- #1

etotheipi

2020 Award

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## Homework Statement:

- See below

## Relevant Equations:

- N/A

I've got to do an experiment that essentially involves rolling a ball bearing down a (frictional) ramp and measuring its acceleration. It's quoted in the manual that the linear acceleration of a ball bearing rolling down a ramp at angle ##\theta## is ##a = \frac{5}{9} g \sin{\theta}##. When I worked it out myself, I wrote down two equations$$mg\sin{\theta} - F = ma$$ $$rF = \frac{Ia}{r} \implies F = \frac{Ia}{r^2}$$which then gives$$mg\sin{\theta} = a \left(m + \frac{I}{r^2} \right) \implies a = \frac{g\sin{\theta}}{1 + \frac{I}{mr^2}}$$For a ball bearing, a sphere, ##I = \frac{2}{5} mr^2##, so we should get$$a = \frac{5}{7} g\sin{\theta}$$which is a different coefficient to the one quoted. I thought it was pretty unlikely that there's a mistake in the handbook, so I presumed there was an aspect of the model that I haven't included in the analysis above. I wondered if anyone could tell how to get to the quoted equation? Thanks!