A spring of force constant `k` is cut into two equal parts, and mass 'm' is suspended from parallel combination of the springs. Then frequency of small osciollations is

Options

A. `f = (1)/(2pi) sqrt((K)/(m))`
B. `f - (1)/(2pi) sqrt((K)/(2m))`
C. `f = (1)/(2pi) sqrt((4K)/(m))`
D. `f = (1)/(2pi) sqrt((2K)/(m))`

Explanation

To solve the problem of finding the frequency of small oscillations for a mass suspended from two equal parts of a spring in parallel, we can follow these steps:

### Step 1: Understand the Spring Constant
Initially, we have a spring with a force constant \( k \). When this spring is cut into two equal parts, each part will have a new spring constant. The spring constant \( k \) is inversely proportional to the length of the spring. Therefore, if the original length \( L \) is cut in half, the new spring constant \( k' \) for each half will be:

\[ k' = \frac{k}{\frac{L}{2}} = 2k \]

### Step 2: Determine the Equivalent Spring Constant
When the two halves of the spring are placed in parallel, the equivalent spring constant \( k_{eq} \) can be calculated by adding the spring constants of the two springs:

\[ k_{eq} = k' + k' = 2k + 2k = 4k \]

### Step 3: Use the Formula for Frequency
The frequency \( f \) of small oscillations for a mass \( m \) attached to a spring is given by the formula:

\[ f = \frac{1}{2\pi} \sqrt{\frac{k_{eq}}{m}} \]

Substituting the equivalent spring constant \( k_{eq} = 4k \) into the frequency formula gives:

\[ f = \frac{1}{2\pi} \sqrt{\frac{4k}{m}} \]

### Step 4: Simplify the Expression
We can simplify the expression for frequency:

\[ f = \frac{1}{2\pi} \cdot 2 \sqrt{\frac{k}{m}} = \frac{2}{2\pi} \sqrt{\frac{k}{m}} = \frac{1}{\pi} \sqrt{\frac{k}{m}} \]

### Final Answer
Thus, the frequency of small oscillations when the spring is cut into two equal parts and arranged in parallel is:

\[ f = \frac{1}{\pi} \sqrt{\frac{k}{m}} \]

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