Question

You measure two quantities as \( A = 1.0 \,m \pm 0.2 \,m \), \( B = 2.0 \,m \pm 0.2 \,m \). We should report the correct value for \( \sqrt{AB} \) as:

• A. \( 1.4 \,m \pm 0.4 \,m \)
• B. \( 1.41 \,m \pm 0.15 \,m \)
• C. \( 1.4 \,m \pm 0.3 \,m \)
• D. \( 1.4 \,m \pm 0.2 \,m \)

Hint:

For multiplication or division, relative errors are added. The formula for error propagation in square roots is: \[ \frac{\Delta Y}{Y} = \frac{1}{2} \left( \frac{\Delta A}{A} + \frac{\Delta B}{B} \right). \]

Answer 1

The Correct Option is D

Step 1: {Given values}
Input data:
\[A = 1.0 \,m \pm 0.2 \,m, \quad B = 2.0 \,m \pm 0.2 \,m\]Derived quantity:
\[Y = \sqrt{AB}\]Step 2: {Calculate the value of \( Y \)}
Calculation of \( Y \):
\[Y = \sqrt{(1.0)(2.0)}\]\[= \sqrt{2.0} = 1.414 \,m\]Step 3: {Determine the uncertainty in \( Y \)}
Uncertainty propagation formula:
\[\frac{\Delta Y}{Y} = \frac{1}{2} \left( \frac{\Delta A}{A} + \frac{\Delta B}{B} \right)\]Substitution of values:
\[\frac{\Delta Y}{1.4} = \frac{1}{2} \left( \frac{0.2}{1.0} + \frac{0.2}{2.0} \right)\]Step 4: {Simplify the expression}
Simplification:
\[\frac{\Delta Y}{1.4} = \frac{1}{2} \left( 0.2 + 0.1 \right)\]\[\frac{\Delta Y}{1.4} = \frac{1}{2} \times 0.3 = 0.15\]\[\Delta Y = 0.15 \times 1.4 = 0.21\]Step 5: {Round off to one significant digit}
Rounding \( \Delta Y \) to one significant digit:
\[\Delta Y = 0.2 \,m\]Final result:
\[Y = 1.4 \,m \pm 0.2 \,m\]Step 6: {Verify the options}
Comparison with options: The correct option is (D).