A uniform metal rod of length 1 and mass m is rotating with an angular velocity of magnitude o about an axis passing through its centre and perpendicular to the rod. If the temperature of the rod increases by T^@C , then change in its magnitude of angular velocity varies directly with

A uniform metal rod of length 1 and mass m is rotating with an angular

1.	A uniform metal rod of length 1 and mass m is rotating with an angular velocity of magnitude o about an axis passing through its centre and perpendicular to the rod. If the temperature of the rod increases by T^@C , then change in its magnitude of angular velocity varies directly with
Answer» `1/omega ` `sqrt omega ` `omega ` `omega^2` Solution :When the temperature of the rod increases by `T^@C` , length of the rod BECOMES `L. = l (1 + alpha T)` where a is the coefficient of linear expansion. From the law of CONSERVATION of angular momentum, we have `I omega = I. omega. ` ` rArr 1/12 ml^2 omega = 1/12 ml^2 omega.` ` rArr (omega.)/(omega) = ( (l)/(l.))^2 = (1)/((1+ alpha T)^2) = (1+ alpha T)^(-2)` For GIVEN VALUE of` T, (1 + alpha T)^(-2)`is constant, say, C ` therefore (omega.)/(omega) = C` `(omega. -omega) /(omega) = C- 1` ` rArr omega. - omega = (C - 1) omega` ` rArr omega. - omega prop omega `