Name: Exercise 5: Approximating the Period of a Pendulum using the Binomial and Geometric Series
Uploaded: 2024-09-20T04:15:08+00:00
Duration: 42 min 7 s
Description: 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 fo

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Health & Science Education taylor series taylor polynomial science education taylor approximation calculus STEM proof math proof

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 sine^2n, 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 sine^2n: 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