A spring of force constant `k` is cut into two parts whose lengths are in the ratio `1:2`. The two parts are now connected in parallel and a block of mass `m` is suspended at the end of the combined spring. The period of oscillation of block is

Options

A. `2pi sqrt((2m)/(9K))`
B. `2pi sqrt((m)/(9K))`
C. `2pi sqrt((2m)/(5K))`
D. `2pi sqrt((m)/(5K))`

Explanation

To solve the problem, we need to follow these steps:

### Step 1: Understand the Problem
We have a spring with a spring constant \( k \) that is cut into two parts in the ratio \( 1:2 \). We need to find the period of oscillation when these two parts are connected in parallel and a mass \( m \) is suspended from them.

### Step 2: Determine the Lengths of the Spring Parts
Let the total length of the spring be \( L \). Since the spring is cut in the ratio \( 1:2 \):
- Length of the first part \( L_1 = \frac{1}{3}L \)
- Length of the second part \( L_2 = \frac{2}{3}L \)

### Step 3: Calculate the Spring Constants of Each Part
The spring constant \( k \) is inversely proportional to the length of the spring. Therefore:
- For the first part:
\[ k_1 = \frac{k \cdot L}{L_1} = \frac{k \cdot L}{\frac{1}{3}L} = 3k \]
- For the second part:
\[ k_2 = \frac{k \cdot L}{L_2} = \frac{k \cdot L}{\frac{2}{3}L} = \frac{3k}{2} \]

### Step 4: Find the Effective Spring Constant
When springs are connected in parallel, the effective spring constant \( k_{\text{effective}} \) is the sum of the individual spring constants:
\[ k_{\text{effective}} = k_1 + k_2 = 3k + \frac{3k}{2} \]
To add these, we can find a common denominator:
\[ k_{\text{effective}} = 3k + 1.5k = \frac{6k}{2} + \frac{3k}{2} = \frac{9k}{2} \]

### Step 5: Calculate the Period of Oscillation
The period \( T \) of oscillation for a mass \( m \) attached to a spring with spring constant \( k_{\text{effective}} \) is given by the formula:
\[ T = 2\pi \sqrt{\frac{m}{k_{\text{effective}}}} \]
Substituting \( k_{\text{effective}} = \frac{9k}{2} \):
\[ T = 2\pi \sqrt{\frac{m}{\frac{9k}{2}}} = 2\pi \sqrt{\frac{2m}{9k}} \]

### Final Answer
Thus, the period of oscillation of the block is:
\[ T = 2\pi \sqrt{\frac{2m}{9k}} \]

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