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Phy 201
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Chapter 1 Text Question #11
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Chapter 1, question number 11, for a Phy 201 student:
11) What roughly is the percent uncertainty in the volume of a spherical beach ball whose radius = 2.86 +/- 0.09m
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I understand that you find the percent uncertainty by dividing the estimated uncertainty (0.09m) by the measured value (the derived value for volume, using the equation volume of a sphere is equal to 4/3*pi*r^3), quantity multiplied by 100, to find the percent. I keep coming up a decimal or two short of the given answer. I come up with .9%, what am I missing? I also noticed that the estimated uncertainty is in meters, while the derived volume is in m^3, could this be my issue?
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That is the crux of the issue.
If you divide a quantity with units of meters by a quantity with units of meters^3, you get a quantity with units m^-2, not a pure number. A percent uncertainty is a pure number. This is the same as saying that your uncertainty and the quantity for which you are finding the uncertainty have to be the same type of quantity. One other way of saying this: if the volume is, say 15 m^3 with a 2% uncertainty, then since 2% of 15 m^3 is 0.3 m^3, the uncertainty has the same units. A given percent of a quantity is another quantity with the same units.
In this case you could figure out the volume of a ball of radius 2.86 m, then the volume of a ball of radius 2.86 m + 0.09 m, and the volume of a ball of radius 2.86 m - 0.09 m.
Since .09 m is much less than 2.86 m, the added volume you get by adding 0.09 m to the radius is very nearly the same as the volume you would lose by decreasing the radius by 0.09 m, and if you have calculated the three volumes you will see that this is the case.
If you then divide the change in volume that results from a 0.09 m change in radius, and convert the result to a percent, you will get the correct percent change.
There is actually a simpler alternative. You can easily verify that 0.09 m is about 3% of 2.86. So there is a 3% uncertainty in the radius.
A 3% increase in the radius will give you 1.03 * original radius. The volume depends on the cube of the radius, and the cube of the increased radius is (1.03)^3 * (original radius)^3.
Since 1.03^3 is about 1.09, the cubed radius increased by about 9%. This will increase the volume by about 9%.
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