Example 

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  ${\displaystyle T_{}}$ t h e or y = 2 π 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

  
        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 t h e or y = 2 π L g

Suggested Answer 1:

 angle θ  ≈ 10 degrees for ${\displaystyle e}$ r r or = ( 2.010 - 2.006 ) 2.010 ( 100 ) = 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

YouTube

http://youtu.be/BRbCW2MsL94?t=2m16s This video shows many pendulums that goes in phase and out of phase with each other pendulum to creating a visually stunning effect.

Model:

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