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Leaving Cert Applied Maths: Vertical circular motion

How often Vertical circular motion comes up on the Applied Maths papers, every year it was asked, and questions to try.

HL Asked on 3 of the last 3 Higher Level papers, most recently in 2025. banker

Vertical circular motion, Higher Level(8 marks)

Quick ones on Vertical circular motion.

(a)A particle whirls in a vertical circle on a string. When does the string go slack?
  1. When the speed becomes zero
  2. When the tension becomes zero
  3. When the tension equals mg
(b)Minimum speed at the top for a particle on a string to stay on a vertical circle of radius r?
  1. √(2gr)
  2. 0
  3. √(gr)
(c)Tension at the lowest point of a vertical circle: mass m on a string, speed v, radius r?
  1. T = mg + mv²/r
  2. T = mg
  3. T = mv²/r − mg
Show the answers

(a) When the tension becomes zero

(b) √(gr)

(c) T = mg + mv²/r

Your turn: Higher Level questions on Vertical circular motion.

Higher Level

Asked on 3 of the last 3 Higher Level papers, most recently in 2025. banker

Every paper, year by year

YearWhere it came up
2025Q4
2024Q4
2023Q10

Links open the State Examinations Commission’s paper for that year.

More Vertical circular motion questions

Vertical circular motion, 3 marks

Vertical circle of radius r: speed u at the bottom. Speed v when level with the centre?

  1. v² = u² − 4gr
  2. v² = u² − gr
  3. v² = u² − 2gr
Show the answer

v² = u² − 2gr

Level with the centre is a rise of r. Energy: ½mu² = ½mv² + mgr, so v² = u² − 2gr. The 4gr version is for the top, a rise of 2r.

Vertical circular motion, 3 marks

Mass m on a string, radius r, speed v, level with the centre of a vertical circle. Tension?

  1. T = mv²/r − mg
  2. T = mv²/r
  3. T = mg + mv²/r
Show the answer

T = mv²/r

There the string is horizontal and the weight acts vertically, along the tangent. So only the tension points to the centre: T = mv²/r. Gravity changes the speed, not the radial force.

Vertical circular motion, 3 marks

Why can energy conservation be used for a particle on a string moving in a vertical circle?

  1. The tension is perpendicular to the motion, so does no work
  2. The tension is constant all the way round the circle
  3. The speed is constant all the way round the circle
Show the answer

The tension is perpendicular to the motion, so does no work

Tension acts along the string, at right angles to the velocity, so it does no work. Only gravity does work, and gravity's work is in the PE term. So KE + PE is constant.

Other Applied Maths topics

All of Leaving Cert Applied Maths