Question

A physical quantity \(P\) is related to four observables \(a\), \(b\), \(c\), and \(d\) as \[ P=\frac{\sqrt{ab}\cdot d^{\alpha}}{\sqrt{c}} \] (\(\alpha\) is a constant). The percentage errors in \(a\), \(b\), \(c\), and \(d\) are \(0.5\%\) each. If the percentage error in \(P\) is \(2\%\), then \(\alpha\) is

• A. \(\frac{5}{2}\)
• B. \(\frac{2}{5}\)
• C. \(\frac{3}{4}\)
• D. \(\frac{3}{2}\)

Hint:

For products and quotients, \[ Q=x^ay^bz^c, \] the maximum percentage error is obtained by adding the absolute values of the powers multiplied by their respective percentage errors: \[ \%\Delta Q = |a|\%\Delta x + |b|\%\Delta y + |c|\%\Delta z. \]

Answer 1

The Correct Option is A

Step 1: Write P in terms of powers.
\( P = \frac{\sqrt{ab}\, d^{\alpha}}{\sqrt{c}} = a^{1/2}\, b^{1/2}\, c^{-1/2}\, d^{\alpha} \)

Step 2: Apply percentage error formula.
\( \frac{\Delta P}{P}\times 100 = \frac{1}{2}(0.5) + \frac{1}{2}(0.5) + \frac{1}{2}(0.5) + \alpha(0.5) \)
\( 2 = 0.25 + 0.25 + 0.25 + 0.5\alpha \)
\( 0.5\alpha = 1.25 \Rightarrow \alpha = 2.5 \)

\[ \boxed{\alpha = \tfrac{5}{2}} \]