SHM Chapter 01b pendulum SHM?

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:

1.     Define the question in your own words
2.     Plan an investigation to explore angle of release to record the actual period T and theoretical period  ${\displaystyle T_{the\text{or}y}=2\pi \sqrt{\frac{L}{g}}}$  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/s²
3.     A suggested record of the results could look like this

   

angle / degree	T /s	Ttheory / s	${\displaystyle err\text{or}=\frac{(T-T_{the\text{or}y})}{T}100\%}$
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  ≈ ${\displaystyle T_{the\text{or}y}=2\pi \sqrt{\frac{L}{g}}}$

Suggested Answer 1:

 angle θ  ≈ 10 degrees for ${\displaystyle err\text{or}=\frac{(2.010-2.006)}{2.010}\left(100\right)=0.2\%}$, 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.

Real Life Application of Small Angle Approximations:

Astronomical applications of the Small Angle Approximation

Model:

http://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHM01b/SHM01b_Simulation.xhtml