# Chapter SHM Example 01_02

## Q1: what is the maximum angle of release before the motion is not accurately described as a simple harmonic motion for the case of a simple free pendulum?

Example 1: Simple pendulum A pendulum bob given an initial horizontal displacement and released to swing freely to produce to and fro motion## Suggested Inquiry Steps:

- Define the question in your own words
- Plan an investigation to explore angle of release to record the
actual period T and theoretical period $T}_{the\phantom{\rule{1ex}{0ex}}\text{or}\phantom{\rule{1ex}{0ex}}y}=2\pi \sqrt{\frac{L}{g}$
where L is the length of the mass pendulum of mass, m and g is the gravitational acceleration of which the mass is experiencing, on Earth's
surface g = 9.81 m/s
^{2} - A suggested record of the results could look like this

angle / degree | T /s | T_{theory} / s |
$err\phantom{\rule{1ex}{0ex}}\text{or}\phantom{\rule{1ex}{0ex}}=\frac{(T-{T}_{the\phantom{\rule{1ex}{0ex}}\text{or}\phantom{\rule{1ex}{0ex}}y})}{T}100\%$ |

05 | |||

10 | |||

15 | |||

20 | |||

30 | |||

40 | |||

50 | |||

60 | |||

70 | |||

80 | |||

90 |

With the evidences collected or otherwise, suggests what the conditions of which the angle of oscillation can the actual period T be approximated to theoretical period such that T ≈ $T}_{the\phantom{\rule{1ex}{0ex}}\text{or}\phantom{\rule{1ex}{0ex}}y}=2\pi \sqrt{\frac{L}{g}$

## Suggested Answer 1:

angle θ ≈ 10 degrees for $err\phantom{\rule{1ex}{0ex}}\text{or}\phantom{\rule{1ex}{0ex}}=\frac{(2.010-2.006)}{2.010}\left(100\right)=0.2\%$, depending on what is the error acceptable, small angle is typically about less than 10 degree of swing from the vertical.

## Conclusion:

Motion approximates simple harmonic motion when the angle of oscillation is small < 10 degrees..

## Other Interesting fact(s):

Motion approximates SHM when the spring does not exceed limit of proportionality during oscillations.