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 Initial Conditions**:
- 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 Block**:
- When the block is immersed in the liquid, three forces act on it:
- The gravitational force \( F_g = m g \) acting downwards.
- The buoyant force \( F_b = \rho_L V g \) acting upwards, where \( V \) is the volume of the block.
- The spring force \( F_s = -kx \) acting upwards when the block is displaced.

3. **Calculate the Effective Weight of the Block**:
- The effective weight \( W_{\text{eff}} \) of the block when submerged is given by:
\[ W_{\text{eff}} = F_g - F_b = mg - \rho_L V g \]
- Since \( m = \rho_B V \), we can substitute:
\[ W_{\text{eff}} = \rho_B V g - \rho_L V g = V g (\rho_B - \rho_L) \]
- Substituting \( \rho_L = \frac{1}{2} \rho_B \):
\[ W_{\text{eff}} = V g \left( \rho_B - \frac{1}{2} \rho_B \right) = V g \left( \frac{1}{2} \rho_B \right) \]

4. **Determine the New Effective Mass**:
- The effective mass \( m' \) that contributes to the oscillation is now half of the original mass:
\[ m' = \frac{1}{2} m = \frac{1}{2} \rho_B V \]

5. **Relate the New Time Period to the Original Time Period**:
- The time period of oscillation for a spring-mass system is given by:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
- The new time period \( T' \) can be expressed as:
\[ T' = 2\pi \sqrt{\frac{m'}{k}} = 2\pi \sqrt{\frac{\frac{1}{2} m}{k}} = 2\pi \sqrt{\frac{m}{2k}} = \frac{T_0}{\sqrt{2}} \]
- Since \( T_0 = 7 \text{ s} \):
\[ T' = \frac{7}{\sqrt{2}} \approx 4.95 \text{ s} \]

6. **Final Result**:
- The new time period of oscillation when the spring-mass system is immersed in the liquid is approximately \( 4.95 \text{ s} \).

### Summary:
The new time period of the oscillations of the spring-mass system immersed in the liquid is approximately \( 4.95 \text{ s} \).