Question

the diameter of tennis ball is 3.774 , the error is 0.012 , the percentage error of tennis ball is 0.318% . how to get the percentage uncertainty on the volume of the tennis ball

Answer 1

To find uncertainty for volume for tennis ball, we must consider the path we need to take to get the volume.

1) From diameter to radius
2) Radius to volume

To find radius from diameter, we have to divide the diameter by 2. This also means that the error for radius is divided by 2.

Thus, now:
\[R=1.887\pm0.006 units\] Notice that fractional uncertainty does not change here.

Now, I quote without proof the uncertainty for power:
\[For\:q=x^{m}y^{n}, Fractionl\:Uncertainty\:\frac{\delta q}{q}=\sqrt{(m\frac{\delta x}{x})^{2}+(n\frac{\delta y}{y})^{2}}\]

Here, we let y=1 so there isn't uncertainty and let r be the x in the formula. Since volume is to the R^3, m=3.
\[\frac{\delta r}{r}=\sqrt{(3\frac{\delta r}{r})^{2}}=3\frac{0.006}{1.887}=9.54*10^{-3}\:units^{3}\]

Answer 2

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