Question

A physical quantity A is related to four observables a, b, c and d as follows, $A = \dfrac{{{a^2}{b^3}}}{{c\sqrt d }}$, the percentage error of measurements in a, b, c and d are $1\%$, $3\%$, $2\%$ and $2\%$ respectively. What is the percentage error in the quantity A?
A. $12\%$
B. $7\%$
C. $5\%$
D. $14\%$

Answer 1

Use the formula for the percentage error in the measurement of a physical quantity. This formula gives the relation between the percentage error, absolute error and actual error. Determine the absolute error in the measurement of the four observables and derive the formula for percentage error of four observables. Using these values, determine the total percentage error in the measurement of the physical quantity A.

Formula used:
The percentage error in the measurement of a physical quantity is given by
${\text{Percentage error}} = \dfrac{{{\text{Absolute error}}}}{{{\text{Actual error}}}} \times 100$ …… (1)

Complete step by step answer:
We have given that the physical quantity A is given by,
$A = \dfrac{{{a^2}{b^3}}}{{c\sqrt d }}$
The percentage error of measurements in a, b, c and d are $1\%$, $3\%$, $2\%$ and $2\%$ respectively.The absolute error of measurements in a, b, c and d are $\Delta a$, $\Delta b$, $\Delta c$ and $\Delta d$ respectively. From equation (1), the percentage errors in the measurements of a, b, c and d can be written as
$1\% = \dfrac{{\Delta a}}{a} \times 100$
$\Rightarrow 3\% = \dfrac{{\Delta b}}{b} \times 100$
$\Rightarrow 2\% = \dfrac{{\Delta c}}{c} \times 100$
$\Rightarrow 2\% = \dfrac{{\Delta d}}{d} \times 100$

Let us now calculate the total percentage error in the measurement of the physical quantity A.
$\dfrac{{\Delta A}}{A} \times 100 = \left( {2 \times \dfrac{{\Delta a}}{a} \times 100} \right) + \left( {3 \times \dfrac{{\Delta b}}{b} \times 100} \right) + \left( {1 \times \dfrac{{\Delta c}}{c} \times 100} \right) + \left( {\dfrac{1}{2} \times \dfrac{{\Delta d}}{d} \times 100} \right)$
Substitute $1\%$ for $\dfrac{{\Delta a}}{a} \times 100$, $3\%$ for $\dfrac{{\Delta b}}{b} \times 100$, $2\%$ for $\dfrac{{\Delta c}}{c} \times 100$ and $2\%$ for $\dfrac{{\Delta d}}{d} \times 100$ in the above equation.
$\dfrac{{\Delta A}}{A} \times 100 = \left( {2 \times 1\% } \right) + \left( {3 \times 3\% } \right) + \left( {1 \times 2\% } \right) + \left( {\dfrac{1}{2} \times 2\% } \right)$
$\Rightarrow \dfrac{{\Delta A}}{A} \times 100 = \left( {2\% } \right) + \left( {9\% } \right) + \left( {2\% } \right) + \left( {1\% } \right)$
$\therefore \dfrac{{\Delta A}}{A} \times 100 = 14\%$

Therefore, the total error in the measurement of the physical quantity A is $14\%$. Hence, the correct option is D.

Note: The students may think why the powers of the four observables used in the formula for physical quantity A are used in the formula for determination of total percentage error in the measurement of A. But the students should keep in mind that we should use the multiple factor (power of the variables) which denotes the number of time that observable is used while calculating the total percentage error in measurement of A.