A physical quantity P is related to four observations a, b, c and d as follows:

$P=a^3b^2 / c\sqrt{d}$

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

Solution:

The given physical quantity is $P = a^3b^2 / (c\sqrt{d})$.

This can be written as $P = a^3 b^2 c^{-1} d^{-1/2}$.

For a quantity $Q = x^A y^B z^C$, the maximum percentage error in Q is given by:

$(\Delta Q / Q) \times 100\% = |A| (\Delta x / x) \times 100\% + |B| (\Delta y / y) \times 100\% + |C| (\Delta z / z) \times 100\%$

Applying this rule to P:

Percentage error in P $= (\Delta P / P) \times 100\%$

$= |3| (\Delta a / a) \times 100\% + |2| (\Delta b / b) \times 100\% + |-1| (\Delta c / c) \times 100\% + |-1/2| (\Delta d / d) \times 100\%$

Given percentage errors:

$(\Delta a / a) \times 100\% = 1\%$

$(\Delta b / b) \times 100\% = 3\%$

$(\Delta c / c) \times 100\% = 2\%$

$(\Delta d / d) \times 100\% = 4\%$

Substitute the values:

Percentage error in P $= 3 \times (1\%) + 2 \times (3\%) + 1 \times (2\%) + (1/2) \times (4\%)$

$= 3\% + 6\% + 2\% + 2\%$

$= 13\%$

Thus, the percentage error in the quantity P is 13%.