A force F is applied on a square plate of side L. If the percentage error in the determination of L is 2% and that in F is 4%. What is the permissible error in pressure?

Options

A. 0.08
B. 0.06
C. 0.04
D. 0.02

Explanation

To find the permissible error in pressure when a force \( F \) is applied on a square plate of side \( L \), we can follow these steps:

### Step 1: Understand the relationship between pressure, force, and area.
Pressure \( P \) is defined as the force \( F \) applied per unit area \( A \). For a square plate with side length \( L \), the area \( A \) is given by:
\[ A = L^2 \]
Thus, the pressure can be expressed as:
\[ P = \frac{F}{A} = \frac{F}{L^2} \]

### Step 2: Determine the formula for permissible error in pressure.
The permissible error in pressure can be calculated using the formula for relative error:
\[ \frac{\Delta P}{P} = \frac{\Delta F}{F} + 2 \cdot \frac{\Delta L}{L} \]
where:
- \( \Delta P \) is the absolute error in pressure,
- \( P \) is the pressure,
- \( \Delta F \) is the absolute error in force,
- \( F \) is the force,
- \( \Delta L \) is the absolute error in length,
- \( L \) is the length of the side of the square.

### Step 3: Substitute the given percentage errors.
From the problem, we know:
- The percentage error in \( L \) is \( 2\% \), which means:
\[ \frac{\Delta L}{L} = \frac{2}{100} = 0.02 \]
- The percentage error in \( F \) is \( 4\% \), which means:
\[ \frac{\Delta F}{F} = \frac{4}{100} = 0.04 \]

### Step 4: Plug in the values into the formula.
Now, substituting these values into the permissible error formula:
\[ \frac{\Delta P}{P} = 0.04 + 2 \cdot 0.02 \]
Calculating the second term:
\[ 2 \cdot 0.02 = 0.04 \]
So, we have:
\[ \frac{\Delta P}{P} = 0.04 + 0.04 = 0.08 \]

### Step 5: Convert the relative error to percentage.
To express the permissible error in pressure as a percentage, we multiply by 100:
\[ \text{Permissible Error in Pressure} = 0.08 \times 100 = 8\% \]

### Final Answer:
The permissible error in pressure is \( 8\% \).

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