A bar is 500 mm long and is stretched to 505 mm with a force of 50 kN. The bar is 10 mm diameter. Calculate the stress and strain
The material has remained within the elastic limit. Determine the modulus of elasticity.
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A bar is 500 mm long and is stretched to 505 mm with a force of 50 kN. The bar is 10 mm diameter. Calculate the stress and strain
To calculate the stress, we'll use the formula:
Stress (σ) = Force (F) / Area (A)
Given:
Force (F) = 50 kN = 50,000 N
Diameter of the bar (d) = 10 mm
Radius (r) = 10 mm / 2 = 5 mm = 0.005 m
Area (A) of the bar can be calculated using the formula for the area of a circle:
A = π * r^2
Substituting the values:
A = π * (0.005 m)^2 = 0.00007854 m²
Now, we can calculate the stress:
Stress (σ) = 50,000 N / 0.00007854 m²
σ ≈ 637,949,778.69 Pa (or N/m²)
Next, to calculate the strain, we'll use the formula:
Strain (ε) = (Change in length (ΔL)) / Original length (L₀)
Given:
Original length (L₀) = 500 mm = 0.5 m
Change in length (ΔL) = 505 mm - 500 mm = 5 mm = 0.005 m
Now, we can calculate the strain:
Strain (ε) = 0.005 m / 0.5 m
ε = 0.01 (no units, as it is a pure ratio)
Since the material remains within the elastic limit, we can use Hooke's law to calculate the modulus of elasticity (Young's modulus):
Modulus of elasticity (E) = Stress (σ) / Strain (ε)
Substituting the values:
E = 637,949,778.69 Pa / 0.01
E ≈ 63,794,977,869 Pa (or N/m²)
Therefore, the modulus of elasticity (Young's modulus) is approximately 63,794,977,869 Pa (or N/m²).
To calculate the stress and strain of the bar, we can use the following formulas:
Stress (σ) = Force (F) / Area (A)
Strain (ε) = Change in Length (ΔL) / Original Length (L)
First, let's calculate the stress:
1. Calculate the area of the bar:
The bar is in the shape of a cylinder, so we can use the formula for the area of a circle:
Area (A) = π * (Diameter/2)^2
A = π * (10 mm / 2)^2
A = 78.54 mm^2
2. Convert the force from kN to N:
Since the area is in square millimeters, we need to convert the force from kilonewtons to newtons:
Force (F) = 50 kN * 1000 N/kN
F = 50,000 N
3. Calculate the stress:
Stress (σ) = F/A
σ = 50,000 N / 78.54 mm^2
Next, let's calculate the strain:
4. Calculate the change in length:
The change in length can be found by subtracting the original length from the stretched length:
Change in Length (ΔL) = Stretched Length - Original Length
ΔL = 505 mm - 500 mm
5. Calculate the strain:
Strain (ε) = ΔL / L
ε = (505 mm - 500 mm) / 500 mm
Finally, let's determine the modulus of elasticity:
The modulus of elasticity, also known as Young's modulus (E), can be defined as the ratio of stress to strain within the elastic limit.
6. Calculate the modulus of elasticity:
E = Stress / Strain
E = σ / ε
By plugging in the values we calculated for stress and strain, we can find the modulus of elasticity.
It is important to note that to accurately determine the modulus of elasticity, you would need to perform multiple tests on the material and take the average value. However, in this case, since we are given that the material remained within the elastic limit, we can assume the stress-strain relationship is linear and use the calculated values for stress and strain.