A spring mass system osciallates with a time period 7s. The entiresystem is immersed in a liquid whose density at halt that of the material of the block. Find the new time period ( in s ) of osciallations.

Answer

To solve the problem of finding the new time period of a spring-mass system immersed in a liquid, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Given Information:**
- The initial time period \( T_0 \) of the spring-mass system is given as \( 7 \, \text{s} \).
- The density of the liquid \( \rho_l \) is half that of the density of the block \( \rho_b \), i.e., \( \rho_l = \frac{1}{2} \rho_b \).

2. **Identify the Forces Acting on the Mass:**
- The forces acting on the mass when it is submerged in the liquid are:
- The gravitational force \( F_g = mg \) acting downwards.
- The buoyant force \( F_b = \rho_l V g \) acting upwards, where \( V \) is the volume of the block.

3. **Calculate the Buoyant Force:**
- Since the volume of the block \( V \) can be expressed in terms of its mass \( m \) and density \( \rho_b \) as \( V = \frac{m}{\rho_b} \), we can substitute this into the buoyant force equation:
\[ F_b = \rho_l \left(\frac{m}{\rho_b}\right) g = \frac{1}{2} \rho_b \left(\frac{m}{\rho_b}\right) g = \frac{mg}{2} \]

4. **Determine the Effective Weight of the Block in the Liquid:**
- The effective weight \( W_{eff} \) of the block in the liquid can be calculated as:
\[ W_{eff} = F_g - F_b = mg - \frac{mg}{2} = \frac{mg}{2} \]

5. **Calculate the New Mass for Oscillation:**
- The new effective mass \( m_{eff} \) that will oscillate can be considered as \( \frac{m}{2} \) because of the buoyancy effect.

6. **Relate the Time Period to the New Mass:**
- The time period of oscillation for a spring-mass system is given by:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
- For the new time period \( T' \) with the effective mass, we have:
\[ T' = 2\pi \sqrt{\frac{m_{eff}}{k}} = 2\pi \sqrt{\frac{\frac{m}{2}}{k}} = 2\pi \sqrt{\frac{m}{2k}} = \sqrt{2} T_0 \]

7. **Substituting the Initial Time Period:**
- Since \( T_0 = 7 \, \text{s} \):
\[ T' = \sqrt{2} \cdot 7 \approx 9.9 \, \text{s} \]

### Final Answer
Thus, the new time period \( T' \) of oscillations is approximately \( 9.9 \, \text{s} \).

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