A circular metal column is to support a load of 500 Tonne and it must not compress more than 0.1 mm. The modulus of elasticity is 210 GPa. the column is 2 m long.
Calculate the cross sectional area and the diameter.
A= d^2 * 3.14/4
To calculate the cross-sectional area and diameter of the circular metal column, we can use the formula for stress:
Stress = Force / Area
Given:
Load (Force) = 500 Tonne = 500,000 kg
Compression (ΔL) = 0.1 mm = 0.1 x 10^-3 m
Modulus of Elasticity (E) = 210 GPa = 210 x 10^9 Pa
Length (L) = 2 m
We can use the formula for stress:
Stress = E * Strain
where Strain (ε) = ΔL / L
Rearranging this equation, we can isolate the cross-sectional area:
Area = Force / (E * Strain)
Let's calculate the cross-sectional area of the column first.
Step 1: Convert the load to Newtons:
Force = 500,000 kg x 9.8 m/s^2
Force = 4,900,000 Newtons
Step 2: Convert the modulus of elasticity to Pascals:
E = 210 GPa x 10^9 Pa/GPa
E = 210 x 10^9 Pa
Step 3: Calculate the strain:
Strain (ε) = ΔL / L
Strain (ε) = 0.1 x 10^-3 m / 2 m
Step 4: Calculate the cross-sectional area:
Area = Force / (E * Strain)
Area = 4,900,000 N / (210 x 10^9 Pa * (0.1 x 10^-3 m / 2 m))
Now let's calculate the diameter using the formula for the area of a circle:
Area = π * (diameter)^2 / 4
Step 5: Rearrange the formula to solve for the diameter:
Diameter = √(4 * Area / π)
Step 6: Plug in the calculated cross-sectional area and solve for the diameter:
Diameter = √(4 * Area / π)
Now, we can solve for the cross-sectional area and diameter.
To calculate the cross-sectional area and diameter of the circular metal column, we can use the formula:
A = F / (σ * L)
Where:
A = cross-sectional area
F = load or force applied on the column
σ = stress (maximum allowable compression)
L = length of the column
Let's plug in the values given:
F = 500 Tonnes = 500,000 kg (since 1 Tonne = 1000 kg)
σ = 0.1 mm = 0.1 * 10^-3 m
L = 2 m
First, we need to convert the units of σ from mm to meters:
σ = 0.1 * 10^-3 m
Now, we can calculate the cross-sectional area:
A = (500,000 kg) / (210 * 10^9 Pa * 2 m)
To convert the modulus of elasticity from GigaPascals (GPa) to Pascals (Pa), we multiply it by 10^9.
A = 500,000 kg / (210 * 10^9 Pa * 2 m)
Simplifying further:
A ≈ 1.19 * 10^-6 m^2
Now, we can calculate the diameter using the formula:
A = π * (d^2 / 4)
Rearranging the equation to solve for the diameter (d):
d = √((4 * A) / π)
Substituting the value for A:
d ≈ √((4 * 1.19 * 10^-6 m^2) / π)
Calculating the diameter:
d ≈ 4.37 * 10^-4 m
Therefore, the approximate cross-sectional area of the column is 1.19 * 10^-6 square meters, and the diameter is approximately 4.37 * 10^-4 meters.