A thin uniform copper rod of length l and mass m rotates uniformly with an angular velocity `omega` in a horizontal plane about a vertical axis passing through one of its ends. Determine the tension in the rod as a function of the distance r from the rotation axis. Find the elongation of the rod.

Explanation

Choose a small element of width `dx` at a distance a from one end of the rod . The mass of the element `dm = (m)/(l) dx` Let `T` is the tension to the rod at a distance x by Newton's second law , from the motion of element of mass `dm` , we have
`T - (T + dT) = dm omega^(2) x or - dT = ((m)/(l) dx) omega^(2) x `
Interating above equation we get
`- int_(T)^(0) dT = (m omega^(2))/(l) int_(x)^(1) xx dx`
`or - |T|_(T)^(0) = (m omega^(2))/(2l)|x^(2) |_(x)^(0)`
`or - (0 - T) = (m omega^(2))/(2l)(t^(2) - x^(2))`
`or T = (m omega^(2))/(2l)(t^(2) - x^(2))`