Question

A force F is applied on a square plate of side L. If the percentage error in F is 3% and in L is 2%, then the percentage error in pressure is \dots

• A. 7%
• B. 5%
• C. 3%
• D. 2%

Hint:

Remember the golden rule of error analysis: Errors never cancel out; they ALWAYS accumulate! Whether a variable is multiplied ($AB$) or divided ($A/B$), you must always add their percentage errors together.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are dealing with error propagation in a derived physical quantity. We must calculate the maximum possible percentage error in pressure ($P$) given the known percentage errors in the measurements of force ($F$) and length ($L$).

Step 2: Key Formula or Approach:
1. The mathematical definition of Pressure is Force divided by Area: $P = \frac{F}{A}$.
2. For a square plate, Area ($A$) is $L^2$. Therefore, $P = \frac{F}{L^2}$.
3. For a derived quantity $Z = \frac{A^x}{B^y}$, the maximum relative percentage error is the sum of the absolute individual percentage errors, multiplied by their respective exponent powers:
$$\frac{\Delta Z}{Z} % = x \left( \frac{\Delta A}{A} % \right) + y \left( \frac{\Delta B}{B} % \right)$$

Step 3: Detailed Explanation:
From the formula $P = F \cdot L^{-2}$, we can write the relative error equation.
(Errors ALWAYS add together to find the maximum possible boundary; negative powers become positive multipliers in error analysis).
$$\frac{\Delta P}{P} % = 1 \cdot \left( \frac{\Delta F}{F} % \right) + 2 \cdot \left( \frac{\Delta L}{L} % \right)$$
We are given:
Percentage error in Force $\left( \frac{\Delta F}{F} % \right) = 3%$
Percentage error in Length $\left( \frac{\Delta L}{L} % \right) = 2%$
Substitute these values into the error equation:
$$\text{Error in } P = 3% + 2 \times (2%)$$
$$\text{Error in } P = 3% + 4%$$
$$\text{Error in } P = 7%$$

Step 4: Final Answer:
The total percentage error in pressure is 7%, matching option (a).