The time period of a simple pendulum, in the form of a hollow metallic sphere, is T. When it is filled with sand and mercury, then its time periods are `T_(1) and T_(2)` respectively. When it is partially filled with sand then its time period is `T_(3)`. The correct relation between `T_(1), T_(2) & T_(3)` will be

Options

A. `T = T_(1) = T_(2) = T_(3)`
B. `T = T_(1) = T_(2) lt T_(3)`
C. `T = T_(1) gt T_(2) gt T_(3)`
D. `T gt T_(1) gt T_(2) = T_(3)`

Answer

To solve the problem, we need to analyze how the time period of a simple pendulum changes when the mass distribution of the pendulum changes. The time period \( T \) of a simple pendulum is given by the formula:

\[ T = 2\pi \sqrt{\frac{L}{g}} \]

where:
- \( L \) is the distance from the pivot point to the center of mass of the pendulum,
- \( g \) is the acceleration due to gravity.

### Step 1: Understanding the Initial Time Period \( T \)

For the hollow metallic sphere, the center of mass is at its midpoint. The time period is given as \( T \).

### Step 2: Time Period When Filled with Sand \( T_1 \)

When the hollow sphere is filled with sand, the center of mass will still be at the center of the sphere (since sand fills it uniformly). Therefore, the effective length \( L \) remains the same, and thus:

\[ T_1 = T \]

### Step 3: Time Period When Filled with Mercury \( T_2 \)

Similarly, when the hollow sphere is filled with mercury, which is denser than sand, the center of mass is still at the center of the sphere. Thus, the effective length \( L \) remains unchanged, and we have:

\[ T_2 = T \]

### Step 4: Time Period When Partially Filled with Sand \( T_3 \)

When the hollow sphere is partially filled with sand, the center of mass shifts downward. This means that the effective length \( L \) increases. Since the time period is directly related to the square root of the effective length, we can conclude that:

\[ T_3 > T \]

### Step 5: Establishing Relationships

From the above analysis, we can summarize the relationships as follows:

- \( T_1 = T \)
- \( T_2 = T \)
- \( T_3 > T \)

Thus, since \( T_1 \) and \( T_2 \) are equal and both are less than \( T_3 \), we can write the final relationship as:

\[ T_1 = T_2 < T_3 \]

### Final Relation

The correct relation between \( T_1, T_2, \) and \( T_3 \) is:

\[ T_1 = T_2 < T_3 \]