The Young modulus (Y) of a material is given by `Y=(WL)/(pir^(2)l)`. If the percentage error in W, L, r and l are `0.5%, 1%, 3%" and "4%` respectively then maximum percentage error in Y is -

Options

A. `7.5%`
B. `9%`
C. `11.5%`
D. `13%`

Answer

To find the maximum percentage error in Young's modulus (Y), we start with the formula for Young's modulus given by:

\[ Y = \frac{WL}{\pi r^2 l} \]

Where:
- \( W \) is the weight,
- \( L \) is the length,
- \( r \) is the radius,
- \( l \) is the original length of the material.

### Step 1: Identify the variables and their percentage errors
The percentage errors in the measurements are given as follows:
- Percentage error in \( W \) = \( 0.5\% \)
- Percentage error in \( L \) = \( 1\% \)
- Percentage error in \( r \) = \( 3\% \)
- Percentage error in \( l \) = \( 4\% \)

### Step 2: Write the formula for maximum percentage error
Using error propagation, the maximum percentage error in \( Y \) can be calculated using the formula:

\[ \frac{\Delta Y}{Y} = \frac{\Delta W}{W} + \frac{\Delta L}{L} + 2 \frac{\Delta r}{r} + \frac{\Delta l}{l} \]

### Step 3: Substitute the percentage errors into the formula
Now we substitute the given percentage errors into the formula:

\[ \frac{\Delta Y}{Y} = 0.5\% + 1\% + 2(3\%) + 4\% \]

### Step 4: Calculate the total percentage error
Calculating the right side:

\[ \frac{\Delta Y}{Y} = 0.5 + 1 + 6 + 4 \]

\[ \frac{\Delta Y}{Y} = 11.5\% \]

### Step 5: Conclusion
Thus, the maximum percentage error in Young's modulus \( Y \) is:

\[ \text{Maximum percentage error in } Y = 11.5\% \]

### Final Answer
The maximum percentage error in Young's modulus \( Y \) is **11.5%**.

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