The relative density of a material of a body is found by weighing it first in air and then in water. If the weight of he body in air is `W_(1) = 8.00 +- N` and the weight in water is `W_(2) = 6.00 +- 0.05 N`, then the relative density `rho_(r ) = W_(1)//( W_(1) - W_(2))` with the maximum permissible eror is

Options

A. `4.00 +- 0.62%`
B. `4.00 +- 0.82%`
C. `4.00 +- 3.2%`
D. `4.00+- 5.62%`

Explanation

To solve the problem of finding the relative density and its maximum permissible error, we will follow these steps:

### Step 1: Calculate the Relative Density
The formula for relative density (\( \rho_r \)) is given by:
\[ \rho_r = \frac{W_1}{W_1 - W_2} \]
Where:
- \( W_1 = 8.00 \, \text{N} \) (weight in air)
- \( W_2 = 6.00 \, \text{N} \) (weight in water)

Substituting the values:
\[ \rho_r = \frac{8.00}{8.00 - 6.00} = \frac{8.00}{2.00} = 4.00 \]

### Step 2: Determine the Errors in Measurements
The errors in the measurements are:
- Error in \( W_1 \): \( \Delta W_1 = 0.05 \, \text{N} \)
- Error in \( W_2 \): \( \Delta W_2 = 0.05 \, \text{N} \)

### Step 3: Calculate the Error in \( W_1 - W_2 \)
The error in the difference \( W_1 - W_2 \) can be calculated as:
\[ \Delta (W_1 - W_2) = \Delta W_1 + \Delta W_2 = 0.05 + 0.05 = 0.10 \, \text{N} \]

### Step 4: Calculate the Error in Relative Density
Using the formula for relative density, we can find the error in \( \rho_r \):
\[ \rho_r = \frac{W_1}{W_1 - W_2} \]
The relative density can be expressed as:
\[ \rho_r = W_1 \cdot (W_1 - W_2)^{-1} \]
The error in \( \rho_r \) can be calculated using the formula for propagation of errors:
\[ \frac{\Delta \rho_r}{\rho_r} = \frac{\Delta W_1}{W_1} + \frac{\Delta (W_1 - W_2)}{(W_1 - W_2)} \]
Substituting the values:
\[ \frac{\Delta \rho_r}{4} = \frac{0.05}{8.00} + \frac{0.10}{2.00} \]
Calculating each term:
\[ \frac{0.05}{8.00} = 0.00625 \]
\[ \frac{0.10}{2.00} = 0.05 \]
Adding these:
\[ \frac{\Delta \rho_r}{4} = 0.00625 + 0.05 = 0.05625 \]

### Step 5: Calculate the Maximum Permissible Error
Now, multiply by \( \rho_r \) to find \( \Delta \rho_r \):
\[ \Delta \rho_r = 4 \times 0.05625 = 0.225 \]

### Step 6: Express the Result
Thus, the relative density with its maximum permissible error is:
\[ \rho_r = 4.00 \pm 0.225 \]