$A$ particle is moving in a uniform circular motion with angular momentum $L$. If the frequency of motion is doubled and its kinetic energy is halved,find the new value of its angular momentum.

$A$ ball is thrown up with a certain velocity at an angle $\theta$ to the horizontal. The kinetic energy $KE$ of the ball varies with horizontal displacement $x$ as

$A$ projectile crosses two walls of equal height $H$ symmetrically as shown in the figure. The time of flight $T$ is given by ........ $\text{s}$.

$A$ 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 $\%$)

$A$ particle moving in the $xy$ plane experiences a velocity-dependent force $\overrightarrow{F} = k(v_y \hat{i} + v_x \hat{j})$,where $v_x$ and $v_y$ are the $x$ and $y$ components of its velocity $\overrightarrow{v}$. If $\overrightarrow{a}$ is the acceleration of the particle,then which of the following statements is true for the particle?

Two stones of masses $m$ and $2\,m$ are whirled in horizontal circles,the heavier one in a radius $\frac{r}{2}$ and the lighter one in radius $r$. The tangential speed of the lighter stone is $n$ times that of the heavier stone when they experience the same centripetal force. The value of $n$ is