A mass m attached to a spring oscillates with a period of 3 second. If the mass is increased by 2 kg, the period increases by one second. Calculate the initial mass m. (Assume that elastic limit is not crossed)

Explanation

To solve the problem step by step, we will use the formula for the period of oscillation of a mass-spring system, which is given by:

\[ T = 2\pi \sqrt{\frac{m}{k}} \]

where \( T \) is the period, \( m \) is the mass, and \( k \) is the spring constant.

### Step 1: Set up the equations for the initial and modified conditions

1. **Initial Condition**: The period \( T_1 \) is 3 seconds.
\[ T_1 = 2\pi \sqrt{\frac{m}{k}} \quad \text{(1)} \]
Substituting \( T_1 = 3 \):
\[ 3 = 2\pi \sqrt{\frac{m}{k}} \]

2. **Modified Condition**: When the mass is increased by 2 kg, the new period \( T_2 \) becomes 4 seconds.
\[ T_2 = 2\pi \sqrt{\frac{m + 2}{k}} \quad \text{(2)} \]
Substituting \( T_2 = 4 \):
\[ 4 = 2\pi \sqrt{\frac{m + 2}{k}} \]

### Step 2: Rearranging the equations

From equation (1):
\[ \frac{3}{2\pi} = \sqrt{\frac{m}{k}} \implies \left(\frac{3}{2\pi}\right)^2 = \frac{m}{k} \implies m = k \left(\frac{3}{2\pi}\right)^2 \quad \text{(3)} \]

From equation (2):
\[ \frac{4}{2\pi} = \sqrt{\frac{m + 2}{k}} \implies \left(\frac{4}{2\pi}\right)^2 = \frac{m + 2}{k} \implies m + 2 = k \left(\frac{4}{2\pi}\right)^2 \quad \text{(4)} \]

### Step 3: Equating the two expressions for \( k \)

From equation (3):
\[ k = \frac{m}{\left(\frac{3}{2\pi}\right)^2} \]

From equation (4):
\[ k = \frac{m + 2}{\left(\frac{4}{2\pi}\right)^2} \]

Setting the two expressions for \( k \) equal:
\[ \frac{m}{\left(\frac{3}{2\pi}\right)^2} = \frac{m + 2}{\left(\frac{4}{2\pi}\right)^2} \]

### Step 4: Cross-multiplying and simplifying

Cross-multiplying gives:
\[ m \left(\frac{4}{2\pi}\right)^2 = (m + 2) \left(\frac{3}{2\pi}\right)^2 \]

Expanding both sides:
\[ m \cdot \frac{16}{4\pi^2} = (m + 2) \cdot \frac{9}{4\pi^2} \]

Cancelling \( \frac{1}{4\pi^2} \) from both sides:
\[ 16m = 9(m + 2) \]

### Step 5: Solving for \( m \)

Expanding the right side:
\[ 16m = 9m + 18 \]

Rearranging gives:
\[ 16m - 9m = 18 \implies 7m = 18 \implies m = \frac{18}{7} \approx 2.57 \text{ kg} \]

### Final Answer
The initial mass \( m \) is approximately \( 2.57 \) kg.

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