$A$ metal wire of length $0.5\; m$ and cross-sectional area $10^{-4}\; m^{2}$ has a breaking stress of $5 \times 10^{8}\; N/m^{2}$. $A$ block of mass $10\; kg$ is attached to one end of the wire and is rotated in a horizontal circle. The maximum linear velocity of the block will be $v\; m/s$. Find $v$.

Two blocks of masses $1 \,kg$ and $2 \,kg$ are connected by a metal wire going over a smooth pulley. The breaking stress of metal is $\frac{40}{3 \pi} \times 10^6 \,N m^{-2}$. What should be the minimum radius of the wire used if it should not break (in $mm$)? $(g = 10 \,m s^{-2})$

Two strips of metal are riveted together at their ends by four rivets,each of diameter $6.0\; mm$. What is the maximum tension that can be exerted by the riveted strip if the shearing stress on the rivet is not to exceed $6.9 \times 10^{7}\; Pa$? Assume that each rivet is to carry one quarter of the load.

$A$ body of mass $10 \, kg$ is attached to the end of a wire of length $0.3 \, m$. If the breaking stress of the wire is $4.8 \times 10^7 \, N/m^2$ and its cross-sectional area is $10^{-6} \, m^2$,what is the maximum angular velocity in $rad/sec$ with which it can be rotated in a horizontal plane without breaking the wire?

$A$ string $AB$ of unstretched length $L$ is stretched by applying a force $F$ at the mid-point $C$ such that the segments $AC$ and $BC$ make an angle $\theta$ with $AB$ as shown in the figure. The string may be considered as an elastic element with a force to elongation ratio $K$. The force $F$ is given by

$A$ uniform wire of length $l$ and weight $w$ is suspended from the roof with a weight $W$ attached at the other end. The stress in the wire at a distance $l/3$ from the top is given by $(\frac{W}{A} + \gamma \frac{w}{A})$, where $A$ is the cross-sectional area of the wire. The value of $\gamma$ is . . . . . . .