The relative density of a material is found by weighing the body first in air and then in water . If the weight in air is `( 10.0 +- 0.1) gf` and the weight in water is `( 5.0 +- 0.1) gf`, then the maximum permissible percentage error in relative density is

Options

A. `1`
B. `2`
C. `3`
D. `5`

Explanation

To solve the problem of finding the maximum permissible percentage error in relative density, we can follow these steps:

### Step 1: Understand the formula for relative density
The relative density (RD) of a material is calculated using the formula:
\[ \text{Relative Density} = \frac{W_a}{W} \]
where \(W_a\) is the weight of the body in air and \(W\) is the weight of the body in water.

### Step 2: Identify the given values
From the problem, we have:
- Weight in air, \(W_a = 10.0 \, \text{gf} \pm 0.1 \, \text{gf}\)
- Weight in water, \(W = 5.0 \, \text{gf} \pm 0.1 \, \text{gf}\)

### Step 3: Calculate the weight difference
The weight of the body in water is given as:
\[ W = W_a - W_f \]
where \(W_f\) is the buoyant force (weight of the water displaced). Here, we can assume \(W_f = W\) since it is already given.

### Step 4: Calculate the total error in weight
The total error in weight when measuring in water is:
\[ \Delta W = \Delta W_a + \Delta W \]
Substituting the values:
\[ \Delta W = 0.1 \, \text{gf} + 0.1 \, \text{gf} = 0.2 \, \text{gf} \]

### Step 5: Calculate the relative density
Using the values of \(W_a\) and \(W\):
\[ \text{Relative Density} = \frac{10.0 \, \text{gf}}{5.0 \, \text{gf}} = 2.0 \]

### Step 6: Calculate the percentage error in relative density
The maximum permissible percentage error in relative density can be calculated using:
\[ \frac{\Delta \rho}{\rho} \times 100 = \frac{\Delta W_a}{W_a} \times 100 + \frac{\Delta W}{W} \times 100 \]

Substituting the values:
1. For the first term:
\[ \frac{\Delta W_a}{W_a} \times 100 = \frac{0.1}{10.0} \times 100 = 1\% \]
2. For the second term:
\[ \frac{\Delta W}{W} \times 100 = \frac{0.2}{5.0} \times 100 = 4\% \]

### Step 7: Combine the percentage errors
Now, adding the two percentage errors:
\[ \text{Total Percentage Error} = 1\% + 4\% = 5\% \]

### Final Answer
The maximum permissible percentage error in relative density is:
\[ \boxed{5\%} \]