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If there are two cylinders (A and B) both with the same volume. B's radius is half of A's, so the length of B must be 4 ($2^2$) times that of A.

The uncertainty for the radius of A is the same as the for the radius of B, and the uncertainty is the same for the lengths. But the uncertainty of the volume isn't necessarily the same as for the radius)

Which cylinder A or B would have the greatest uncertainty for volume? Or would they be the same because as the percentage uncertainty for radius for B is larger than A but the percentage uncertainty for length B is less that for A? Do the uncertainties need to be know?

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What did you try? – Bernhard Mar 14 '13 at 11:28
I tried to calculate them but got in such a mess, I couldn't post them coherently. I tried using letters c and k to see which is greater but literally got so confused I couldn't make any sense of it – Jake Mar 14 '13 at 12:11

It looks like a homework question so I will not give an exact formula, just some general thoughts.

Measurement uncertainty is absolute, not relative, meaning that we can measure radius with, say, 1 mm precision, regardless if $r=50 mm$ or $r=5mm$. However, relative to the radius value, our error is different: 2% in the former case and 20% in the latter. Same applies to the length of the cylinder, by the way.

This topic is called Error propagation and it is not difficult to understand if you derivatives, but in general you cannot measure precisely the volume of too short or too thin cylinder.

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"Measurement uncertainty is absolute, not relative [...]" Well, that depends on the instrument. The uncertainty associated with reading a length off a ruler and similar tasks is absolute. – dmckee Mar 14 '13 at 14:11

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