The pressure on a circular plate is measured by measuring the force on the plate and the radius of the plate. If the errors in measurement of force and radius are 5% and 3% , respectively , then the percentage of error in the measurement of pressure is

Options

A. 8
B. 14
C. 11
D. 12

Answer

To find the percentage error in the measurement of pressure on a circular plate, we can follow these steps:

### Step 1: Understand the formula for pressure
The pressure \( P \) on a circular plate is given by the formula:
\[ P = \frac{F}{A} \]
where \( F \) is the force applied on the plate and \( A \) is the area of the plate.

### Step 2: Calculate the area of the circular plate
The area \( A \) of a circular plate can be expressed in terms of its radius \( r \):
\[ A = \pi r^2 \]
Thus, we can rewrite the pressure formula as:
\[ P = \frac{F}{\pi r^2} \]

### Step 3: Identify the errors in measurements
We are given:
- The percentage error in force \( F \) is \( 5\% \).
- The percentage error in radius \( r \) is \( 3\% \).

### Step 4: Use the formula for error propagation
To find the percentage error in pressure \( P \), we need to consider how errors in \( F \) and \( r \) affect \( P \). The formula for the propagation of errors in this case is:
\[ \frac{\Delta P}{P} \times 100 = \frac{\Delta F}{F} \times 100 - 2 \times \frac{\Delta r}{r} \times 100 \]
Here, the negative sign for the radius indicates that an increase in radius will decrease the pressure.

### Step 5: Substitute the known values
Substituting the known percentage errors:
\[ \% \text{Error in } P = \% \text{Error in } F - 2 \times \% \text{Error in } r \]
\[ \% \text{Error in } P = 5\% - 2 \times 3\% \]
\[ \% \text{Error in } P = 5\% - 6\% \]
\[ \% \text{Error in } P = 5\% + 6\% = 11\% \]

### Step 6: Conclusion
The percentage error in the measurement of pressure is:
\[ \boxed{11\%} \]