$A$ metal rod of Young's modulus $Y$ and coefficient of linear expansion $\alpha$ has its temperature raised by $\Delta \theta$. The linear stress required to prevent the expansion of the rod is:

If the given graph shows the load $(W)$ attached to and the elongation $(\Delta l)$ produced in a wire of length $1 \,m$ and area of cross-section $1 \,mm^2$, then the Young's modulus of the material of the wire is

Three bars having lengths $l, 2l$,and $3l$ and areas of cross-section $A, 2A$,and $3A$ are joined rigidly end to end. The compound rod is subjected to a stretching force $F$. The total increase in the length of the rod is (Young's modulus of the material is $Y$ and the bars are massless).

$A$ meter scale of mass $m$,Young's modulus $Y$,and cross-sectional area $A$ is hung vertically from the ceiling at the zero mark. The separation between the $30\ cm$ and $70\ cm$ marks will be: (Assume $\frac{mg}{AY}$ is dimensionless)

$A$ steel wire of length $L$ at $40^{\circ} C$ is suspended from the ceiling and then a mass $m$ is hung from its free end. The wire is cooled down from $40^{\circ} C$ to $30^{\circ} C$ to regain its original length $L$. The coefficient of linear thermal expansion of the steel is $\alpha = 10^{-5} /^{\circ} C$,Young's modulus of steel is $Y = 10^{11} N/m^2$,and the radius of the wire is $r = 1 \ mm$. Assume that $L \gg$ diameter of the wire. Then the value of $m$ in $kg$ is nearly:

$A$ wire of length $2L$ is fixed between two walls. When a weight $W = mg$ is applied at its midpoint,it sags by a distance $x$ $(x << L)$. Then $m$ is equal to: