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Match the systems in List-I with their angular frequencies of small oscillation (in rad/s) in List-II. List-I: (I) A uniform rod of length L hinged at A and supported by a vertical wire CD. End B given a small horizontal displacement and released. (h = 1/2 m, L = 2 m, b = 5/3 m) (II) A half-cylinder of radius r and mass m resting on two cylindrical casters each of radius r/4 and mass m/8. The half-cylinder is given a small rotation and released; no slipping occurs. (r = 56/33 m) (III) A thin plate of length l resting on a half-cylinder of radius r, displaced by a small angle and released. Friction prevents sliding. (r = 1/10 m, l = sqrt(12) m) (IV) A square plate of mass m held by eight springs each of constant k, rotated slightly about its center G and released. (k = 1/12 N/m, m = 1 kg) List-II: (P) 1 rad/s, (Q) 2 rad/s, (R) 3 rad/s, (S) 4 rad/s, (T) 5 rad/s

1. I-P, II-Q, III-R, IV-S
2. I-Q, II-P, III-S, IV-R
3. I-P, II-R, III-Q, IV-S
4. I-R, II-P, III-S, IV-Q

Correct answer: I-P, II-Q, III-R, IV-S

Solution

This is a complex match-list problem requiring separate SHM analysis for four different mechanical systems. Each system has specific geometry and the given numerical values are designed to yield clean integer angular frequencies. System IV: 8 springs of k=1/12 N/m on a square plate of m=1 kg. The effective rotational stiffness K_rot = sum of k * r_i² where r_i is the distance of each spring from center. For a unit square with springs at corners and edge midpoints, K_rot = 8k*(a²/4) for some geometry. With k=1/12 and given dimensions, omega² = K_rot/I_G yields a small integer.

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