A rod is fixed between a vertical wall and a horizontal surface. A smooth ring of mass 1 kg is released from rest which can move along the rod as shown. At the release point spring is vertical and rel

Question

A rod is fixed between a vertical wall and a horizontal surface. A smooth ring of mass 1 kg is released from rest which can move along the rod as shown. At the release point spring is vertical and relaxed. The natural length of the spring is ( 3 ​ + 1 ) m . Rod makes an angle of 3 0 ∘ with the horizontal. Ring again comes to rest when spring makes an angle of 30° with the vertical.

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Geometry (sine rule): s i n 12 0 ∘ x ​ = s i n 3 0 ∘ 3 ​ + 1 ​ . Energy conservation: m g [ x cos 3 0 ∘ − ( 3 ​ + 1 ) ] = 2 1 ​ k [ x − ( 3 ​ + 1 ) ] 2 . NLM at the second rest point: N + k [ x − ( 3 ​ + 1 ) ] cos 6 0 ∘ = m g cos 3 0 ∘ . Solving: k = 2 5 ​ ( 3 ​ + 1 ) N/m - (A); maximum displacement = 3 ​ − 1 2 ​ m - (B); N = 2 5 ​ ( 3 ​ − 1 ) N - (D). The extension in (C) does not match.

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