A solid cylinder of mass `M = 10kg` and cross - sectional area `A = 20cm^(2)` is suspended by a spring of force contant `k = 100 N//m` and hangs partically immersed in water. Calculate the period of small oscillation of the cylinder.

Explanation

To calculate the period of small oscillation of a solid cylinder partially immersed in water, we will follow these steps:

### Step 1: Identify the given values
- Mass of the cylinder, \( M = 10 \, \text{kg} \)
- Cross-sectional area, \( A = 20 \, \text{cm}^2 = 20 \times 10^{-4} \, \text{m}^2 = 0.002 \, \text{m}^2 \)
- Spring constant, \( k = 100 \, \text{N/m} \)
- Density of water, \( \rho = 1000 \, \text{kg/m}^3 \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)

### Step 2: Calculate the upthrust (buoyant force)
The upthrust \( F_b \) can be calculated using the formula:
\[ F_b = \rho \cdot A \cdot g \]
Substituting the values:
\[ F_b = 1000 \, \text{kg/m}^3 \cdot 0.002 \, \text{m}^2 \cdot 10 \, \text{m/s}^2 = 20 \, \text{N} \]

### Step 3: Calculate the effective spring constant
The effective spring constant \( k_{\text{net}} \) is the sum of the spring constant \( k \) and the equivalent spring constant due to the buoyant force:
\[ k_{\text{net}} = k + F_b \]
Substituting the values:
\[ k_{\text{net}} = 100 \, \text{N/m} + 20 \, \text{N} = 120 \, \text{N/m} \]

### Step 4: Calculate the period of oscillation
The period \( T \) of small oscillations is given by the formula:
\[ T = 2\pi \sqrt{\frac{M}{k_{\text{net}}}} \]
Substituting the values:
\[ T = 2\pi \sqrt{\frac{10 \, \text{kg}}{120 \, \text{N/m}}} \]
Calculating the value:
\[ T = 2\pi \sqrt{\frac{10}{120}} = 2\pi \sqrt{\frac{1}{12}} \approx 2\pi \cdot 0.2887 \approx 1.81 \, \text{s} \]

### Conclusion
The period of small oscillation of the cylinder is approximately \( T \approx 1.81 \, \text{s} \).

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