A uniform rod of length 2 m is placed on a smooth horizontal table and is rotated about a vertical axis passing through one of its ends at a constant angular velocity of 20 rad s ⁻¹ . The density of the material of the rod is 7500 kg m ⁻³ and its Young's modulus is 2 10¹¹ N m ⁻² . The total elongation of the rod is x 10⁻⁵ m . The value of x is _____.

Step-by-step solution

Consider an element of length dr at a distance r from the axis of rotation. The mass of this element is dm = A dr , where A is the cross-sectional area and is the density. The tension T(x) at a distance x from the axis is responsible for providing the centripetal force to the portion of the rod from x to L : T(x) = _ x ^ L (dm) ^2 r = _ x ^ L ( A dr) ^2 r T(x) = A ^2 [ r^2 2 ]_ x ^ L = A ^2 2 (L^2 - x^2) The elongation d( l) of a small element of length dx at distance x is: d( l) = T(x) dx AY The total elongation l