Question

A physical quantity X is given by $X = \dfrac{{2{k^3}{l^2}}}{{m\sqrt n }}$ .The percentage error in the measurements of k, l, m and n are 1%, 2%, 3%, 4% respectively. The value of X is uncertain by:
A) 8%
B) 10%
C) 12%
D) None of these

Answer 1

Error: uncertainty in a measurement is called error. It is the difference between the measured value and true value.
$\varepsilon = \dfrac{{{A_m} - {A_t}}}{{{A_t}}}$ , $\varepsilon$ denotes the error, ${A_m}$ denotes the measured value and ${A_t}$ measures the true value of the quantity being measured.
In the question above we are provided with % of the error then we will use:
$\varepsilon = \dfrac{{\Delta t}}{t} \times 100$

Complete step by step solution:
Our calculation comes as we have;
$X = \dfrac{{2{k^3}{l^2}}}{{m\sqrt n }}$(Given in the question)
Therefore, we can write;
$\Rightarrow \dfrac{{\Delta X}}{X} = 2\dfrac{{\Delta l}}{l} + 3\dfrac{{\Delta k}}{k} + \dfrac{{\Delta m}}{m} + \dfrac{1}{2} \times \dfrac{{\Delta n}}{n}$ ( the powers of all the parameters have been multiplied)
We have to find the error in percentage:
$\dfrac{{\Delta X}}{X} \times 100$ ( will be equal to the below mentioned equation)
$\Rightarrow \dfrac{{\Delta X}}{X} \times 100 = (2\dfrac{{\Delta l}}{l} + 3\dfrac{{\Delta k}}{k} + \dfrac{{\Delta m}}{m} + \dfrac{1}{2} \times \dfrac{{\Delta n}}{n}) \times 100$
Substituting the values of each error
$\Rightarrow \dfrac{{\Delta X}}{X} \times 100 = (2 \times 2\% + 3 \times 1\% + 3\% + \dfrac{1}{2} \times 4\% ) \times 100$
Perform the simple calculation of multiplication and division to solve the above equation;
$\Rightarrow \dfrac{{\Delta X}}{X} \times 100 = (4\% + 3\% + 3\% + 2\% ) \\\ \Rightarrow \dfrac{{\Delta X}}{X} \times 100 = 12\% \\\$
Error obtained is 12%.
Option (C) is correct.

Note: We have different types of error, which are stated below:
Constant errors: errors which keep on repeating every time are constant errors.
Systematic errors: errors which occur according to a certain pattern or system and are classified into 4 parts: instrumental errors, personal errors, errors due to external sources, errors due to external sources or errors due to imperfection.
Gross errors: errors which occur due to improper setting of the instrument, recording observations wrongly, not to take precautions into account, using some wrong value in calculations. Random errors: it is common experience that the repeated measurement of a quantity gives values which are slightly different from each other.