When a ceiling fan is switched off its angular velocity reduces to $50\%$ while it makes $36$ rotations. How many more rotation will it make before coming to rest (Assume uniform angular retardation):-

The angular speed of truck wheel is increased from $900 \mathrm{rpm}$ to $2460 \mathrm{rpm}$ in $26$ seconds. The number of revolutions by the truck engine during this time is ______.

(Assuming the acceleration to be uniform).

A solid disc of radius $a$ and mass $m$ rolls down without slipping on an inclined plane making an angle $\theta$ with the horizontal. The acceleration of the disc will be $\frac{2}{\mathit{ }b}g\sin \theta$, where $b$ is _______.  (Round off to the nearest integer)

($g=$ acceleration due to gravity and $\theta =$ angle as shown in figure)

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

A rigid massless rod of length $3l$ has two masses attached at each end as shown in the figure. The rod is pivoted at point $P$ on the horizontal axis. When released from the initial horizontal position, its instantaneous angular acceleration will be

A rod of length $50 cm$ is pivoted at one end. It is raised such that it makes an angle of ${30}^o$ from the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in $rad s^{-1}$ ) will be $\left(g=10 m{s}^{-2}\right)$

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