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(A) Consider a uniform rod of length $L$,mass $M$,and cross-sectional area $A$ rotating about one end with constant angular velocity $\omega$. Conside

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(A) Consider a uniform rod of length $L$,mass $M$,and cross-sectional area $A$ rotating about one end with constant angular velocity $\omega$. Consider an element of length $dx$ at a distance $x$ from the axis of rotation. The mass of this element is $dm = (M/L) dx$. The centripetal force required for this element is $dF = (dm) \omega^2 x = (M/L) \omega^2 x dx$. The tensile force $F(x)$ at distance $x$ is the force required to rotate the portion of the rod from $x$ to $L$. $F(x) = \int_x^L (M/L) \omega^2 r dr = (M/L) \omega^2 [r^2/2]_x^L = (M \omega^2 / 2L) (L^2 - x^2)$. The tensile stress $\sigma(x)$ is given by $F(x)/A$. $\sigma(x) = (M \omega^2 / 2AL) (L^2 - x^2)$. This equation represents a downward-opening parabola where $\sigma$ is maximum at $x = 0$ and $\sigma = 0$ at $x = L$. This matches the curve shown in graph $A$.

$A$ uniform rod is rotating in a gravity-free region with a certain constant angular velocity $\omega$. The variation of tensile stress $\sigma$ with distance $x$ from the axis of rotation is best represented by which of the following graphs?

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