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Christd · Apr 17, 2013, 09:09 AM

Hello, I was asked to design an experiment to measure the decay of Amplitude of a ball rolling in simple harmonic motion in a bowl. How would I go about measuring the amplitude of this? Would I just measure the displacement from equilibrium in the horizontal plane and not take into account the height? Thank you

ebaines · Apr 17, 2013, 09:23 AM

If you have a way of measuring horizontal displacement accurately, and if you know the shape of the bowl (is it a hemisphere?) you can calculate the corresponding amplitude either as a displacement angle or as arc length from the equilibrium point.

Christd · Apr 17, 2013, 09:33 AM

Originally Posted by ebaines

If you have a way of measuring horizontal displacement accurately, and if you know the shape of the bowl (is it a hemisphere?) you can calculate the corresponding amplitude either as a displacement angle or as arc length from the equilibrium point.

Thanks for the reply, yes it is a hemisphere. After measuring the horizontal displacement, what other measurements, if any, would I have to take to get the arc length?

ebaines · Apr 17, 2013, 09:44 AM

All you need to know is the radius of the hemisphere and the magnituide of horzontal displacement. For horizontal displacement x you have:

$\sin \theta = \frac x R$

So:

$\theta = sin^{-1} (\frac x R)$

and arc length S is:

$S = R \theta = R sin^{-1} (\frac x R)$

Christd · Apr 17, 2013, 09:58 AM

Originally Posted by ebaines

All yuo need to know is the radius of the hemisphere and the magnituide of horzontal displacement. For horizontal displacement x you have:

$\sin \theta = \frac x R$

So:

$\theta = sin^{-1} (\frac x R)$

and arc length S is:

$S = R \theta = R sin^{-1} (\frac x R)$

Thank You! :)

Christd · Apr 17, 2013, 10:48 AM

One last thing, if the bowl was shallow could I measure the horizontal displacement and ignore the height and draw a graph of amplitude against time to show an exponential decrease or would that not work?

ebaines · Apr 17, 2013, 12:06 PM

Sure it would work. If all you're trying to demonstrate is decaying amplitude i don't think it really matters whether you measure arc length or horizontal displacement. I would recommend that you try and keep the angle of displacement $\theta$ fairly small, so that $\sin \theta \approx \theta$ - this makes it a lot easier to develop the equation of motion for the ball rolling in the bowl than if $\theta$ is large.