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, we will follow these steps:

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

### Step 2: Identify the initial conditions
In this case, the initial mass \( m_1 \) is \( 1 \, \text{kg} \). Therefore, the initial frequency \( f_1 \) can be expressed as:
\[ f_1 = \frac{1}{2\pi} \sqrt{\frac{k}{1}} = \frac{1}{2\pi} \sqrt{k} \]

### Step 3: Determine the new frequency
We want to reduce the frequency to half of the initial frequency. Thus, the new frequency \( f_2 \) is:
\[ f_2 = \frac{f_1}{2} = \frac{1}{2} \left(\frac{1}{2\pi} \sqrt{k}\right) = \frac{1}{4\pi} \sqrt{k} \]

### Step 4: Relate the new frequency to the new mass
Using the frequency formula for the new mass \( m_2 \), we have:
\[ f_2 = \frac{1}{2\pi} \sqrt{\frac{k}{m_2}} \]
Setting the two expressions for \( f_2 \) equal gives:
\[ \frac{1}{4\pi} \sqrt{k} = \frac{1}{2\pi} \sqrt{\frac{k}{m_2}} \]

### Step 5: Solve for the new mass
Cross-multiplying to eliminate the fractions:
\[ \sqrt{k} = 2 \sqrt{\frac{k}{m_2}} \]
Squaring both sides:
\[ k = 4 \frac{k}{m_2} \]
Dividing both sides by \( k \) (assuming \( k \neq 0 \)):
\[ 1 = \frac{4}{m_2} \]
Thus, solving for \( m_2 \):
\[ m_2 = 4 \, \text{kg} \]

### Step 6: Determine the additional mass needed
Since the initial mass is \( 1 \, \text{kg} \), the additional mass \( m_a \) that needs to be added is:
\[ m_a = m_2 - m_1 = 4 \, \text{kg} - 1 \, \text{kg} = 3 \, \text{kg} \]

### Final Answer
The mass that needs to be added to reduce the frequency by half is **3 kg**.

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