A circular metal colunn 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 colunn is
2 m long.
Calculate the cross sectional area and the diameter.
To calculate the cross-sectional area and diameter of the circular metal column, we can use the formula for compressive stress:
Stress = Force / Area
Given:
Load = 500 Tonne
Compression limit = 0.1 mm
Modulus of Elasticity = 210 GPa = 210,000 MPa
Length of column = 2 m
Step 1: Convert the load from tonnes to newtons.
1 tonne = 1000 kg
1 kg = 9.81 N
So, 500 tonnes = 500 x 1000 x 9.81 N = 4,905,000 N
Step 2: Calculate the compressive stress.
Stress = Load / Area
0.1 mm = 0.1 x 10^-3 m (convert mm to m)
Stress = 4,905,000 N / Area
Step 3: Calculate the cross-sectional area.
We can rearrange the above equation to calculate the area:
Area = Load / Stress
Area = 4,905,000 N / Stress
Step 4: Calculate the diameter using the cross-sectional area formula.
Area = π * (diameter^2) / 4
diameter^2 = (4 * Area) / π
diameter = √((4 * Area) / π)
Now, let's substitute the values into the equations and calculate the results.
Step 2: Calculate the compressive stress.
Stress = 4,905,000 N / (0.1 x 10^-3 m) = 49,050,000 N/m^2 (or Pascals)
Step 3: Calculate the cross-sectional area.
Area = 4,905,000 N / (49,050,000 N/m^2) = 0.1 m^2
Step 4: Calculate the diameter.
diameter = √((4 * 0.1 m^2) / π) = √(0.4 / π) ≈ 0.356 m (rounded to 3 decimal places)
Therefore, the cross-sectional area of the column is 0.1 m^2, and the diameter is approximately 0.356 meters.