A ring of radius $r$ (made of wire of density $\rho$ ) placed on a smooth horizontal surface is rotated about a stationary vertical axis passing through its centre and perpendicular to the plane of the ring as shown in figure. Determine the angular velocity in $\mathrm{rad}/\mathrm{s}$ at which the ring breaks. The wire breaks at a tensile stress \sigma. (given $\sigma /\rho =9 \mathrm{N}-\mathrm{m}/\mathrm{kg}$ and $r=1 \mathrm{m}$)

(i) Estimate the force with which a karate master strike a board, assuming the hand’s speed at the moment of impact is 10.0 m $s-1$, decreasing to $1.00\mathrm{}\mathrm{m}\mathrm{}\mathrm{s}-1$ during a $0.002\mathrm{}\mathrm{s}$ time internal of contact between the hand and the board. The mass of his hand and arm is $1.00\mathrm{}\mathrm{k}\mathrm{g}$.

(ii) Estimate the shear, assuming this force it exerted on a $1.00 \mathrm{cm}$ thick pine board that is $10.0 \mathrm{cm}$ wide.
(iii) If the maximum shear stress that a pine board can support before breaking is $3.60\times {10}^6 \mathrm{N}/{\mathrm{m}}^2$, will the board break?

A rod pulled with forces $F_1$ and $f_2$ at its ends. Find the:

(i) Tension at a distance $x$ from left and at the rod,

(ii) longitudinal and shearing stresses at the given cross-section $PQ$ if the cross-section of the rod is $A_0$,
(iii) conditions for maximum longitudinal and shearing stresses at the given cross-section $PQ$.