At the end of Lecture 13 on Simple Harmonic Oscillators, what is the reason for the period of the marble being longer than the period of the airtrac?

Question

anonymous · Answer

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anonymous · Answer

If you were to buy an air track to show SHM you would spend a lot of money.  You could however as a teacher explain the concept of low angle approximation and only need to buy a section of string and find a rock from the yard and you will very closely approximate the same SHM at angles under 15 degrees although I think WL uses 5 degrees which is even better.

anonymous · Answer

The period of the air trac is longer (21.5 s) than the period of the marble (1.85s).  The question is: why is the SHO approximation for the period valid in the case of the air trac but invalid for the case of the marble?  The reason is the SHO applies to one dimensional motion only.  Even though the air trac is two-dimensional, the curve is so flat it is excellently approximated by a straight line.  So, the SHO approximation for the period of the motion is very close to the actual period.  As the curvature of the surface increases, it can no longer be ignored. Now, the length of the circular arc becomes noticeably greater than the length of the distance between the ends, causing the period to deviate (increase) from the period predicted by the SHO model.

anonymous · Answer

The period is different because hat marble is not only translating but also rolling. And While rolling the net translational accelaration is different than when the body is only translating. 
Hence, the difference in time period.