$A$ wheel of radius $r$ rolls without slipping with a speed $v$ on a horizontal road. When it is at a point $A$ on the road,a small lump of mud separates from the wheel at its highest point $B$ and drops at point $C$ on the road. The distance $AC$ will be

$A$ uniform rod of mass $M$ is hinged at its upper end. $A$ particle of mass $m$ moving horizontally strikes the rod at its mid-point elastically. If the particle comes to rest after the collision,find the value of $M/m$.

$A$ disc of mass $M$ and radius $R$ is free to rotate about its vertical axis as shown in the figure. $A$ battery-operated motor of negligible mass is fixed to this disc at a point on its circumference. Another disc of the same mass $M$ and radius $R/2$ is fixed to the motor's thin shaft. Initially,both the discs are at rest. The motor is switched on so that the smaller disc rotates at a uniform angular speed $\omega$. If the angular speed at which the large disc rotates is $\omega/n$,then the value of $n$ is. . . . .

$A$ ring starting from rest rotates under a constant angular acceleration of $8 \ rad \ s^{-2}$ due to an applied torque. How many revolutions will the ring complete in $5 \ s$? How many revolutions will it complete in the $6^{th}$ second? If the torque becomes zero after $6 \ s$,how many revolutions will the ring complete in the $7^{th}$ second?

Which one of the following four graphs best depicts the variation with $x$ of the moment of inertia $I$ of a uniform triangular lamina about an axis parallel to its base at a distance $x$ from it?

$A$ block of mass $m = 1 \, kg$ slides with velocity $v = 6 \, m/s$ on a frictionless horizontal surface and collides with a uniform vertical rod and sticks to it as shown. The rod is pivoted about $O$ and swings as a result of the collision,making an angle $\theta$ before momentarily coming to rest. If the rod has mass $M = 2 \, kg$ and length $l = 1 \, m$,the value of $\theta$ is approximately (Take $g = 10 \, m/s^2$) (in $^{\circ}$)