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

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%.