A circular platform is free to rotate in a horizontal plane about a vertical axis passing through its centre. A tortoise is sitting at the edge of the platform. Now the platform is given an angular velocity ${\omega }_0$. When the tortoise moves along a chord of the platform with a constant velocity (with respect to the platform) the angular velocity of the platform $\omega \left(\mathrm{t}\right)$ will vary with time $t$ as

A horizontal circular platform of radius $0.5 \mathrm{m}$ and mass $0.45 \mathrm{kg}$ is free to rotate about its axis. Two massless spring toy guns, each carrying a steel ball of mass $0.05 \mathrm{kg}$ are attached to the platform at a distance $0.25 \mathrm{m}$ from the centre on either side along its diameter (see figure). Each gun simultaneously fires the balls horizontally and perpendicular to the diameter in opposite directions. After leaving the platform, the balls have the horizontal speed of $9{ \mathrm{m} \mathrm{s}}^{-1}$ with respect to the ground. The rotational speed of the platform in ${\mathrm{rad}}^{-1}$ after the balls leave the platform is

A uniform circular disc of mass $1.5 \mathrm{kg}$ and radius $0.5 \mathrm{m}$ is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude $F=0.5 \mathrm{N}$ are applied simultaneously along the three sides of an equilateral triangle $XYZ$ its vertices on the perimeter of the disc (see figure) One second after applying the forces, the angular speed of the disc in $\mathrm{rad} {\mathrm{s}}^{-1}$ is

A straight rod of negligible mass is mounted on a frictionless pivot and masses $2.5 \mathrm{kg}$ and $1 \mathrm{kg}$ are suspended at distances $40 \mathrm{cm}$ and $100 \mathrm{cm}$ respectively from the pivot as shown. The rod is held at an angle $\theta$ with the horizontal and released.