A spring has a certain mass suspended from it and its period for vertical oscillations is `T_(1)`. The spring is now cut into two equal halves and the same mass is suspended from one of the half. The period of vertical oscillation is now `T_(2)`. The ratio of `T_(2)//T_(1)` is

Options

A. `1//2`
B. `1//sqrt(2)`
C. `sqrt(2)`
D. `2`

Explanation

To solve the problem, we need to find the ratio of the periods of oscillation \( T_2 \) and \( T_1 \) when a mass is suspended from a spring and then from half of that spring. Let's break it down step by step.

### Step 1: Write the expression for the period of oscillation for the original spring.

The period of oscillation \( T_1 \) for a mass \( m \) suspended from a spring with spring constant \( k \) is given by the formula:
\[ T_1 = 2\pi \sqrt{\frac{m}{k}} \]

### Step 2: Analyze the spring when it is cut into two halves.

When the spring is cut into two equal halves, each half will have a new spring constant \( k' \). The relationship between the spring constant and the length of the spring is given by:
\[ k' = \frac{k}{\frac{L}{L/2}} = 2k \]
This means that the spring constant of each half is double that of the original spring.

### Step 3: Write the expression for the period of oscillation for one half of the spring.

Now, when the same mass \( m \) is suspended from one half of the spring, the period of oscillation \( T_2 \) can be expressed as:
\[ T_2 = 2\pi \sqrt{\frac{m}{k'}} \]
Substituting \( k' = 2k \) into the equation gives:
\[ T_2 = 2\pi \sqrt{\frac{m}{2k}} = 2\pi \sqrt{\frac{m}{k}} \cdot \frac{1}{\sqrt{2}} \]

### Step 4: Relate \( T_2 \) to \( T_1 \).

From the expression for \( T_1 \):
\[ T_1 = 2\pi \sqrt{\frac{m}{k}} \]
We can substitute this back into the equation for \( T_2 \):
\[ T_2 = T_1 \cdot \frac{1}{\sqrt{2}} \]

### Step 5: Find the ratio \( \frac{T_2}{T_1} \).

Now, we can find the ratio of the two periods:
\[ \frac{T_2}{T_1} = \frac{T_1 \cdot \frac{1}{\sqrt{2}}}{T_1} = \frac{1}{\sqrt{2}} \]

### Final Answer:
Thus, the ratio \( \frac{T_2}{T_1} \) is:
\[ \frac{T_2}{T_1} = \frac{1}{\sqrt{2}} \]

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