Question

A physical quantity C is related to four other quantities p, q, r and s as follows $ C = \frac{pq^2}{r^3 \sqrt{s}} $ The percentage errors in the measurement of p, q, r and s are 1%, 2%, 3% and 2% respectively. The percentage error in the measurement of C will be _______ %.

Hint:

For a physical quantity \( C \) related to other quantities by \( C = p^a q^b r^c s^d \), the maximum percentage error in \( C \) is given by \( |a| (\% \text{ error in } p) + |b| (\% \text{ error in } q) + |c| (\% \text{ error in } r) + |d| (\% \text{ error in } s) \). Apply this rule directly to the given relation and percentage errors.

Answer 1

Correct Answer: 15

To determine the percentage error in the measurement of C, we must first understand error propagation within the given formula. The expression for C is:

C = \(\frac{pq^2}{r^3 \sqrt{s}}\)

The formula for percentage error in a quantity \(Z = \frac{A^m B^n}{C^p D^q}\) is:

\(\frac{\Delta Z}{Z} \times 100 \approx m \frac{\Delta A}{A} \times 100 + n \frac{\Delta B}{B} \times 100 + p \frac{\Delta C}{C} \times 100 + q \frac{\Delta D}{D} \times 100\)

Applying this to our expression:

• The exponent for p (m) is 1.
• The exponent for q (n) is 2.
• The exponent for r (p) is 3.
• The exponent for √s (q) is 0.5, as \(\sqrt{s} = s^{0.5}\).

The percentage error in C is calculated as:

\(\frac{\Delta C}{C} \times 100 \approx 1 \cdot 1\% + 2 \cdot 2\% + 3 \cdot 3\% + 0.5 \cdot 2\%\)

The individual errors are:

• 1% for p
• 4% for q
• 9% for r
• 1% for s (resulting from 0.5 × 2%)

Summing these errors yields:

1% + 4% + 9% + 1% = 15%

Therefore, the percentage error in the measurement of C is 15%, which falls within the expected range of 15,15.