Two blocks of masses $3 \, kg$ and $5 \, kg$ are connected by a metal wire going over a smooth pulley. The breaking stress of the metal is $\frac{24}{\pi} \times 10^{2} \, Nm^{-2}$. What is the minimum radius of the wire? (Take $g = 10 \, ms^{-2}$) (in $cm$)

$A$ metal block of mass $m$ is suspended from a rigid support through a metal wire of diameter $14\,mm$. The tensile stress developed in the wire under equilibrium state is $7 \times 10^5\,N m^{-2}$. The value of mass $m$ is $......kg$.
(Take,$g = 9.8\,m s^{-2}$ and $\pi = \frac{22}{7}$)

Match the details of Column-$II$ with the given information in Column-$I$ appropriately.

Column-$I$	Column-$II$
$(a)$ Stress is proportional to strain.	$(i)$ Elastic limit
$(b)$ When the load on the wire is removed, the body regains its original dimension.	$(ii)$ Limit of proportionality
	$(iii)$ Plastic deformation

$A$ force of $1000 \, N$ is required to break a copper wire of radius $1 \, mm$. What force is required to break a copper wire of radius $2 \, mm$?

The elastic limit for a gas

$A$ force $F$ is needed to break a copper wire having radius $R$. The force needed to break a copper wire of radius $2R$ will be