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

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.

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?

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)

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! :)

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?

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.