List-I shows four planar structures made of uniform solid rods each of mass m and length l . In the List-II the possible moment of inertia of these structures about an axis OCO' , which lies in the plane of the structures, are given. Choose the option that describes the correct match between the entries in List-I to those in List-II. List-I List-II (P) (1) 5 4 m l^2 (Q) (2) 1 6 m l^2 (R) (3) 1 12 m l^2 (S) (4) 2 3 m

Step-by-step solution

The moment of inertia of a uniform rod of mass m and length l about an axis passing through one end and making an angle with the rod is given by I = m l^2 3 ^2 . For structure (P): The axis OO' makes an angle of 45^ with both rods CA and CB . I_P = I_ CA + I_ CB = m l^2 3 ^2(45^ ) + m l^2 3 ^2(45^ ) = 2 m l^2 3 1 2 = m l^2 3 . This matches with (5). For structure (Q): The axis OO' makes an angle of 60^ with rods CA and CB . Rod AB is parallel to the axis at a distance d = l (60^ ) = 3 2 l . I_Q = I_ CA + I_ CB + I_