A thin uniform rod of length 2 m , cross sectional area A and density d is rotated about an axis passing through the centre and perpendicular to its length with angular velocity ω . If value of ω in terms of its rotational kinetic energy E is α E A d , then the value of α is ______.

Step-by-step solution

Rotational kinetic energy is given by E = 1 2 I ω 2 , here, I is moment of inertia of rod about an axis through its centre and perpendicular to it. So, E = 1 2 m l 2 12 ω 2 ⇒ E = 1 2 d A l 3 12 ω 2         ∵ mass = density × volume ⇒ E = d A ( 2 ) 3 24 ω 2 ⇒ ω = 3 E d A Thus, the value of α = 3 .