A simple pendulum with iron bob has a time period T. The bob is now immersed in a non-viscous liquid and oscillated, if the density of liquid is `(1)/(12)`th that of iron, then new time period will be

Options

A. `T sqrt((8)/(7))`
B. `T sqrt((12)/(13))`
C. `T sqrt((12)/(11))`
D. `T sqrt((6)/(5))`

Answer

To find the new time period of a simple pendulum with an iron bob immersed in a non-viscous liquid, we can follow these steps:

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

### Step 2: Determine the Effective Gravity in the Liquid
When the pendulum bob is immersed in a liquid, the effective acceleration due to gravity \( g' \) is modified due to the buoyant force acting on the bob. The new effective gravity can be calculated using the formula:
\[ g' = g \left(1 - \frac{\rho_{\text{liquid}}}{\rho_{\text{iron}}}\right) \]
where:
- \( \rho_{\text{liquid}} \) is the density of the liquid,
- \( \rho_{\text{iron}} \) is the density of the iron bob.

Given that the density of the liquid is \( \frac{1}{12} \)th that of iron, we can express this as:
\[ \rho_{\text{liquid}} = \frac{1}{12} \rho_{\text{iron}} \]

### Step 3: Substitute the Density Values
Substituting the value of \( \rho_{\text{liquid}} \) into the equation for \( g' \):
\[ g' = g \left(1 - \frac{1/12 \cdot \rho_{\text{iron}}}{\rho_{\text{iron}}}\right) = g \left(1 - \frac{1}{12}\right) = g \left(\frac{11}{12}\right) \]

### Step 4: Calculate the New Time Period
Now, we can substitute \( g' \) back into the formula for the time period:
\[ T' = 2\pi \sqrt{\frac{L}{g'}} = 2\pi \sqrt{\frac{L}{g \left(\frac{11}{12}\right)}} \]
This simplifies to:
\[ T' = 2\pi \sqrt{\frac{12L}{11g}} = \sqrt{\frac{12}{11}} \cdot 2\pi \sqrt{\frac{L}{g}} = \sqrt{\frac{12}{11}} \cdot T \]

### Final Answer
Thus, the new time period \( T' \) of the pendulum bob immersed in the liquid is:
\[ T' = \sqrt{\frac{12}{11}} T \]