Question

A physical quantity \( Q \) is related to four observables \( a \), \( b \), \( c \), and \( d \) as follows: \[ Q = \frac{a b^4}{c d^2} \] Where:
- \( a = (60 \pm 3) \, \text{Pa} \),
- \( b = (20 \pm 0.1) \, \text{m} \),
- \( c = (40 \pm 0.2) \, \text{N·s/m}^2 \),
- \( d = (50 \pm 0.1) \, \text{m} \).
Then the percentage error in \( Q \) is:

Hint:

When calculating the percentage error for a product or quotient, sum the individual errors for each quantity involved, considering the powers to which they are raised.

Answer 1

Correct Answer: 7

The percentage error in \( Q \) is calculated using the formula for error propagation: \[ \frac{\Delta Q}{Q} = \frac{\Delta a}{a} + 4 \frac{\Delta b}{b} + 2 \frac{\Delta c}{c} + 2 \frac{\Delta d}{d} \] Substituting the values: \[ \frac{\Delta Q}{Q} = \frac{3}{60} + 4 \times \frac{0.1}{20} + 2 \times \frac{0.2}{40} + 2 \times \frac{0.1}{50} \] \[ \frac{\Delta Q}{Q} = 0.05 + 0.02 + 0.01 + 0.008 = 0.07 \] Thus, the percentage error in \( Q \) is 7%.