Pendulum Length Discovery

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Concept: A pendulum’s motion depends on its length and gravity. If gravity is known, we can use the period of the swing to determine the length of the pendulum. See how close your calculations are to the actual length!

For small swings:

T = 2π √(L / g)

We can rearrange to solve for length:

L = (g × T²) / (4π²)

Where:

  • T = period (time for one full back-and-forth swing)
  • L = length of pendulum (meters)
  • g = acceleration due to gravity (~9.8 m/s²)

Overall goal

To measure the period of a pendulum and use it to calculate its length, then compare with the directly measured length.

Need

  • String or light cord
  • Small weight (washer, key, small metal object, etc.)
  • Meter stick or measuring tape
  • Stopwatch (or phone timer)
  • Support to hang pendulum (door frame, clamp, etc.)

Safety

  • Keep swings small to avoid hitting objects or people
  • Ensure the support is stable before releasing the pendulum
  • Do not stand directly in the swing path

Procedure

  1. Set up a pendulum of unknown length (do not measure it yet).
  2. Pull the pendulum back slightly and release without pushing.
  3. Measure the time for multiple full swings (recommend 10 swings for accuracy). A full swing is a cycle from where the object starts, across to the others side, and back to the original place. One 'back and forth'. 
  4. Divide total time by number of swings to find the period (T). How many seconds per swing?
  5. Repeat timing 2–3 times and average your result.

  6. Use the formula to calculate length:

    L = (g × T²) / (4π²)
    (Use g ≈ 9.8 m/s² unless otherwise instructed)

  7. Now measure the actual physical length of the pendulum and compare.

Observations / Data Table

  • Trial 1 time (10 swings): ______ s   Time per swing: _____
  • Trial 2 time (10 swings): ______ s   Time per swing: _____
  • Trial 3 time (10 swings): ______ s   Time per swing:_____
  • Average period, or time per swing (T): ______ s
  • Calculated length (L): ______ m
  • Measured length: ______ m

Math Check

Using your measured values:

L = (g × T²) / (4π²)

Interpret your result:

  • How close is your calculated length to the measured length?
  • If they differ, consider possible sources of error:
    • reaction time in timing
    • non-small swing angle
    • unit conversion problems (remember your answer will be in meters, and if you use a measuring tape most are in cm, so if your answer is .4, then you should measure 40 cm)

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