Question

If the measured values of two quantities are $A \pm \Delta A$ and $B \pm \Delta B$, $\Delta A$ and $\Delta B$ being the mean absolute errors. What is the maximum possible error in A+B?
Show that if $Z = \dfrac{A}{B}{\text{ then }}\dfrac{{\Delta Z}}{Z} = \dfrac{{\Delta A}}{A} + \dfrac{{\Delta B}}{B}$.

Answer 1

Hint: Whenever we measure the physical quantity experimentally, we repeat the experiment many times and take many readings to eliminate the errors. Usually errors will be added when we add two physical quantities because we always care about the maximum error one can obtain.

Formula used:
$\ln \left[ {\dfrac{A}{B}} \right] = \ln \left[ A \right] - \ln \left[ B \right]$
$d(\ln z) = \dfrac{{dz}}{z}$

Complete step by step answer:
Usually errors can be broadly classified into two categories. They are systematic errors and random errors. Zero errors and improperly calibrated instrument errors come under systematic errors. Zero errors can be further classified into two types. They are positive zero error and negative zero error. If there is a positive zero error then to the final reading we should subtract the error and if it is a negative zero error then in the final reading we should add the error. By properly calibrating the instrument we can reduce the systematic error.
The proper way of representing the physical quantity measured is a combination of actual value and its error.
We were given with $A \pm \Delta A$ and $B \pm \Delta B$
If we add A and B, then we should add errors too to get the maximum possible error in A+B.
Hence the maximum possible error in A+B will be $\Delta A + \Delta B$
We were given with
$Z = \dfrac{A}{B}{\text{ }}$
By applying logarithm on bo0th sides we will get
$\eqalign{
  & \Rightarrow \ln \left[ {\dfrac{A}{B}} \right] = \ln \left[ A \right] - \ln \left[ B \right] \cr
  & \therefore \ln \left[ Z \right] = \ln \left[ A \right] - \ln \left[ B \right] \cr} $
$d(\ln z) = \dfrac{{dz}}{z}$
By differentiating of both sides we will get
$ \dfrac{{\Delta Z}}{Z} = \dfrac{{\Delta A}}{A} + \dfrac{{\Delta B}}{B}$
The reason why we added errors is because we always find out the maximum error.
Hence proved.

Note:
The proof for the second statement can also be done without applying logarithm on both sides too. There is the formula to find out the differential of ratio of two quantities. We will get the same answer by that method too. Even in the experiment to determine the time period of the pendulum, we add the errors.