Two circular loops P and Q of radii 'r' and 'nr' are made respectively from a uniform wire. Moment of inertia of loop Q about its axis is 4 times that of loop P about its axis. The value of n is

Step-by-step solution

Linear mass density is constant for both loops. The mass of each loop equals times its circumference. For loop P with radius r , mass M_P = 2 r and moment of inertia about its axis I_P = M_P r^2 = 2 r^3 . For loop Q with radius nr , mass M_Q = 2 nr and moment of inertia I_Q = M_Q (nr)^2 = 2 n^3 r^3 . Given I_Q = 4I_P , substituting yields 2 n^3 r^3 = 4(2 r^3) . Canceling 2 r^3 from both sides gives n^3 = 4 , so n = 4^ 1/3 = (2^2)^ 1/3 = 2^ 2/3 . The value n = 2^ 2/3 corresponds to option B.