Question

You measure two quantities as \(A = 1.0 \pm 0.2\,\text{m}\), \(B = 2.0 \pm 0.2\,\text{m}\). What should be the correctly reported value for \(\sqrt{AB}\)?

• A. \(1.4 \pm 0.4\,\text{m}\)
• B. \(1.41 \pm 0.51\,\text{m}\)
• C. \(1.4 \pm 0.3\,\text{m}\)
• D. \(1.4 \pm 0.2\,\text{m}\)

Hint:

For error propagation:
Add fractional errors for multiplication/division
Multiply fractional error by power for roots or powers
Round uncertainty to one significant figure

Answer 1

The Correct Option is C

Step 1: Calculate the mean value. \[ \sqrt{AB} = \sqrt{(1.0)(2.0)} = \sqrt{2} \approx 1.41 \approx 1.4 \]
Step 2: Find fractional errors. For a quantity \(Q = A^{1/2}B^{1/2}\), \[ \frac{\Delta Q}{Q} = \frac{1}{2}\left(\frac{\Delta A}{A} + \frac{\Delta B}{B}\right) \] \[ \frac{\Delta A}{A} = \frac{0.2}{1.0} = 0.2, \quad \frac{\Delta B}{B} = \frac{0.2}{2.0} = 0.1 \] \[ \frac{\Delta Q}{Q} = \frac{1}{2}(0.2 + 0.1) = 0.15 \]
Step 3: Find absolute error. \[ \Delta Q = 0.15 \times 1.41 \approx 0.21 \approx 0.3 \] Final Answer: \[ \boxed{\sqrt{AB} = (1.4 \pm 0.3)\,\text{m}} \]