Exercise 5: Approximating the Period of a Pendulum using the Binomial and Geometric Series

In this video I go over the formula for the period (time it takes to go back and forth) of a pendulum and approximate it using the binomial and geometric series. I first rewrite the pendulum period formula using the binomial series and given integral for sin²ⁿ(x), and approximate it using the first 2 terms. Since all the terms of the series are positive, the actual pendulum period is larger than the 2nd order approximation. I then show that the binomial series form of the pendulum period is actually less than a geometric series. Thus we have an upper and lower bound for our approximation. I go over several examples and determine the % accuracy as well.

Timestamps:

• Exercise 5: Pendulum Period: 0:00
• Solution to (a): Binomial series approximation of the pendulum period: 3:54
  • Plugging in our given formula for sin²ⁿ(x): 11:53
  • Pendulum period formula as an infinite binomial series: 18:27
• Solution to (b): Period inequality using geometric series: 19:13
  • Pendulum period is greater than the first term approximation because all the terms are positive: 21:30
  • Recap on geometric series: 23:56
  • Comparing our Taylor series with geometric series: 24:57
  • Pendulum period is less than sum of a geometric series: 30:14
  • Pendulum inequality formula: 33:21
• Solution to (c): Estimate pendulum period of length 1 meter: 34:47
  • Estimate for 10 degrees is 0.2% less than the actual 2.01 second period: 38:02
  • Estimate for 42 degrees: 39:43
    • Estimate for 42 degrees is 3.4% less than the actual 2.01 second period: 41:07

Full video, notes, and playlists:

• Full video and playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0F76sIU8xm09oqBTq1mlry3
• HIVE notes: @mes/infinite-sequences-and-series-applications-of-taylor-polynomials
• Infinite Sequences and Series: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz .

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