How to determine uncertainty in derived quantity when multiplication or division is performed

This is the second part of a series of post on finding uncertainty in derived quantities. You can find the index here and the part on addition and subtraction here.

Now very often when you performed an experiment you meet quantities that need to be processed to obtain a derived quantity such as g, the acceleration due to gravity, or any other derived quantities.

You have already seen the first part on addition and subtraction and now you will see how to do it for multiplication and division.

If in an experiment is performed and the following quantities are measured with their uncertainties.

A = 10.2+-0.2 cm

B = 5.4 +-0.4 cm

Now if you need to process these quantities to find AB and A/B with their uncertainties how would you do it?

Uncertainty in multiplication

How would you determine the value of AB and its uncertainty?

You will have to calculate the value of AB first.

AB = 10.2 *5.4 = 55.o8 =55 (2 sf )

To determine the uncertainty in AB you will have to use the equation below.

Rearranging the equation will give you

Hence AB = 55+-5 cm²

Remember the uncertainty in A and B are to 1 sf hence the uncertainty in AB must be given to 1 sf.

Uncertainty in division

You are now going to determine the value of A/B and its uncertainty.

You will have to determine the value of A/B first.

A/B = 10.2 /5.4 =1.888 = 1.9

To determine the uncertainty of A/B  you will determine the equation below

Rearranging the equation will give you

Remember the uncertainty in A/B is given to 1 sf since the uncertainties in A and B are given to 1 sf.

Hence A/B = 1.9 +-0.2

With these two formula you can thus determine the uncertainty in any derived quantities that involves multiplication and division.

Now you can move to the next part where the method to determine uncertainty of derived quantities where powers,  square root, etc are involved.

Measurement, errors and uncertainties

Calculation

Calculation in Physics

Performing addition and subtraction

Performing multiplication and division

Performing Calculation with lg and ln

Uncertainties and errors

Uncertainty and How to process it

How to determine uncertainty in a derived quantity when addition or subtraction is performed?

How to determine uncertainty in derived quantity when multiplication or division is performed.

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

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.

Example

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

=20.4³

=8489

=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

Then the fractional uncertainty is

What if area

Then can you deduce the fractional uncertainty?

Yes it is

Can you deduce what would be the fractional uncertainty if?

Yes it is )

Lastly you deduce the fractional uncertainty if ?

Where K is a dimensionless constant with no uncertainty.

Yes it is also )

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

Question

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 is a dimentionless constant with no uncertainty and hence will play no part in thecalculations.