A cylinder is rolling without slipping on an inclined plane. Show that to prevent slipping the condition ${\mu }_s\ge \frac{1}{3}\tan \theta$

At time $t=0$, a disk of radius $1 \mathrm{m}$ starts to roll without slipping on a horizontal plane with an angular acceleration of $\alpha =\frac{2}{3} \mathrm{rad} {\mathrm{s}}^{-2}$. A small stone is stuck to the disk. At $t=0$, it is at the contact point of the disk and the plane. Later, at time $t=\sqrt{\pi } \mathrm{s}$, the stone detaches itself and flies off tangentially from the disk. The maximum height (in $\mathrm{m}$) reached by the stone measured from the plane is $\frac{1}{2}+\frac{x}{10}$. The value of $x$ is [Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$.]

If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.

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 mass $M$ and radius $R$ rolls without slipping on a horizontal surface (see figure).

If the speed of its centre is $v_0$, then the magnitude of the angular momentum of the disc about a fixed point $P$ at a height $5\mathrm{R}/2$ above the horizontal surface

A small roller of diameter $20 \mathrm{cm}$ has an axle of diameter $10 \mathrm{cm}$ (see figure below on the left). It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle (see figure on the right). The scale is now pushed slowly on the axle so that it moves without slipping on the axle, and the roller starts rolling without slipping. After the roller has moved $50 \mathrm{cm},$ the position of the scale will look like (figures are schematic and not drawn to scale)

A sphere of radius $a$ and mass $m$ rolls along a horizontal plane with constant speed $v_0$. It encounters an inclined plane at angle $\theta$ and climbs upward. Assuming that it rolls without slipping, how far up the sphere will travel?

A disc is performing pure rolling on a smooth stationary surface with constant angular velocity as shown in Figure. At any instant, for the lower most point of the disc,

Velocity of the centre of a small cylinder is $v$. There is no slipping anywhere. The velocity of the centre of the larger cylinder is