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Sagot :
Sure, let's work through the problem step-by-step to estimate the percentage elongation of a circular bar with a diameter of 2.5 cm, subjected to an axial tension of 20 kN, with a Young's modulus of 70 GPa.
### Step 1: Convert Units
1. Diameter: Convert diameter from centimeters to meters.
- Diameter in meters, [tex]\( d \)[/tex]:
[tex]\[ d = 2.5 \, \text{cm} \times \frac{1 \, \text{m}}{100 \, \text{cm}} = 0.025 \, \text{m} \][/tex]
2. Tension: Convert tension from kilonewtons to newtons.
- Tension in newtons, [tex]\( F \)[/tex]:
[tex]\[ F = 20 \, \text{kN} \times 1000 \, \frac{\text{N}}{\text{kN}} = 20000 \, \text{N} \][/tex]
3. Young's Modulus: Convert Young's modulus from GPa to Pascals.
- Young's modulus in Pascals, [tex]\( E \)[/tex]:
[tex]\[ E = 70 \, \text{GPa} \times 10^9 \, \frac{\text{Pa}}{\text{GPa}} = 70 \times 10^9 \, \text{Pa} = 70000000000 \, \text{Pa} \][/tex]
### Step 2: Calculate the Cross-sectional Area
The bar is circular, so its cross-sectional area, [tex]\( A \)[/tex], can be calculated using the formula for the area of a circle:
[tex]\[ A = \pi \left( \frac{d}{2} \right)^2 \][/tex]
Plug in the diameter in meters:
[tex]\[ A = \pi \left( \frac{0.025}{2} \right)^2 \approx 0.000490873852 \, \text{m}^2 \][/tex]
### Step 3: Calculate the Stress
Stress, [tex]\( \sigma \)[/tex], is defined as the force [tex]\( F \)[/tex] divided by the cross-sectional area [tex]\( A \)[/tex]:
[tex]\[ \sigma = \frac{F}{A} \][/tex]
Plug in the values of tension and area:
[tex]\[ \sigma = \frac{20000 \, \text{N}}{0.000490873852 \, \text{m}^2} \approx 40743665.4315252 \, \text{Pa} \][/tex]
### Step 4: Calculate the Strain
Strain, [tex]\( \varepsilon \)[/tex], is defined as the stress [tex]\( \sigma \)[/tex] divided by Young's modulus [tex]\( E \)[/tex]:
[tex]\[ \varepsilon = \frac{\sigma}{E} \][/tex]
Plug in the values of stress and Young's modulus:
[tex]\[ \varepsilon = \frac{40743665.4315252 \, \text{Pa}}{70000000000 \, \text{Pa}} \approx 0.000582052363 \][/tex]
### Step 5: Calculate the Percentage Elongation
Percentage elongation is the strain multiplied by 100:
[tex]\[ \text{Percentage Elongation} = \varepsilon \times 100 \][/tex]
Plug in the value of strain:
[tex]\[ \text{Percentage Elongation} \approx 0.000582052363 \times 100 \approx 0.058205236 \][/tex]
So, the estimated percentage elongation of the circular bar under the given conditions is approximately [tex]\( 0.0582\% \)[/tex].
### Step 1: Convert Units
1. Diameter: Convert diameter from centimeters to meters.
- Diameter in meters, [tex]\( d \)[/tex]:
[tex]\[ d = 2.5 \, \text{cm} \times \frac{1 \, \text{m}}{100 \, \text{cm}} = 0.025 \, \text{m} \][/tex]
2. Tension: Convert tension from kilonewtons to newtons.
- Tension in newtons, [tex]\( F \)[/tex]:
[tex]\[ F = 20 \, \text{kN} \times 1000 \, \frac{\text{N}}{\text{kN}} = 20000 \, \text{N} \][/tex]
3. Young's Modulus: Convert Young's modulus from GPa to Pascals.
- Young's modulus in Pascals, [tex]\( E \)[/tex]:
[tex]\[ E = 70 \, \text{GPa} \times 10^9 \, \frac{\text{Pa}}{\text{GPa}} = 70 \times 10^9 \, \text{Pa} = 70000000000 \, \text{Pa} \][/tex]
### Step 2: Calculate the Cross-sectional Area
The bar is circular, so its cross-sectional area, [tex]\( A \)[/tex], can be calculated using the formula for the area of a circle:
[tex]\[ A = \pi \left( \frac{d}{2} \right)^2 \][/tex]
Plug in the diameter in meters:
[tex]\[ A = \pi \left( \frac{0.025}{2} \right)^2 \approx 0.000490873852 \, \text{m}^2 \][/tex]
### Step 3: Calculate the Stress
Stress, [tex]\( \sigma \)[/tex], is defined as the force [tex]\( F \)[/tex] divided by the cross-sectional area [tex]\( A \)[/tex]:
[tex]\[ \sigma = \frac{F}{A} \][/tex]
Plug in the values of tension and area:
[tex]\[ \sigma = \frac{20000 \, \text{N}}{0.000490873852 \, \text{m}^2} \approx 40743665.4315252 \, \text{Pa} \][/tex]
### Step 4: Calculate the Strain
Strain, [tex]\( \varepsilon \)[/tex], is defined as the stress [tex]\( \sigma \)[/tex] divided by Young's modulus [tex]\( E \)[/tex]:
[tex]\[ \varepsilon = \frac{\sigma}{E} \][/tex]
Plug in the values of stress and Young's modulus:
[tex]\[ \varepsilon = \frac{40743665.4315252 \, \text{Pa}}{70000000000 \, \text{Pa}} \approx 0.000582052363 \][/tex]
### Step 5: Calculate the Percentage Elongation
Percentage elongation is the strain multiplied by 100:
[tex]\[ \text{Percentage Elongation} = \varepsilon \times 100 \][/tex]
Plug in the value of strain:
[tex]\[ \text{Percentage Elongation} \approx 0.000582052363 \times 100 \approx 0.058205236 \][/tex]
So, the estimated percentage elongation of the circular bar under the given conditions is approximately [tex]\( 0.0582\% \)[/tex].
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