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 find the percentage error in the measurement of C, we first need to understand how errors propagate through the given formula. The expression for C is: 

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

The percentage error formula for a product or quotient involving powers, given a function like \(Z = \frac{A^m B^n}{C^p D^q}\), 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:

• m = 1 for p
• n = 2 for q
• p = 3 for r
• q = 0.5 for √s (since \(\sqrt{s} = s^{0.5}\))

The percentage error in C is:

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

Calculating this gives:

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

Add these errors together:

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

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

Answer 2

The physical quantity C is given by: \[ C = \frac{pq^2}{r^3 s^{1/2}} = p^1 q^2 r^{-3} s^{-1/2} \] The percentage error in C is given by the sum of the percentage errors in each quantity multiplied by the absolute value of their exponents in the expression for C. \[ \left( \frac{\Delta C}{C} \times 100 \right)_{max} = \left| \frac{\partial C}{\partial p} \frac{p}{C} \right| \left( \frac{\Delta p}{p} \times 100 \right) + \left| \frac{\partial C}{\partial q} \frac{q}{C} \right| \left( \frac{\Delta q}{q} \times 100 \right) + \left| \frac{\partial C}{\partial r} \frac{r}{C} \right| \left( \frac{\Delta r}{r} \times 100 \right) + \left| \frac{\partial C}{\partial s} \frac{s}{C} \right| \left( \frac{\Delta s}{s} \times 100 \right) \] Alternatively, using the rule for percentage errors: \[ % \text{ error in } C = |1 \times (% \text{ error in } p)| + |2 \times (% \text{ error in } q)| + |-3 \times (% \text{ error in } r)| + |-\frac{1}{2} \times (% \text{ error in } s)| \] Given percentage errors: % error in p = 1% % error in q = 2% % error in r = 3% % error in s = 2% Substituting these values: \[ % \text{ error in } C = |1 \times 1%| + |2 \times 2%| + |-3 \times 3%| + |-\frac{1}{2} \times 2%| \] \[ % \text{ error in } C = 1% + 4% + 9% + 1% \] \[ % \text{ error in } C = 15% \] The maximum percentage error in the measurement of C will be 15%.