Two beads each of mass m are fixed on a light rigid rod of length 2 l which is free to rotate in a horizontal plane. The bead on the far end is given some velocity v as shown in the figure. If K cm represents the kinetic energy of the centre of mass of the system and K R represents the rotational kinetic energy of the system, then what is the value of K cm K R ?

Step-by-step solution

The distance of the centre of mass from the hinge, r cm = m l + m ( 2 l ) m + m = 3 2 l The angular velocity of the system, ω = v 2 l The velocity of the centre of mass of the system, v cm = 3 l 2 ω = 3 4 v ⇒ K cm = 1 2 2 m v 2 cm = 9 m v 2 16 The moment of inertia of the system about the hinge, I = m l 2 + m 2 l 2 = 5 m l 2 The rotational kinetic energy of the system, K R = 1 2 I ω 2 = 1 2 5 m l 2 v 2 l 2 ⇒ K R = 5 m v 2 8 K cm K R = 9 m v 2 16 × 8 5 m v 2 = 0 . 9