$A$ uniform thin rod of length $2L$ and mass $m$ lies on a horizontal table. $A$ horizontal impulse $J$ is given to the rod at one end. There is no friction. The total kinetic energy of the rod just after the impulse will be

$A$ ring,a solid sphere,and a thin disc of different masses rotate with the same kinetic energy. Equal torques are applied to stop them. Which will make the least number of rotations before coming to rest?

$A$ wheel of radius $0.5 \ m$ and a moment of inertia of $10 \ kg \cdot m^2$ is rotating freely at an angular speed of $70 \ rev/min$. The wheel can be stopped in $5.0 \ s$ by pressing a wet cloth against the rim and exerting a radially inward force of $88 \ N$. The coefficient of kinetic friction between the wheel and wet cloth is:

Two flywheels are connected by a non-slipping belt as shown in the figure. $I_1 = 4 \ kg \ m^2$,$r_1 = 20 \ cm$,$I_2 = 20 \ kg \ m^2$,and $r_2 = 30 \ cm$. $A$ torque of $10 \ Nm$ is applied on the smaller wheel. Match the entries of column $I$ with appropriate entries of column $II$.

Quantities	Their numerical values (in $SI$ units)
a. Angular acceleration of smaller wheel	$1$. $5/3$
b. Torque on the larger wheel	$2$. $100/3$
c. Angular acceleration of larger wheel	$3$. $5/2$

$A$ simple pendulum of length $\ell$ is displaced so that its taut string is horizontal and then released. $A$ uniform bar of length $L$ pivoted at one end is simultaneously released from its horizontal position. If their motions are synchronous (i.e.,they have the same angular velocity at any angle $\theta$ below the horizontal),what is the length $L$ of the bar?

$A$ particle starts from the point $(0\,m, 8\,m)$ and moves with a uniform velocity of $3\, \hat{i} \,m/s$. After $5\,s$,the angular velocity of the particle about the origin will be: