CDS (II) 2023 General Knowledge · Question 34 of 120

Correct — C, The pendulum has minimum potential energy at position R. Gravitational potential energy depends on one thing only — how high the bob is above whatever level you measure from. A swinging pendulum traces an arc, and along that arc the bob is at its lowest when the string hangs vertically, at the mean position it passes through on every swing, and at its highest at the two turning points where it stops for an instant before coming back. So potential energy is least at the bottom of the arc and greatest at the two extremes. The paper printed a sketch of a pendulum at several positions, which the options label P, Q, R, S and T. R is the only one of those five that the options never pair with a partner — P is paired with T, and Q with S — and on a symmetric swing the position with no mirror twin is the vertical one, at the bottom of the arc. There the bob is moving fastest and its kinetic energy is greatest, while its potential energy is at its minimum. At P and T the bob is momentarily at rest, so kinetic energy is zero and potential energy is at its maximum. Q and S lie in between and carry intermediate values of both. The total of the two stays constant if air resistance and friction are ignored, which is the whole point the item is built on.

• (a)The pendulum has minimum potential energy at positions P and T. — Exactly inverted. P and T are the two extreme positions of the swing, the highest points the bob reaches, where its speed drops to zero. Potential energy there is at its maximum and kinetic energy at its minimum.
• (b)The pendulum has minimum potential energy at positions Q and S. — Q and S sit part-way along the arc on either side of the vertical. The bob is higher there than at R and lower than at the extremes, so their potential energy is intermediate — neither the maximum nor the minimum.
• (d)The pendulum has same potential energy at all positions. — That would require the bob to stay at one height, in which case it would not be swinging at all. It is precisely the rise and fall in height, and the exchange between potential and kinetic energy that goes with it, that keeps the pendulum oscillating.

A simple pendulum is the standard classroom demonstration of energy conversion. Gravitational potential energy is mgh, so it tracks height alone; kinetic energy is half mv squared, so it tracks speed. As the bob falls from an extreme towards the vertical, height falls and speed rises, converting potential energy into kinetic; past the vertical the exchange runs the other way. If there were no air resistance the sum would be constant and the swing would never die. In reality a little energy leaks away each cycle, which is why the amplitude of a real pendulum slowly decreases.

Two habits make this item safe. The first is to remember that potential energy is about position and kinetic energy is about motion, so the question 'where is potential energy least?' is really the question 'where is the bob lowest?'. The second is that the choice of reference level does not matter: potential energy is always measured from some chosen zero, and wherever that zero is placed, the lowest point of the arc carries the least. One thing to note honestly about this card is that the paper's sketch is not reproduced here, so the labelling has to be read from the stem and the options. The options themselves settle it — they pair P with T and Q with S as symmetric partners, which leaves R alone in the middle, and the only position without a mirror twin on a symmetric swing is the vertical one. A second useful cross-check is that at the extremes the bob is instantaneously at rest, and a body at rest at the top of its travel is exactly where stored energy peaks.

• Gravitational potential energy is mgh and depends only on height; kinetic energy is half mv squared and depends only on speed.
• A pendulum bob is lowest at its mean position, where the string is vertical, and highest at its two extreme positions.
• At an extreme position the bob's velocity is momentarily zero, so kinetic energy is zero and potential energy is maximum.
• At the mean position speed is maximum and potential energy minimum; ignoring friction, the sum of the two is the same everywhere.
• A real pendulum loses a little energy each cycle to air resistance, so its amplitude decreases slowly — which is why the exchange is only approximately lossless.

• Swapping the two extremes for the mean position; the bob is slowest and highest at the extremes, fastest and lowest at the vertical.
• Thinking that because the bob is momentarily at rest at an extreme it must have the least energy there — it has the least kinetic energy but the most potential energy.
• Assuming acceleration is zero wherever velocity is zero. At the extreme position velocity is zero but the restoring acceleration is at its largest.

As a labelled-position item on a pendulum or a projectile, asking where potential or kinetic energy is greatest or least, or as a statements item on speed and acceleration at the mean and extreme positions.

• practice — not a real PYQ

  At which position of a swinging simple pendulum is the kinetic energy of the bob the greatest?

  • (a)At either extreme position
  • (b)At the mean position
  • (c)Half-way between the mean and an extreme position
  • (d)It is the same at every position

  Answer(b) At the mean position — the bob is lowest and fastest there, so potential energy is minimum and kinetic energy maximum.

• practice — not a real PYQ

  Ignoring air resistance, which quantity remains constant throughout the swing of a simple pendulum?

  • (a)Its potential energy
  • (b)Its kinetic energy
  • (c)The sum of its potential and kinetic energy
  • (d)Its velocity

  Answer(c) The sum of its potential and kinetic energy — in the absence of friction the total mechanical energy is conserved, even though each part changes continuously.