A simple pendulum has a time period T in vacuum. Its time period when it is completely immersed in a liquid of density one-eight of the density of material of the bob is

Options

A. `sqrt((7)/(9)) T`
B. `sqrt((5)/(8))` T
C. `sqrt((3)/(8))T`
D. `sqrt((8)/(7))T`

Explanation

To solve the problem, we need to determine the time period of a simple pendulum when it is completely immersed in a liquid with a specific density. Here's a step-by-step solution:

### Step 1: Understand the Time Period in Vacuum
The time period \( T \) of a simple pendulum in a vacuum is given by the formula:
\[ T = 2\pi \sqrt{\frac{l}{g}} \]
where \( l \) is the length of the pendulum and \( g \) is the acceleration due to gravity.

### Step 2: Identify the Density of the Liquid
We are given that the density of the liquid is one-eighth of the density of the material of the bob. If we denote the density of the bob as \( \rho \), then the density of the liquid \( \rho_{liquid} \) is:
\[ \rho_{liquid} = \frac{\rho}{8} \]

### Step 3: Calculate the Buoyant Force
When the bob is submerged in the liquid, it experiences a buoyant force. The buoyant force \( F_b \) can be calculated using Archimedes' principle:
\[ F_b = V \cdot \rho_{liquid} \cdot g = V \cdot \left(\frac{\rho}{8}\right) \cdot g \]
where \( V \) is the volume of the bob.

### Step 4: Calculate the Weight of the Bob
The weight \( W \) of the bob is given by:
\[ W = V \cdot \rho \cdot g \]

### Step 5: Determine the Effective Weight in the Liquid
The effective weight \( W_{effective} \) of the bob when submerged in the liquid is the actual weight minus the buoyant force:
\[ W_{effective} = W - F_b = V \cdot \rho \cdot g - V \cdot \left(\frac{\rho}{8}\right) \cdot g \]
Factoring out \( V \cdot g \):
\[ W_{effective} = V \cdot g \left(\rho - \frac{\rho}{8}\right) = V \cdot g \cdot \frac{7\rho}{8} \]

### Step 6: Calculate the Effective Gravitational Acceleration
The effective gravitational acceleration \( g_{effective} \) can be expressed as:
\[ g_{effective} = \frac{W_{effective}}{V \cdot \rho} = \frac{V \cdot g \cdot \frac{7\rho}{8}}{V \cdot \rho} = \frac{7g}{8} \]

### Step 7: Find the New Time Period in the Liquid
Now, we can find the new time period \( T' \) when the pendulum is submerged in the liquid:
\[ T' = 2\pi \sqrt{\frac{l}{g_{effective}}} = 2\pi \sqrt{\frac{l}{\frac{7g}{8}}} = 2\pi \sqrt{\frac{8l}{7g}} \]
This can be rewritten as:
\[ T' = \sqrt{\frac{8}{7}} \cdot 2\pi \sqrt{\frac{l}{g}} = \sqrt{\frac{8}{7}} \cdot T \]

### Step 8: Final Result
Thus, the time period of the pendulum when completely immersed in the liquid is:
\[ T' = \sqrt{\frac{8}{7}} \cdot T \]

### Conclusion
The correct answer is:
\[ \boxed{\sqrt{\frac{8}{7}} \cdot T} \]