Question: 42. A physical quantity P is related to four observations a, b, c and d as follows:
P=a^{3} b^{2} / c \sqrt{d}The percentage errors of measurement in a, b, c and d are 1 \%, 3 \%, 2 \%, and 4 \% respectively.
The percentage error in the quantity P is
(1) 10 \%
(2) 2 \%
(3) 13 \%
(4) 15 \%
Answer: Option (3)
Explanation:
When a physical quantity depends on measured quantities raised to powers,
the percentage error in the result is obtained by adding
the absolute percentage errors multiplied by their respective powers.
The given relation is:
P = \frac{a^{3} b^{2}}{c \sqrt{d}}This can also be written as:
P = a^{3} b^{2} c^{-1} d^{-1/2}The percentage error in P is:
\frac{\Delta P}{P} \times 100 = 3 \left(\frac{\Delta a}{a} \times 100\right) + 2 \left(\frac{\Delta b}{b} \times 100\right) + 1 \left(\frac{\Delta c}{c} \times 100\right) + \frac{1}{2} \left(\frac{\Delta d}{d} \times 100\right)Substituting the given percentage errors:
= 3 \times 1 + 2 \times 3 + 1 \times 2 + \frac{1}{2} \times 4 = 3 + 6 + 2 + 2 = 13 \%Therefore, the percentage error in the quantity P is 13 \%,
which corresponds to Option (3).