Match the angular frequency (in rad / s ) in list-II with the system in list-I. List - I List - II (I) A uniform rod of length L hinged at A and by a vertical wire CD. End B is given a small horizonta

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Match the angular frequency (in rad / s ) in list-II with the system in list-I. List - I List - II (I) A uniform rod of length L hinged at A and by a vertical wire CD. End B is given a small horizontal displacement and then released. (Take h = 1/2 m , L = 2 m , b = 5/3 m ) (P) 1 (II) A half section of uniform cylinder of radius r and mass m rests on two casters A and B, each of which is a uniform cylinder of radius r/4 and mass m/8. The half cylinder is rotated through a small angle and released and that no slipping occurs. (Take : r = 56/33 m ) (Q) 4 (III) A thin plate of length ℓ rests on a half cylinder of radius r . The plate is displaced by a small angle and released. Friction is sufficient to prevent sliding. (Take r = 1/10 m , ℓ = 12 ​ m ) (R) 2 (IV) A square plate of mass m is held by eight springs, each of constant k. The plate is rotated through a small angle about G and released. (Take k = 12 1 ​ N/m , m = 1 kg ) (S) 5 (T) 3

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(I) Wire tension by torque balance: T C ​ b = m gL /2 For a small angle, point C rises by 2 h b 2 θ 2 ​ . Δ U = T C ​ 2 h b 2 θ 2 ​ = 4 h m gL b ​ θ 2 I A ​ = 3 m L 2 ​ ω I 2 ​ = 2 h L 3 g b ​ = 25 s − 2 ω I ​ = 5 rad/s ⇒ S (II) The half-cylinder centre remains fixed; its centre of mass is 3 π 4 r ​ below it. I half ​ = 2 1 ​ m r 2 No slip makes each caster spin four times as fast. I eff ​ = 2 1 ​ m r 2 + 2 [ 2 1 ​ ( 8 m ​ ) ( 4 r ​ ) 2 × 4 2 ] = 8 5 ​ m r 2 U = m g 3 π 4 r ​ ( 1 − cos θ ) ω II 2 ​ = 15 π r 32 g ​ = 4 s − 2 ω II ​ = 2 rad/s ⇒ R (III) Source length: 12 ​ m . 2 1 ​ m g r θ 2 = 2 1 ​ ( 12 m ℓ 2 ​ ) θ ˙ 2 ω III 2 ​ = ℓ 2 12 g r ​ = 12 12 ( 10 ) ( 0.1 ) ​ = 1 s − 2 ω III ​ = 1 rad/s ⇒ P (IV) Each spring extends by 2 ℓ θ ​ . U = 8 [ 2 1 ​ k ( 2 ℓ θ ​ ) 2 ] = k ℓ 2 θ 2 I G ​ = 6 m ℓ 2 ​ ω I V 2 ​ = I G ​ 2 k ℓ 2 ​ = m 12 k ​ = 1 s − 2 ω I V ​ = 1 rad/s ⇒ P I → S , II → R , III → P , I V → P ⇒ A

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List - I	List - II
(I) A uniform rod of length L hinged at A and by a vertical wire CD. End B is given a small horizontal displacement and then released. (Take $h = 1/2 m, L = 2 m, b = 5/3 m$ )	(P) 1
(II) A half section of uniform cylinder of radius r and mass m rests on two casters A and B, each of which is a uniform cylinder of radius r/4 and mass m/8. The half cylinder is rotated through a small angle and released and that no slipping occurs. (Take : $r = 56/33 m$ )	(Q) 4
(III) A thin plate of length $ℓ$ rests on a half cylinder of radius $r$ . The plate is displaced by a small angle and released. Friction is sufficient to prevent sliding. (Take $r = 1/10 m, ℓ = 12 m$ )	(R) 2
(IV) A square plate of mass m is held by eight springs, each of constant k. The plate is rotated through a small angle about G and released. (Take $k = \frac{1}{12} N/m, m = 1 kg$ )	(S) 5
	(T) 3