A block $X$ of mass $0.5 \mathrm{kg}$ is held by a long massless string on a frictionless inclined plane of inclination $30°$ to the horizontal. The string is wound on a uniform solid cylindrical drum $Y$ of mass $2 \mathrm{kg}$ and radius $0.2 \mathrm{m}$ as shown in the diagram. The drum is given an initial angular velocity such that the block $X$ starts moving up the plane. Find the tension in the string during motion. At a certain instant of time, the magnitude of the angular velocity of $Y$ is $10 \mathrm{rad} {\mathrm{s}}^{-1}$. Calculate the distance travelled by $X$ from that instant of time unit it comes to rest.

One end of a rod of length $L=1 \mathrm{m}$ is fixed to a point on the circumference of a wheel of radius $R=\frac{1}{\sqrt{3}} \mathrm{m}.$ The other end is sliding freely along a straight channel passing through the center $O$ of the wheel as shown in the figure below. The wheel is rotating with a constant angular velocity $\omega$ (in radian per second) about $O.$

The speed of the sliding end $P$ when $\theta =60°$ is

A uniform rod of length ' ${\ell }^{\text{'}}$ is pivoted at one of its ends on a vertical shaft of negligible radius. When the shaft rotates at angular speed $\omega$ the rod makes an angle $\theta$with it (see figure). To find $\theta$ equate the rate of change of angular momentum (direction going into the paper) $\frac{m{\ell }^2}{12}{\omega }^2\sin \theta$ about the centre of mass (CM) to the torque provided by the horizontal and vertical forces $F_H$and $F_v$ about the CM. The value of $\theta$ is then such that:

A ball is rolling without slipping in a spherical shallow bowl (radius $R$ ) as shown in the figure and is executing simple harmonic motion. If the radius of the ball is doubled, then the time period of oscillation