Motion in a vertical circle — A/L Combined Mathematics

How do I practise Motion in a vertical circle?

1 question with a worked solution

Check before you answer: Energy Conservation Equation: Writing 1/2 m v² + mg h = constant correctly between benchmark points.

1. A particle of mass m attached to the end of a light inextensible string of length r is whirled in a vertical circle. What is the minimum speed v_top the particle must have at the highest point of the circle so that the string does not become slack (tension T ≥ 0)?
   1. A) √(5 g r)
   2. B) √(g r)
   3. C) √(2 g r)
   4. D) 2 √(g r)
   Check my answer

   Correct answer: B) √(g r)

   At the topmost point, both tension T and weight mg act vertically downwards toward the center: T + mg = m v_top² / r => T = m (v_top² / r - g). For the string not to slacken, T ≥ 0 => v_top² / r ≥ g => v_top ≥ √(g r). Hence, the minimum speed is √(g r).

   Show worked solution

   Step 1: Inward force at top: T + mg = m v² / r. Step 2: Condition for taut string: T ≥ 0 => m v² / r ≥ mg => v ≥ √(gr). Answer: √(g r).
   Modelled on: the A/L Combined Mathematics examination pattern

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Where does Motion in a vertical circle fit in the syllabus?

Year 2 · Term 3