A thin rod of mass m and length l is rotating in the horizontal plane about its one of the ends with constant angular velocity….

Q: A thin rod of mass m and length l is rotating in the horizontal plane about its one of the ends with constant angular velocity ω . The tension at the middle point of the rod is

(a)  mlω2

(b) mlω2/4

(c) 3mlω2/8

(d) mlω2/8

Ans: (c)

Sol: Consider a elemental mass at a distance x from one end is

dm = (m/l)dx

dF = (dm)ω2 x

$ \displaystyle T = \int dF = \int_{l/2}^{l}\frac{m}{l}\omega^2 x dx $

$\displaystyle = \frac{m}{l}\omega^2 \int_{l/2}^{l} xdx $

$ \displaystyle = \frac{m}{l}\omega^2[\frac{x^2}{2}]_{l/2}^{l} $

$ \displaystyle = \frac{3}{8}m\omega^2 l $

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