A material has a Poisson's ratio 0.3. It suffers a uniform longitudinal strain `4.5 xx 10^(-3)`, calculate the percentage change in its volume.

Options

A. `0.15%`
B. `0.02%`
C. `0.24%`
D. `0.18%`

Explanation

To solve the problem, we need to calculate the percentage change in volume of a material given its Poisson's ratio and the longitudinal strain. Here’s a step-by-step solution:

### Step 1: Understand the given values
- Poisson's ratio (σ) = 0.3
- Longitudinal strain (ΔL/L) = 4.5 × 10^(-3)

### Step 2: Relate Poisson's ratio to lateral strain
Poisson's ratio is defined as the negative ratio of lateral strain to longitudinal strain:
\[ \sigma = -\frac{\Delta r / r}{\Delta L / L} \]
Where:
- Δr/r = lateral strain
- ΔL/L = longitudinal strain

### Step 3: Calculate the lateral strain
Rearranging the formula gives:
\[ \Delta r / r = -\sigma \times \Delta L / L \]
Substituting the known values:
\[ \Delta r / r = -0.3 \times 4.5 \times 10^{-3} \]
Calculating this:
\[ \Delta r / r = -1.35 \times 10^{-3} \]

### Step 4: Calculate the fractional change in volume
The fractional change in volume (ΔV/V) can be expressed as:
\[ \frac{\Delta V}{V} = 2 \left(\frac{\Delta r}{r}\right) + \left(\frac{\Delta L}{L}\right) \]
Substituting the values we have:
\[ \frac{\Delta V}{V} = 2 \times (-1.35 \times 10^{-3}) + 4.5 \times 10^{-3} \]
Calculating this:
\[ \frac{\Delta V}{V} = -2.7 \times 10^{-3} + 4.5 \times 10^{-3} = 1.8 \times 10^{-3} \]

### Step 5: Convert fractional change to percentage change
To find the percentage change in volume, we multiply the fractional change by 100:
\[ \text{Percentage change in volume} = \left(\frac{\Delta V}{V}\right) \times 100 \]
Substituting the value we calculated:
\[ \text{Percentage change in volume} = 1.8 \times 10^{-3} \times 100 = 0.18\% \]

### Final Answer
The percentage change in volume is **0.18%**.

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