$A$ uniform wire of length $L$ and radius $r$ is twisted by an angle $\alpha$. If the modulus of rigidity of the wire is $\eta$,then the elastic potential energy stored in the wire is .........

$A$ cylindrical rod of length $1 \ m$ and radius $4 \ cm$ is mounted vertically. It is subjected to a shear force of $10^5 \ N$ at the top. Considering infinitesimally small displacement in the upper edge,the angular displacement $\theta$ of the rod axis from its original position would be: (Shear modulus,$G = 10^{10} \ N/m^2$)

The upper end of a wire of radius $4 \, mm$ and length $100 \, cm$ is clamped and its other end is twisted through an angle of $60^o$. The angle of shear is .......... $^o$.

Given below are two statements: One is labelled as Assertion $(A)$ and the other is labelled as Reason $(R)$.
Assertion $(A)$: The stretching of a spring is determined by the shear modulus of the material of the spring.
Reason $(R)$: $A$ coil spring of copper has more tensile strength than a steel spring of same dimensions.
In the light of the above statements,choose the most appropriate answer from the options given below:

$A$ horizontal aluminium rod of diameter $4 \text{ cm}$ projects $6 \text{ cm}$ from a wall. An object of mass $400 \pi \text{ kg}$ is suspended from the end of the rod. The shearing modulus of aluminium is $3.0 \times 10^{10} \text{ N/m}^2$. The vertical deflection of the end of the rod is (given $g = 10 \text{ m/s}^2$): (in $\text{ mm}$)

$A$ force $F$ of the same magnitude is applied tangentially on the upper and lower faces of a cube in opposite directions. The side of the cube is $L$. The upper face of the cube shifts parallel to itself by a distance $x_{1}$. If another cube of the same material but with side $2L$ is subjected to the same condition,then the displacement of the top layer is: