$A$ body weighs $6$ g when placed in one pan and $24$ g when placed on the other pan of a false balance. If the beam is horizontal when both the pans are empty,the true weight of the body is ....... g.

$A$ pendulum consists of a bob of mass $m=0.1 \ kg$ and a massless inextensible string of length $L=1.0 \ m$. It is suspended from a fixed point at height $H=0.9 \ m$ above a frictionless horizontal floor. Initially,the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. $A$ horizontal impulse $P=0.2 \ kg \cdot m/s$ is imparted to the bob at some instant. After the bob slides for some distance,the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is $J \ kg \cdot m^2/s$. The kinetic energy of the pendulum just after the lift-off is $K$ Joules. $(1)$ The value of $J$ is. . . . . . $(2)$ The value of $K$ is. . . . . Give the answers of the questions $(1)$ and $(2)$.

$A$ rod hinged at one end is released from the horizontal position as shown in the figure. When it becomes vertical,its lower half separates without exerting any reaction at the breaking point. Then the maximum angle '$\theta$' made by the hinged upper half with the vertical is ......... $^o$.

$A$,$B$,and $C$ are a disc,a solid sphere,and a spherical shell respectively,with the same radii $(R)$ and masses $(M)$. These bodies are placed as shown in the figure. The moment of inertia of the given system about the axis $PQ$ is $\frac{x}{15} I$,where $I$ is the moment of inertia of the disc about its diameter. The value of $x$ is . . . . . . .

$A$ thin rod of length '$L$' lies along the $x$-axis with its ends at $x = 0$ and $x = L$. Its linear mass density $\lambda$ varies with $x$ as $\lambda = k{\left( {\frac{x}{L}} \right)^n}$,where $n$ is a non-negative constant. If the position $x_{CM}$ of the center of mass of the rod is plotted against '$n$',which of the following graphs best approximates the dependence of $x_{CM}$ on $n$?

$Assertion$ : $A$ helicopter must necessarily have two propellers.
$Reason$ : Two propellers are provided in a helicopter in order to conserve linear momentum.