A vertical metal cylinder of radius 2 cm and length 2 m is fixed at the lower end and a load of 100 kg is put on it. find a. the stress b. the strain and c. the compression of the cylinder. Young modulus of the metal `=2xx10^11Nm^-2`

Explanation

To solve the problem step by step, we will calculate the stress, strain, and compression of the vertical metal cylinder subjected to a load.

### Given Data:
- Radius of the cylinder (r) = 2 cm = 0.02 m (conversion from cm to m)
- Length of the cylinder (L) = 2 m
- Load (mass) = 100 kg
- Young's modulus (E) = \(2 \times 10^{11} \, \text{N/m}^2\)
- Acceleration due to gravity (g) = \(10 \, \text{m/s}^2\)

### Step 1: Calculate the Force (Weight) Acting on the Cylinder
The force (F) acting on the cylinder due to the load is given by:
\[ F = m \cdot g \]
Substituting the values:
\[ F = 100 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 1000 \, \text{N} \]

### Step 2: Calculate the Cross-Sectional Area (A) of the Cylinder
The cross-sectional area (A) of the cylinder is given by:
\[ A = \pi r^2 \]
Substituting the radius:
\[ A = \pi (0.02 \, \text{m})^2 = \pi \cdot 0.0004 \, \text{m}^2 = \frac{4\pi}{10000} \, \text{m}^2 \]

### Step 3: Calculate the Stress (σ)
Stress (σ) is defined as the force per unit area:
\[ \sigma = \frac{F}{A} \]
Substituting the values:
\[ \sigma = \frac{1000 \, \text{N}}{\frac{4\pi}{10000} \, \text{m}^2} \]
Calculating:
\[ \sigma = \frac{1000 \cdot 10000}{4\pi} = \frac{10000000}{4\pi} \approx 7.96 \times 10^5 \, \text{N/m}^2 \]

### Step 4: Calculate the Strain (ε)
Strain (ε) is defined as the ratio of stress to Young's modulus:
\[ \epsilon = \frac{\sigma}{E} \]
Substituting the values:
\[ \epsilon = \frac{7.96 \times 10^5 \, \text{N/m}^2}{2 \times 10^{11} \, \text{N/m}^2} \]
Calculating:
\[ \epsilon \approx 4 \times 10^{-6} \]

### Step 5: Calculate the Compression (ΔL)
Compression (ΔL) can be calculated using the formula:
\[ \Delta L = \epsilon \cdot L \]
Substituting the values:
\[ \Delta L = 4 \times 10^{-6} \cdot 2 \, \text{m} \]
Calculating:
\[ \Delta L = 8 \times 10^{-6} \, \text{m} \]

### Final Answers:
a. Stress (σ) = \(7.96 \times 10^5 \, \text{N/m}^2\)
b. Strain (ε) = \(4 \times 10^{-6}\)
c. Compression (ΔL) = \(8 \times 10^{-6} \, \text{m}\)