Question

If a uniform rope of mass m and length l is rotated about one of its ends with a constant angular velocity $\omega$ in a free space, find expression of tensile force developed in the rope as function of distance r from fixed end.

Answer 1

T(r) = \frac{m \omega^2}{2l} (l^2 - r^2)

Answer 2

The tensile force $T(r)$ at a distance $r$ from the fixed end is the cumulative centripetal force required for all the mass elements of the rope located between $r$ and the free end $l$. By integrating the infinitesimal centripetal force $dF_c = (\frac{m}{l} dr') \omega^2 r'$ for each element $dr'$ from $r$ to $l$, we obtain the total tension. The integration yields $T(r) = \frac{m \omega^2}{2l} (l^2 - r^2)$.