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A mass m is attached to a string of length L and is swung in a vertical circle.

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Question: A mass m is attached to a string of length L and is swung in a vertical circle. At the highest point of the circle, what is the minimum speed required to keep the mass in circular motion?

Options:

  1. √(gL)
  2. √(2gL)
  3. gL
  4. 2gL

Correct Answer: √(gL)

Solution: 

At the highest point, the centripetal force must equal the weight: mv²/L = mg, thus v = √(gL).

A mass m is attached to a string of length L and is swung in a vertical circle.

Practice Questions

Q1
A mass m is attached to a string of length L and is swung in a vertical circle. At the highest point of the circle, what is the minimum speed required to keep the mass in circular motion?
  1. √(gL)
  2. √(2gL)
  3. gL
  4. 2gL

Questions & Step-by-Step Solutions

A mass m is attached to a string of length L and is swung in a vertical circle. At the highest point of the circle, what is the minimum speed required to keep the mass in circular motion?
  • Step 1: Understand that the mass m is swinging in a vertical circle attached to a string of length L.
  • Step 2: Recognize that at the highest point of the circle, two forces act on the mass: its weight (mg) and the tension in the string.
  • Step 3: Realize that for the mass to stay in circular motion at the highest point, the centripetal force must be provided by the weight of the mass and the tension in the string.
  • Step 4: At the minimum speed, the tension in the string is zero, so the centripetal force is entirely provided by the weight of the mass.
  • Step 5: Write the equation for centripetal force: mv²/L = mg, where v is the speed at the highest point.
  • Step 6: Simplify the equation by canceling m from both sides: v²/L = g.
  • Step 7: Rearrange the equation to solve for v²: v² = gL.
  • Step 8: Take the square root of both sides to find the minimum speed: v = √(gL).
  • Centripetal Force – The force required to keep an object moving in a circular path, directed towards the center of the circle.
  • Gravitational Force – The force of attraction between two masses, which in this case acts on the mass m at the highest point of the circle.
  • Minimum Speed for Circular Motion – The lowest speed at which an object can maintain its circular path without falling due to gravity.
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