A wire has a mass `(0.3+0.003)` gm, radius `(0.5+0.005)` mm and length `(0.6+0.006)` cm. The maximum percentage error in the measurement of density is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the maximum percentage error in the measurement of density, we can follow these steps:

### Step 1: Understand the formula for density
Density (ρ) is defined as mass (m) divided by volume (V). The volume of a cylindrical wire can be expressed as:
\[ V = A \cdot L \]
where \( A \) is the cross-sectional area and \( L \) is the length of the wire. The area \( A \) for a circular cross-section is given by:
\[ A = \pi r^2 \]
Thus, the density can be expressed as:
\[ \rho = \frac{m}{\pi r^2 L} \]

### Step 2: Identify the variables and their uncertainties
Given:
- Mass \( m = 0.3 \, \text{gm} \) with uncertainty \( \Delta m = 0.003 \, \text{gm} \)
- Radius \( r = 0.5 \, \text{mm} \) with uncertainty \( \Delta r = 0.005 \, \text{mm} \)
- Length \( L = 0.6 \, \text{cm} \) with uncertainty \( \Delta L = 0.006 \, \text{cm} \)

### Step 3: Calculate the percentage errors for each measurement
1. **Percentage error in mass**:
\[ \text{Percentage error in } m = \left( \frac{\Delta m}{m} \right) \times 100 = \left( \frac{0.003}{0.3} \right) \times 100 = 1\% \]

2. **Percentage error in radius**:
\[ \text{Percentage error in } r = \left( \frac{\Delta r}{r} \right) \times 100 = \left( \frac{0.005}{0.5} \right) \times 100 = 1\% \]
Since the radius is squared in the formula for density, the contribution to the density error will be doubled:
\[ \text{Effective percentage error in } r = 2 \times 1\% = 2\% \]

3. **Percentage error in length**:
\[ \text{Percentage error in } L = \left( \frac{\Delta L}{L} \right) \times 100 = \left( \frac{0.006}{0.6} \right) \times 100 = 1\% \]

### Step 4: Combine the percentage errors to find the total error in density
The total percentage error in density can be calculated using the formula:
\[ \text{Percentage error in } \rho = \text{Percentage error in } m + \text{Effective percentage error in } r + \text{Percentage error in } L \]
Substituting the values:
\[ \text{Percentage error in } \rho = 1\% + 2\% + 1\% = 4\% \]

### Final Answer
The maximum percentage error in the measurement of density is **4%**.

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