Four identical hollow cylindrical cloumns of steel support a big structure of mass 50.000 kg. the inner and outer radii of each column are 30 cm and 60 cm respectively. Assume the load distribution to be uniform , calculate the compressional strain of each column. the Young's modulus of steel is `2.0 xx 10^(11) Pa`.

Options

A. `(Mg)/(pi(R_(2)^(2)- R_(1)^(2))Y)`
B. `(Mg)/(4pi(R_(2)^(2)- R_(1)^(2))Y)`
C. `(Mg)/(pi(R_(1)^(2)- R_(2)^(2))Y)`
D. `(Mg)/(4pi(R_(1)^(2)- R_(2)^(2))Y)`

Explanation

To solve the problem of calculating the compressional strain of each hollow cylindrical column supporting a structure, we will follow these steps:

### Step 1: Identify the Given Data
- Mass of the structure (M) = 50,000 kg
- Inner radius of the column (R1) = 30 cm = 0.30 m
- Outer radius of the column (R2) = 60 cm = 0.60 m
- Young's modulus of steel (Y) = \(2.0 \times 10^{11} \, \text{Pa}\)

### Step 2: Calculate the Total Force Exerted on the Columns
The total force (F) exerted by the structure is given by:
\[ F = M \cdot g \]
where \( g \) (acceleration due to gravity) is approximately \( 9.81 \, \text{m/s}^2 \).

Calculating the total force:
\[ F = 50,000 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 490,500 \, \text{N} \]

### Step 3: Determine the Force on Each Column
Since there are four identical columns, the force on each column (F_column) is:
\[ F_{\text{column}} = \frac{F}{4} = \frac{490,500 \, \text{N}}{4} = 122,625 \, \text{N} \]

### Step 4: Calculate the Cross-Sectional Area of Each Column
The cross-sectional area (A) of a hollow cylinder is given by:
\[ A = \pi (R_2^2 - R_1^2) \]
Substituting the values:
\[ A = \pi \left((0.60)^2 - (0.30)^2\right) \]
\[ A = \pi \left(0.36 - 0.09\right) = \pi \times 0.27 \approx 0.847 \, \text{m}^2 \]

### Step 5: Calculate the Stress on Each Column
Stress (\( \sigma \)) is defined as the force per unit area:
\[ \sigma = \frac{F_{\text{column}}}{A} \]
Substituting the values:
\[ \sigma = \frac{122,625 \, \text{N}}{0.847 \, \text{m}^2} \approx 144,000 \, \text{Pa} \]

### Step 6: Calculate the Strain Using Young's Modulus
Young's modulus (Y) relates stress and strain (\( \epsilon \)):
\[ Y = \frac{\sigma}{\epsilon} \Rightarrow \epsilon = \frac{\sigma}{Y} \]
Substituting the values:
\[ \epsilon = \frac{144,000 \, \text{Pa}}{2.0 \times 10^{11} \, \text{Pa}} \]
\[ \epsilon \approx 7.2 \times 10^{-7} \]

### Final Answer
The compressional strain of each column is approximately \( 7.2 \times 10^{-7} \).

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