The formula for the physical quantity is \(P = \frac{x^3 y}{z^2}\) and the percentage error in the determination of physical quantities \(x, y, z\) are \(0.6%, 3%\) and \(1.3%\) respectively. The percentage error in the measurement of \(P\) is
Step 1: Understanding the Concept:
When a physical quantity is calculated using a formula involving multiplication and division, the maximum fractional (or percentage) error in the result is the sum of the absolute values of the fractional errors of individual quantities multiplied by their respective powers. Step 2: Key Formula or Approach:
For a quantity \( Q = \frac{A^a B^b}{C^c} \), the maximum percentage error is given by:
\( \frac{\Delta Q}{Q} \times 100% = a\left(\frac{\Delta A}{A} \times 100%\right) + b\left(\frac{\Delta B}{B} \times 100%\right) + c\left(\frac{\Delta C}{C} \times 100%\right) \). Step 3: Detailed Explanation:
The given formula is \( P = \frac{x^3 y^1}{z^2} \).
The given percentage errors are:
For \( x \): \( \frac{\Delta x}{x} \times 100% = 0.6% \)
For \( y \): \( \frac{\Delta y}{y} \times 100% = 3% \)
For \( z \): \( \frac{\Delta z}{z} \times 100% = 1.3% \)
Applying the error propagation formula:
\[ \left( \frac{\Delta P}{P} \times 100% \right)_{\text{max}} = 3\left(\frac{\Delta x}{x} \times 100%\right) + 1\left(\frac{\Delta y}{y} \times 100%\right) + 2\left(\frac{\Delta z}{z} \times 100%\right) \]
Substitute the given values into the equation:
\[ \text{Percentage error in } P = 3(0.6%) + 1(3%) + 2(1.3%) \]
Calculate each term:
\[ \text{Percentage error in } P = 1.8% + 3.0% + 2.6% \]
Add the values together:
\[ \text{Percentage error in } P = 7.4% \]
Step 4: Final Answer:
The percentage error in the measurement of \( P \) is \( 7.4% \).