A small bob tied at one end of a thin string of length 1 m is describing a vertical circle so that the maximum and minimum tension in the string are in the ratio 5 : 1. The velocity of the bob at the highest position is ________ m/s. (Take g = 10 m/s²)

Solution

Physics sir · 4 steps

Write both tensions in terms of the top speed, then let the 5 : 1 ratio fix that speed.

1. Tension at extremes. At the bottom, tension and gravity act in opposite directions, so Tₘₐₓ = mv_b²/L + mg. At the top, both point towards the centre, so Tₘᵢₙ = mvₜ²/L - mg.
2. Link the speeds. Energy conservation over a height drop of 2L gives v_b² = vₜ² + 4gL. Substituting, Tₘₐₓ = mvₜ²/L + 5mg.

   mvₜ²/L + 5mg = 5(mvₜ²/L - mg)

3. Solve. This gives 10mg = 4mvₜ²/L, so vₜ² = 2.5 gL = 2.5 × 10 × 1 = 25 m² s⁻².
4. So. The speed at the highest point is vₜ = 5 m s⁻¹, so the answer is 5.

The idea to keep. In a vertical circle, Tₘₐₓ - Tₘᵢₙ = 6mg always. Express both tensions through the speed at one point using energy conservation, then apply the given condition.

Answer from the NTA final answer key. Solution by PrepWiser.