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 l^2 (5) 1 3 m l^2
A. P 5, Q 1, R 4, S 2
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_ AB = 2 ( m l^2 3 ^2(60^ ) ) + m d^2 I_Q = 2 m l^2 3 3 4 + m ( 3 2 l )^2 = m l^2 2 + 3 m l^2 4 = 5 m l^2 4 . This matches with (1). For structure (R): The axis OO' is the diagonal of the square, making an angle of 45^ with all four rods. I_R = 4 ( m l^2 3 ^2(45^ ) ) = 4 m l^2 3 1 2 = 2 m l^2 3 . This matches with (4). For structure (S): The axis OO' makes an angle of 30^ with both rods CA and CB. I_S = 2 ( m l^2 3 ^2(30^ ) ) = 2 m l^2 3 1 4 = m l^2 6 . This matches with (2). Therefore, the correct matching is P 5, Q 1, R 4, S 2. Answer: P 5, Q 1, R 4, S 2
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