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 find the maximum permissible percentage error in relative density, we will follow these steps:

### Step 1: Understand the formula for Relative Density
Relative density (R) is defined as the ratio of the weight of a body in air (W_A) to its apparent loss of weight when submerged in water. The formula is given by:
\[ R = \frac{W_A}{W_A - W_W} \]
where:
- \( W_A \) = weight in air
- \( W_W \) = weight in water

### Step 2: Identify the values and their uncertainties
From the problem, we have:
- Weight in air, \( W_A = 10.0 \, \text{gf} \) with an uncertainty of \( \Delta W_A = 0.1 \, \text{gf} \)
- Weight in water, \( W_W = 5.0 \, \text{gf} \) with an uncertainty of \( \Delta W_W = 0.1 \, \text{gf} \)

### Step 3: Calculate the apparent loss of weight
The apparent loss of weight when submerged in water is given by:
\[ \text{Apparent Loss in Weight} = W_A - W_W = 10.0 \, \text{gf} - 5.0 \, \text{gf} = 5.0 \, \text{gf} \]

### Step 4: Calculate the uncertainties in the quantities
The uncertainty in the apparent loss of weight (Y) can be calculated as:
\[ \Delta Y = \Delta W_A + \Delta W_W = 0.1 \, \text{gf} + 0.1 \, \text{gf} = 0.2 \, \text{gf} \]

### Step 5: Calculate the relative errors
Now, we can calculate the relative errors for \( W_A \) and the apparent loss of weight:
- For \( W_A \):
\[ \frac{\Delta W_A}{W_A} = \frac{0.1}{10.0} = 0.01 \]

- For the apparent loss of weight:
\[ \frac{\Delta Y}{Y} = \frac{0.2}{5.0} = 0.04 \]

### Step 6: Combine the relative errors
The total relative error in relative density can be expressed as:
\[ \frac{\Delta R}{R} = \frac{\Delta W_A}{W_A} + \frac{\Delta Y}{Y} \]
Substituting the values:
\[ \frac{\Delta R}{R} = 0.01 + 0.04 = 0.05 \]

### Step 7: Convert to percentage error
To find the percentage error, we multiply by 100:
\[ \text{Percentage Error in R} = 0.05 \times 100\% = 5\% \]

### Final Answer
The maximum permissible percentage error in relative density is **5%**.

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