To find the relative density of the material of a body and the maximum permissible percentage error, we can follow these steps:
### Step 1: Identify the weights
We have the following weights:
- Weight in air (W_air) = \(5.00 \, \text{N} \pm 0.05 \, \text{N}\)
- Weight in water (W_water) = \(4.00 \, \text{N} \pm 0.05 \, \text{N}\)
### Step 2: Calculate the loss of weight in water
The loss of weight in water (W_loss) can be calculated as:
\[
W_{\text{loss}} = W_{\text{air}} - W_{\text{water}} = 5.00 \, \text{N} - 4.00 \, \text{N} = 1.00 \, \text{N}
\]
Now, we need to consider the errors. The error in the loss of weight will be the sum of the absolute errors in the weights:
\[
\Delta W_{\text{loss}} = \Delta W_{\text{air}} + \Delta W_{\text{water}} = 0.05 \, \text{N} + 0.05 \, \text{N} = 0.10 \, \text{N}
\]
Thus, the loss of weight in water is:
\[
W_{\text{loss}} = 1.00 \, \text{N} \pm 0.10 \, \text{N}
\]
### Step 3: Calculate the relative density
The relative density (RD) is given by the formula:
\[
\text{Relative Density} = \frac{W_{\text{air}}}{W_{\text{loss}}}
\]
Substituting the values:
\[
\text{Relative Density} = \frac{5.00 \, \text{N}}{1.00 \, \text{N}} = 5.00
\]
### Step 4: Calculate the maximum permissible percentage error
To find the maximum permissible percentage error in the relative density, we use the formula for the propagation of errors:
\[
\frac{\Delta RD}{RD} = \frac{\Delta W_{\text{air}}}{W_{\text{air}}} + \frac{\Delta W_{\text{loss}}}{W_{\text{loss}}}
\]
Substituting the values:
- \(\Delta W_{\text{air}} = 0.05 \, \text{N}\)
- \(\Delta W_{\text{loss}} = 0.10 \, \text{N}\)
Now calculate:
\[
\frac{\Delta RD}{RD} = \frac{0.05}{5.00} + \frac{0.10}{1.00}
\]
Calculating each term:
\[
\frac{0.05}{5.00} = 0.01 \quad \text{(or 1%)} \quad \text{and} \quad \frac{0.10}{1.00} = 0.10 \quad \text{(or 10%)}
\]
Adding these:
\[
\frac{\Delta RD}{RD} = 0.01 + 0.10 = 0.11 \quad \text{(or 11%)}
\]
### Final Result
Thus, the relative density is:
\[
\text{Relative Density} = 5.00 \pm 11\%
\]
