ and sphere of mass $M_{\mathrm{S}}$ are placed at points $A$and $B$of two inclines, respectively. (See Figure). If they roll on the incline without slipping such that their accelerations are the same, then the ratio $\frac{{\mathrm{sin\theta }}_{\mathrm{c}}}{{\mathrm{sin\theta }}_{\text{s}}}=\frac{2n+1}{2n}$. What is the value of $n$?

These questions consists of two statements each printed as Assertion and Reason. While answering these questions you are required to choose any one of the following five responses.

Assertion: A solid sphere cannot roll without slipping on smooth horizontal surface.

Reason: If the sphere is left free on smooth inclined surface, it cannot roll without slipping.

A uniform solid cylindrical roller of mass $m$ is being pulled on horizontal surface with force $F$ parallel to the surface applied at its centre. If the acceleration of the cylinder is $a$ and it is rolling without slipping, then the value of $F$ is

A uniform solid ball of mass '$m$' rolls without sliding on a fixed horizontal surface. The velocity of the lowest point of the ball with respect to the center of the ball is $v$. The total kinetic energy of the ball is:

A ring of radius $3a$ is fixed rigidly on a table. A small ring whose mass is $m$ and radius $a$ rolls without slipping inside it as shown in the figure. The small ring is released from position $A$. When it reaches the lowest point, the speed of the centre of the ring at that time would be,

A small sphere rolls down without slipping from the top of a track in a vertical plane as shown. The track has an elevated section and a horizontal path. The horizontal part is $1.0 \mathrm{m}$ above the ground level and the top of the track is $2.4 \mathrm{m}$ above the ground. Find the distance on the ground with respect to a point $B$ where the sphere lands.

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

In the previous, which of the bodies reaches the ground with maximum rotational kinetic energy?

What is the work done against friction during rolling, if length of the incline is $1$ meter?

A ring of mass $m$ and radius $R$ is acted upon by a force $F$ as shown in the figure. There is sufficient friction between the ring and the ground. The force of friction necessary for pure rolling is

A small pulley of radius $20 \mathrm{cm}$ and moment of inertia $0.32 \mathrm{kg} {\mathrm{m}}^2$ is used to hang a $2 \mathrm{kg}$ mass with the help of massless string. If the block is released, for no slipping condition acceleration of the block will be

A rod of uniform mass and of length $L$ can freely rotate in a vertical plane about an axis passing through $O.$ The angular velocity of the rod when it falls from position $P$ to $P^{\text{'}}$ through an angle $\alpha$ is

Work done by friction in case of pure rolling

A solid sphere rolls without slipping and presses a spring of spring constant $k$ as shown in the figure. Then, the compression in the spring will be

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

A disc is rolling without slipping on a horizontal surface with $C$, as its centre and $Q$ and $P$ the two points equidistant from $C$. Let $v_P, v_Q$ and $v_C$ be the magnitudes of velocities of points $P, Q$ and $C$, respectively, then

A wheel of bicycle is rolling without slipping on a level road. The velocity of the centre of mass is $v_{\mathrm{CM}}$ then true statement is