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%.

Answer 2

Step 1: Given relation.
The physical quantity is given as:
Q = (a × b⁴) / (c × d²)

Given:
a = (60 ± 3) Pa
b = (20 ± 0.1) m
c = (40 ± 0.2) N·s/m²
d = (50 ± 0.1) m

Step 2: Formula for percentage error.
For Q = (a × b⁴) / (c × d²), the total percentage error is:
Percentage error in Q = [ (Δa/a) + 4(Δb/b) + (Δc/c) + 2(Δd/d) ] × 100

Step 3: Calculate individual percentage errors.
Δa/a = 3/60 = 0.05 = 5%
Δb/b = 0.1/20 = 0.005 = 0.5%
Δc/c = 0.2/40 = 0.005 = 0.5%
Δd/d = 0.1/50 = 0.002 = 0.2%

Step 4: Substitute the values.
Total % error = 5 + 4(0.5) + 0.5 + 2(0.2)
= 5 + 2 + 0.5 + 0.4
= 7.9% ≈ 7%

Final Answer:
Percentage error in Q = 7%