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Saturday, November 21, 2009

How to determine uncertainty in a derived quantity when powers, root,etc are involved.

As we have seen in the first two part of this series it is easier to determine the uncertainty when addition and subtraction is involved but also to determine uncertainty when multiplication and division is involved.

Today we are going to see how to determine the uncertainty when a power is involved.

Very often a derived quantity is determined basic or othere derived quantities using powers, roots, etc.

For example



As we are going to see the method to determine the uncertainty in these three cases are similar and can be adapted to other derived quantities.


Let us have a look at how to determine the uncertainty in volume.

A cube has length of L = 20.4+_0.2cm

Determine the uncertainty in volume.


V = L3

V = L*L*L






=249 = 200 (1 sf because of 0.2 )

V = L3



=8500 (2 sf because of 200)

Hence V = 8500+_200 cm3

After you have studied this example are you able to know how to find the uncertainty in any other derived quantities?

If clip_image002[22]

Then the fractional uncertainty is clip_image004[8]

What if area clip_image006[6]

Then can you deduce the fractional uncertainty?

Yes it is clip_image008[6]

Can you deduce what would be the fractional uncertainty ifclip_image010[6]?

Yes it is clip_image012[6])

Lastly you deduce the fractional uncertainty if clip_image014?

Where K is a dimensionless constant with no uncertainty.

Yes it is also clip_image012[7])

When there is a dimensionless constant with no uncertainty it does not enter in the equation to calculate the fractional uncertainty.


The area of a circle is given by the equation


If the radius r =10.1+_0.5 cm

then calculate the fractional uncertainty and the uncertainty in A.

Please not that clip_image002[28]is a dimentionless constant with no uncertainty and hence will play no part in thecalculations.

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