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Let's solve the given problem step-by-step.
### Given Data:
1. Length of the steel wire, [tex]\( L \)[/tex] = 8 meters (m).
2. Diameter of the steel wire, [tex]\( d \)[/tex] = 4 millimeters (mm) = 0.004 meters (m) (since 1 mm = 0.001 m).
3. Change in temperature, [tex]\( \Delta T \)[/tex] = -10 degrees Celsius (°C).
4. Young's Modulus of steel, [tex]\( Y_{\text{steel}} \)[/tex] = 2 × 10¹¹ Newtons per square meter (N/m²).
5. Coefficient of linear expansion of steel, [tex]\( \alpha_{\text{steel}} \)[/tex] = 1.2 × 10⁻⁵ per Kelvin (K⁻¹).
### Step 1: Calculate the cross-sectional area of the wire
The cross-sectional area ([tex]\( A \)[/tex]) of a circular wire is given by:
[tex]\[ A = \pi \left(\frac{d}{2}\right)^2 \][/tex]
Substitute [tex]\( d = 0.004 \)[/tex] m:
[tex]\[ A = \pi \left(\frac{0.004}{2}\right)^2 \approx 1.2566370614359172 \times 10^{-5} \text{ m}^2 \][/tex]
### Step 2: Calculate the change in length due to temperature change
Thermal expansion or contraction in length ([tex]\( \Delta L \)[/tex]) can be calculated using the formula:
[tex]\[ \Delta L = \alpha_{\text{steel}} \times L \times \Delta T \][/tex]
Substitute the given values:
[tex]\[ \Delta L = 1.2 \times 10^{-5} \times 8 \times (-10) \][/tex]
[tex]\[ \Delta L \approx -0.00096 \text{ m} \][/tex]
(Note: The negative sign indicates a contraction.)
### Step 3: Calculate the increase in tension in the wire
The increase in tension ([tex]\( \Delta T \)[/tex]) due to the temperature change can be derived using Young's Modulus:
[tex]\[ \Delta T = Y_{\text{steel}} \times A \times \left( \frac{\Delta L}{L} \right) \][/tex]
Substitute the values:
[tex]\[ \Delta T = 2 \times 10^{11} \times 1.2566370614359172 \times 10^{-5} \times \left( \frac{-0.00096}{8} \right) \][/tex]
[tex]\[ \Delta T \approx -301.59289474462014 \text{ N} \][/tex]
(Note: The negative sign here indicates an increase in tension due to contraction which results in pulling forces in the wire.)
### Result:
The increase in the tension of the wire when the temperature falls by 10°C is approximately 301.59 Newtons.
In conclusion, when the temperature of an 8-meter steel wire with a diameter of 4 mm falls by 10 degrees Celsius, the tension in the wire increases by approximately 301.59 N.
### Given Data:
1. Length of the steel wire, [tex]\( L \)[/tex] = 8 meters (m).
2. Diameter of the steel wire, [tex]\( d \)[/tex] = 4 millimeters (mm) = 0.004 meters (m) (since 1 mm = 0.001 m).
3. Change in temperature, [tex]\( \Delta T \)[/tex] = -10 degrees Celsius (°C).
4. Young's Modulus of steel, [tex]\( Y_{\text{steel}} \)[/tex] = 2 × 10¹¹ Newtons per square meter (N/m²).
5. Coefficient of linear expansion of steel, [tex]\( \alpha_{\text{steel}} \)[/tex] = 1.2 × 10⁻⁵ per Kelvin (K⁻¹).
### Step 1: Calculate the cross-sectional area of the wire
The cross-sectional area ([tex]\( A \)[/tex]) of a circular wire is given by:
[tex]\[ A = \pi \left(\frac{d}{2}\right)^2 \][/tex]
Substitute [tex]\( d = 0.004 \)[/tex] m:
[tex]\[ A = \pi \left(\frac{0.004}{2}\right)^2 \approx 1.2566370614359172 \times 10^{-5} \text{ m}^2 \][/tex]
### Step 2: Calculate the change in length due to temperature change
Thermal expansion or contraction in length ([tex]\( \Delta L \)[/tex]) can be calculated using the formula:
[tex]\[ \Delta L = \alpha_{\text{steel}} \times L \times \Delta T \][/tex]
Substitute the given values:
[tex]\[ \Delta L = 1.2 \times 10^{-5} \times 8 \times (-10) \][/tex]
[tex]\[ \Delta L \approx -0.00096 \text{ m} \][/tex]
(Note: The negative sign indicates a contraction.)
### Step 3: Calculate the increase in tension in the wire
The increase in tension ([tex]\( \Delta T \)[/tex]) due to the temperature change can be derived using Young's Modulus:
[tex]\[ \Delta T = Y_{\text{steel}} \times A \times \left( \frac{\Delta L}{L} \right) \][/tex]
Substitute the values:
[tex]\[ \Delta T = 2 \times 10^{11} \times 1.2566370614359172 \times 10^{-5} \times \left( \frac{-0.00096}{8} \right) \][/tex]
[tex]\[ \Delta T \approx -301.59289474462014 \text{ N} \][/tex]
(Note: The negative sign here indicates an increase in tension due to contraction which results in pulling forces in the wire.)
### Result:
The increase in the tension of the wire when the temperature falls by 10°C is approximately 301.59 Newtons.
In conclusion, when the temperature of an 8-meter steel wire with a diameter of 4 mm falls by 10 degrees Celsius, the tension in the wire increases by approximately 301.59 N.
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