Question

Show that the frictional error in the ${n^{th}}$ power of a quantity is equal to n times the fractional error in the quantity itself. State the general rule for evaluating the error in a combined calculation.

Answer 1

Hint: The frictional error is also known as instrument friction error that is caused by friction in the instrument mechanism. The error is measured by reading the instrument and vibrating it and taking the reading once again. The difference between the two readings is known as the friction error or instrumental friction error. In this problem we need to consider a quantity and we need to prove that frictional error in the${n^{th}}$ power of that quantity is equal to $n$ times the fractional error.

Complete step by step answer:
Let the quantity be $P$
And ${n^{th}}$ power of $P$ is $Z$
That is $Z = {P^n}$ …………$\left( 1 \right)$
Let the error $\Delta Z$ in $Z$ is given as
$Z \pm \Delta Z = {\left( {P \pm \Delta P} \right)^n}$
On simplifying the above equation we get
\[Z \pm \Delta Z = {P^n}{\left( {1 \pm \dfrac{{\Delta P}}{P}} \right)^n}\]………… $\left( 2 \right)$
Substituting equation $\left( 1 \right)$ in equation $\left( 2 \right)$
\[Z \pm \Delta Z = Z{\left( {1 \pm \dfrac{{\Delta P}}{P}} \right)^n}\]

Dividing the above equation by $Z$
\[1 \pm \dfrac{{\Delta Z}}{Z} = \left( {1 \pm n\dfrac{{\Delta P}}{P}} \right)\]
On cancelling $1$ from both sides we get
\[\dfrac{{\Delta Z}}{Z} = n\dfrac{{\Delta P}}{P}\]
Therefore from the above equation it is clear that the frictional error in the ${n^{th}}$ power of a quantity is equal to n times the fractional error in the quantity itself
The general rule for evaluating the error in a combined calculation is given as,
If $Z = \dfrac{{{A^P}{B^P}}}{{{C^r}}}$
Then the maximum fractional error is given as,
\[\dfrac{{\Delta Z}}{Z} = \dfrac{{\Delta A}}{A} + \dfrac{{\Delta B}}{B} + \dfrac{{\Delta C}}{C}\]

Note: The fractional error is defined as the value of the error divided by the quantity.That is $\dfrac{{\Delta X}}{X}$ . The fractional error is also known as absolute error it is also known as ratio of mean absolute error to the mean value of measured quantity. When this fractional error or relative error is multiplied by $100$ we get a percentage error.