$A$ boy throws a cricket ball from the boundary to the wicket-keeper. If the frictional force due to air cannot be ignored,the forces acting on the ball at the position $X$ are represented by

$A$ rod $P$ of length $1 \ m$ is hinged at one end $A$ and there is a ring attached to the other end by a light inextensible thread. Another long rod $Q$ is hinged at $B$ and it passes through the ring. The rod $P$ is rotated about an axis which is perpendicular to the plane in which both the rods are present and the variation between the angles $\theta$ and $\phi$ is plotted as shown. The distance between the hinges $A$ and $B$ is ........ $m$.

$A$ particle of mass $1 \ kg$ is projected with an initial velocity $10 \ ms^{-1}$ at an angle of projection $45^{\circ}$ with the horizontal. The average torque acting on the projectile, between the time at which it is projected and the time at which it strikes the ground, about the point of projection in $Nm$ is:

$A$ particle is projected. If after $2 \, s$ it makes an angle of $30^\circ$ with the horizontal and after $1 \, s$ more it is moving horizontally,find the initial velocity and angle of projection of the particle.

$A$ particle is in uniform circular motion. Its velocity is perpendicular to which of the following?

One end of a rod of length $L=1 \,m$ is fixed to a point on the circumference of a wheel of radius $R=1 / \sqrt{3} \,m$. The other end is sliding freely along a straight channel passing through the centre $O$ of the wheel as shown in the figure. The wheel is rotating with a constant angular velocity $\omega$ about $O$. The speed of the sliding end $P$,when $\theta=60^{\circ}$ is