Establish the relation between angular acceleration and linear acceleration.

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

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

Represent the union of two sets by Venn diagram for each of the following.

$X=\{x | x$ is a prime number between $80$ and $100\}$

$Y=\{y | y$ is an odd number between $90$ and $100\}$