To determine the Young's modulus of a wire using the provided data, we can follow these steps:
### Step 1: Understand the formula for Young's modulus
The formula for Young's modulus (Y) is given by:
\[
Y = \frac{F \cdot L}{A \cdot \Delta L}
\]
Where:
- \( F \) is the force applied (in Newtons),
- \( L \) is the original length of the wire (in meters),
- \( A \) is the cross-sectional area of the wire (in square meters),
- \( \Delta L \) is the extension of the wire (in meters).
### Step 2: Convert the measurements to SI units
1. **Length of the wire (L)**: Given as 2 m (already in SI units).
2. **Extension (\(\Delta L\))**: Given as 0.8 mm, which needs to be converted to meters:
\[
\Delta L = 0.8 \, \text{mm} = 0.8 \times 10^{-3} \, \text{m} = 0.0008 \, \text{m}
\]
3. **Diameter of the wire (d)**: Given as 0.4 mm, which needs to be converted to meters:
\[
d = 0.4 \, \text{mm} = 0.4 \times 10^{-3} \, \text{m} = 0.0004 \, \text{m}
\]
### Step 3: Calculate the force (F)
The force is the weight of the load applied, which can be calculated using:
\[
F = m \cdot g
\]
Where:
- \( m = 1.0 \, \text{kg} \)
- \( g = 9.8 \, \text{m/s}^2 \)
Calculating the force:
\[
F = 1.0 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 9.8 \, \text{N}
\]
### Step 4: Calculate the cross-sectional area (A)
The cross-sectional area of the wire can be calculated using the formula for the area of a circle:
\[
A = \pi \left(\frac{d}{2}\right)^2
\]
Calculating the area:
\[
A = \pi \left(\frac{0.0004}{2}\right)^2 = \pi \left(0.0002\right)^2 = \pi \cdot 0.00000004 \approx 3.14 \cdot 0.00000004 \approx 1.256 \times 10^{-7} \, \text{m}^2
\]
### Step 5: Substitute the values into the Young's modulus formula
Now we can substitute the values into the Young's modulus formula:
\[
Y = \frac{9.8 \, \text{N} \cdot 2 \, \text{m}}{1.256 \times 10^{-7} \, \text{m}^2 \cdot 0.0008 \, \text{m}}
\]
Calculating the denominator:
\[
1.256 \times 10^{-7} \cdot 0.0008 = 1.0048 \times 10^{-10} \, \text{m}^3
\]
Now substituting back:
\[
Y = \frac{19.6}{1.0048 \times 10^{-10}} \approx 1.95 \times 10^{11} \, \text{N/m}^2
\]
### Step 6: Calculate the uncertainty in Young's modulus
The relative uncertainty in Young's modulus can be calculated using the formula:
\[
\frac{\Delta Y}{Y} = 2 \frac{\Delta d}{d} + \frac{\Delta L}{L}
\]
Where:
- \(\Delta d = 0.01 \, \text{mm} = 0.01 \times 10^{-3} \, \text{m} = 0.00001 \, \text{m}\)
- \(\Delta L = 0.05 \, \text{mm} = 0.05 \times 10^{-3} \, \text{m} = 0.00005 \, \text{m}\)
Calculating the relative uncertainties:
\[
\frac{\Delta d}{d} = \frac{0.00001}{0.0004} = 0.025
\]
\[
\frac{\Delta L}{L} = \frac{0.00005}{0.0008} = 0.0625
\]
Now substituting into the relative uncertainty formula:
\[
\frac{\Delta Y}{Y} = 2(0.025) + 0.0625 = 0.1125
\]
Calculating the uncertainty:
\[
\Delta Y = 0.1125 \times 1.95 \times 10^{11} \approx 2.2 \times 10^{10} \, \text{N/m}^2
\]
### Final Result
Thus, the Young's modulus obtained from the reading is:
\[
Y = 1.95 \times 10^{11} \pm 2.2 \times 10^{10} \, \text{N/m}^2
\]
