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A bead of weight w can slide on smooth circular wire in a vertical plane. The bead is attached by a light thread to the highest point of the wire and in equilibrium, the thread is taut and make an angle θ with the vertical then tension of the thread and reaction on the bead are

1. T = w cos θ R = w tan θ
2. T = 2w cos θ R = w
3. T = w R = w sin θ
4. T = w sin θ R = w cot θ

Correct answer: T = w cos θ R = w tan θ

Solution

The correct option states that the tension in the thread (T) is equal to the weight of the bead multiplied by the cosine of the angle θ, which accounts for the vertical component of the forces in equilibrium. The reaction force (R) is equal to the weight of the bead multiplied by the tangent of the angle θ, reflecting the balance of forces acting on the bead in the horizontal direction.

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