Pendulum (Why the angle need to be less than 5 degree --- is it necessary?)

[Fu-Kwun Hwang]

A small angle vibration of pendulum can be approximated modeled with simple harmonic motion (SHM).
The equation is

 d²θ/dt²= - g* sin (θ)/L ≒ -(g/L)θ
And we get a SHM (ignore difference between sin (θ) and θ).
Some text book indicated that the angle should be less than 5 degree for the above approximation to be valid.

Why the angle should be less than 5 degree? What is the condition or restriction to set the angle to be less than 5 degree?
Can we use 6 degree? Can we use 10 degree?

The question should be asked is "what is the error if the initial angle is 10 degree?".
Under what condition the angle of the pendulum should be less than 5 degree?

The following applet let you adjust the initial angle for the pendulum,
 click right error button to start the swing.
The theorical value for gravity g=9.8 is used.
T(SHM) is calculated from 2*π *√(L/g).
The applet will re-measure the period T(real) for each half cycle.
Then, g(calc)= 4*π²*L/T²(real) is calculated , with error =100*(1-g(calc)/g) % is displayed.

The initial angle for the following applet is 20 degree and the error is less then 2%.
You can set to other angle after several period has passed (for better average value for period).
For example: 5 degree will give you almost less than 0.2% error (after several swings).

You can change the initial angle several times and the plot at the right will show you "error vs. angle".

I found a lot of students doing similar real experiment in the lab and when they found the error was too large, say 8 %. They will claim that it is due to the initial angle was too large (may be 10 degree).
But even 20 degree will cause less than 2% of error. Error was coming from some other measurement.

You can drag the pendum directly to change initial angle and length when the simulation is paused!

Embed a running copy of this simulation
<applet code='users.ntnu.fkh.pendulumperiodangle_pkg.pendulumperiodangleApplet.class' archive='http://www.phy.ntnu.edu.tw/ntnujava/ejsuser/2/ejs_pendulumperiodangle.jar' name='pendulumperiodangle3989' id='pendulumperiodangle3989' width='763' height='492' mayscript=true><param name='draggable' value='true'></applet>
Embed a running copy link(show simulation in a popuped window)
<a href='http://www.phy.ntnu.edu.tw/ntnujava/msg.php?smfejs=3989' target='smfejs'>Run simulation</a>

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