The metallic bob of a simple pendulum has the relative density `rho`. The time period of this pendulum is `T`. If the metallic bob is immersed in water ,then the new time period is given by

Options

A. `T(rho- 1)/(rho)`
B. `T(rho)/(rho - 1)`
C. `T sqrt((rho - 1)/(rho))`
D. `T sqrt((rho)/(rho - 1))`

Explanation

To find the new time period of a simple pendulum when the metallic bob is immersed in water, we can follow these steps:

### Step 1: Understand the initial conditions
The time period \( T \) of a simple pendulum 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 effect of buoyancy
When the metallic bob is immersed in water, the effective weight of the bob changes due to the buoyant force acting on it. The buoyant force \( F_b \) can be calculated using Archimedes' principle:
\[ F_b = \rho_{liquid} \cdot V \cdot g \]
where \( \rho_{liquid} \) is the density of the liquid (water in this case), \( V \) is the volume of the bob, and \( g \) is the acceleration due to gravity.

### Step 3: Relate the densities
The relative density \( \rho \) of the bob is defined as:
\[ \rho = \frac{\text{Density of solid}}{\text{Density of liquid}} = \frac{\rho_{solid}}{\rho_{liquid}} \]
From this, we can express the density of the solid bob as:
\[ \rho_{solid} = \rho \cdot \rho_{liquid} \]

### Step 4: Calculate the effective weight
The effective weight \( W' \) of the bob when immersed in water is given by:
\[ W' = mg - F_b \]
Substituting the expression for buoyant force:
\[ W' = mg - \rho_{liquid} \cdot V \cdot g \]
Since \( V = \frac{m}{\rho_{solid}} \):
\[ W' = mg - \rho_{liquid} \cdot \left(\frac{m}{\rho_{solid}}\right) \cdot g \]
Substituting \( \rho_{solid} = \rho \cdot \rho_{liquid} \):
\[ W' = mg - \rho_{liquid} \cdot \left(\frac{m}{\rho \cdot \rho_{liquid}}\right) \cdot g \]
This simplifies to:
\[ W' = mg - \frac{mg}{\rho} \]
Factoring out \( mg \):
\[ W' = mg \left(1 - \frac{1}{\rho}\right) \]

### Step 5: Calculate the new acceleration
The new acceleration \( g' \) can be expressed as:
\[ g' = \frac{W'}{m} = g \left(1 - \frac{1}{\rho}\right) \]

### Step 6: Substitute into the time period formula
Now, substituting \( g' \) into the time period formula:
\[ T' = 2\pi \sqrt{\frac{L}{g'}} \]
Substituting for \( g' \):
\[ T' = 2\pi \sqrt{\frac{L}{g \left(1 - \frac{1}{\rho}\right)}} \]

### Step 7: Relate \( T' \) to \( T \)
Using the original time period \( T = 2\pi \sqrt{\frac{L}{g}} \):
\[ T' = T \sqrt{\frac{1}{1 - \frac{1}{\rho}}} \]
This can be rewritten as:
\[ T' = T \sqrt{\frac{\rho}{\rho - 1}} \]

### Final Result
Thus, the new time period \( T' \) when the metallic bob is immersed in water is given by:
\[ T' = T \sqrt{\frac{\rho}{\rho - 1}} \]

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