A mass of `1 kg` attached to the bottom of a spring has a certain frequency of vibration. The following mass has to be added to it in order to reduce the frequency by half :

Options

A. `1 kg`
B. `2 kg`
C. `3 kg`
D. `4 kg`

Explanation

To solve the problem of how much mass needs to be added to reduce the frequency of a mass-spring system by half, we can follow these steps:

### Step 1: Understand the frequency formula
The frequency of a mass-spring system is given by the formula:
\[ F = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \]
where:
- \( F \) is the frequency,
- \( k \) is the spring constant,
- \( m \) is the mass attached to the spring.

### Step 2: Define the initial conditions
Let:
- \( m_1 = 1 \, \text{kg} \) (the initial mass),
- \( F_1 \) be the initial frequency.

Using the frequency formula, we can express \( F_1 \) as:
\[ F_1 = \frac{1}{2\pi} \sqrt{\frac{k}{m_1}} = \frac{1}{2\pi} \sqrt{\frac{k}{1}} = \frac{1}{2\pi} \sqrt{k} \]

### Step 3: Define the new conditions
Let \( m_2 \) be the new mass after adding an additional mass \( m \):
\[ m_2 = m_1 + m \]
We want to reduce the frequency to half, so:
\[ F_2 = \frac{F_1}{2} \]

### Step 4: Write the equation for the new frequency
Using the frequency formula for the new mass:
\[ F_2 = \frac{1}{2\pi} \sqrt{\frac{k}{m_2}} = \frac{1}{2\pi} \sqrt{\frac{k}{m_1 + m}} \]
Setting \( F_2 = \frac{F_1}{2} \):
\[ \frac{1}{2\pi} \sqrt{\frac{k}{m_1 + m}} = \frac{1}{2} \left( \frac{1}{2\pi} \sqrt{k} \right) \]

### Step 5: Simplify the equation
Cancel \( \frac{1}{2\pi} \) from both sides:
\[ \sqrt{\frac{k}{m_1 + m}} = \frac{1}{2} \sqrt{k} \]
Squaring both sides gives:
\[ \frac{k}{m_1 + m} = \frac{1}{4} k \]
Now, we can cancel \( k \) (assuming \( k \neq 0 \)):
\[ \frac{1}{m_1 + m} = \frac{1}{4} \]

### Step 6: Solve for \( m_1 + m \)
Cross-multiplying gives:
\[ 1 = \frac{1}{4}(m_1 + m) \]
Thus:
\[ m_1 + m = 4 \]

### Step 7: Solve for the additional mass \( m \)
Substituting \( m_1 = 1 \, \text{kg} \):
\[ 1 + m = 4 \]
So:
\[ m = 4 - 1 = 3 \, \text{kg} \]

### Final Answer
The mass that needs to be added is:
\[ \boxed{3 \, \text{kg}} \]