Two projectiles $A$ and $B$ are thrown with the same speed such that $A$ makes an angle $\theta$ with the horizontal and $B$ makes an angle $\theta$ with the vertical. Then:

$A$ particle of mass $m$ performs uniform circular motion of radius $r$ with linear speed $v$ under the application of force $F$. If $m, v$ and $r$ are all increased by $50 \%$,the necessary change in force required to maintain the particle in uniform circular motion is (in $\%$)

$A$ ball of mass $160 \, g$ is thrown up at an angle of $60^{\circ}$ to the horizontal at a speed of $10 \, m/s$. The angular momentum of the ball at the highest point of the trajectory with respect to the point from which the ball is thrown is nearly $\left(g=10 \, m/s^{2}\right)$ (in $kg \cdot m^{2}/s$).

$A$ particle is moving in a circle with uniform speed '$v$'. In moving from a point to another diametrically opposite point:

An aircraft is flying at a height of $H$ above the ground at a constant speed $V$. What is the maximum angle subtended at a ground observation point by the aircraft's path after time $T$?

$A$ 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 ........ $m$.