A thin uniform rod of length ' L ' and mass ' M ' is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is ' '. Its centre of mass rises to a maximum height of ( g= acceleration due gravity)

Step-by-step solution

Maximum height of center of mass using conservation of mechanical energy The lowest position serves as the reference level for potential energy, so PE _ initial = 0 . At maximum angular speed , the kinetic energy is purely rotational: KE _ initial = 1 2 I ^2 . For a thin uniform rod of length L and mass M rotating about its end, the moment of inertia is I = 1 3 M L^2 , yielding KE _ initial = 1 6 M L^2 ^2 . At the highest point, angular speed is zero: KE _ final = 0 and PE _ final = M g h . By conservation of mecha