To solve the problem of finding the maximum possible percentage error in relative density, we can follow these steps:
### Step 1: Understand the Concept of Relative Density
Relative density (RD) is defined as the ratio of the weight of an object in air (W) to the apparent weight loss of that object when submerged in water (W - W'). Mathematically, it can be expressed as:
\[ \text{RD} = \frac{W}{W - W'} \]
### Step 2: Identify Given Values
From the problem, we have:
- Weight of the body in air, \( W = 5.00 \, \text{N} \) with an uncertainty of \( \Delta W = 0.05 \, \text{N} \)
- Weight of the body in water, \( W' = 4.00 \, \text{N} \) with an uncertainty of \( \Delta W' = 0.05 \, \text{N} \)
### Step 3: Calculate Apparent Weight Loss
The apparent weight loss when the body is submerged in water is given by:
\[ \text{Apparent Weight Loss} = W - W' = 5.00 \, \text{N} - 4.00 \, \text{N} = 1.00 \, \text{N} \]
### Step 4: Calculate Uncertainty in Apparent Weight Loss
The uncertainty in the apparent weight loss is calculated by adding the uncertainties of the weights:
\[ \Delta (W - W') = \Delta W + \Delta W' = 0.05 \, \text{N} + 0.05 \, \text{N} = 0.10 \, \text{N} \]
### Step 5: Calculate Relative Density
Now, we can calculate the relative density:
\[ \text{RD} = \frac{W}{W - W'} = \frac{5.00 \, \text{N}}{1.00 \, \text{N}} = 5.00 \]
### Step 6: Calculate Relative Errors
The relative error in the measurement of relative density can be calculated using:
\[ \text{Relative Error} = \frac{\Delta W}{W} + \frac{\Delta (W - W')}{W - W'} \]
Substituting the values:
- \( \Delta W = 0.05 \, \text{N} \)
- \( W = 5.00 \, \text{N} \)
- \( \Delta (W - W') = 0.10 \, \text{N} \)
- \( W - W' = 1.00 \, \text{N} \)
Calculating each term:
1. \( \frac{\Delta W}{W} = \frac{0.05}{5.00} = 0.01 \)
2. \( \frac{\Delta (W - W')}{W - W'} = \frac{0.10}{1.00} = 0.10 \)
Thus, the total relative error is:
\[ \text{Total Relative Error} = 0.01 + 0.10 = 0.11 \]
### Step 7: Convert to Percentage
To find the maximum possible percentage error in relative density:
\[ \text{Percentage Error} = \text{Total Relative Error} \times 100 = 0.11 \times 100 = 11\% \]
### Conclusion
The maximum possible percentage error in relative density is **11%**.
