In the figure shown , the tension is strings are `T_1` and `T_2` for figure (1) and (2) respectively and acceleration of mass m are `a_1` and `a_2` for figure (1) and (2) respectively.Select the correct option

Options

A. `T_1gtT_2 and a_2gta_1`
B. T_2gtT_1 and a_1gta_2`
C. `T_1gtT_2 and a_1gta_2`
D. `T_2gtT_1 and a_2gta_1`

Answer

To solve the problem, we need to analyze the two figures provided and derive the relationships between the tensions \( T_1 \) and \( T_2 \), and the accelerations \( a_1 \) and \( a_2 \).

### Step 1: Analyze Figure 1
In Figure 1, we have a mass \( m \) connected to a pulley system. The forces acting on the mass are:
- The gravitational force acting downward: \( mg \)
- The tension in the string acting upward: \( T_1 \)

According to Newton's second law, the net force acting on the mass is equal to the mass times its acceleration:
\[ T_1 - mg = ma_1 \]
Rearranging this gives:
\[ T_1 = mg + ma_1 \]

### Step 2: Analyze Figure 2
In Figure 2, we have a different configuration where a mass \( 2m \) is also connected to the pulley system. The forces acting on the mass \( 2m \) are:
- The gravitational force acting downward: \( 2mg \)
- The tension in the string acting upward: \( T_2 \)

Applying Newton's second law again:
\[ T_2 - 2mg = -2ma_2 \]
This indicates that the acceleration \( a_2 \) is downward, hence the negative sign. Rearranging gives:
\[ T_2 = 2mg - 2ma_2 \]

### Step 3: Relate the Accelerations
From the analysis of both figures, we have two equations:
1. \( T_1 = mg + ma_1 \)
2. \( T_2 = 2mg - 2ma_2 \)

### Step 4: Solve for the Accelerations
From the first equation, we can express \( a_1 \):
\[ a_1 = \frac{T_1 - mg}{m} \]

From the second equation, we can express \( a_2 \):
\[ a_2 = \frac{2mg - T_2}{2m} \]

### Step 5: Compare the Tensions and Accelerations
To compare \( a_1 \) and \( a_2 \), we can analyze the relationship between \( T_1 \) and \( T_2 \).

From the equations we derived:
1. \( T_1 = mg + ma_1 \)
2. \( T_2 = 2mg - 2ma_2 \)

Since \( a_1 \) is the acceleration of mass \( m \) upwards and \( a_2 \) is the acceleration of mass \( 2m \) downwards, we can conclude that:
- If \( a_1 \) is greater than \( a_2 \), then \( T_1 \) must be greater than \( T_2 \).

### Conclusion
From the analysis, we find that:
- \( a_1 > a_2 \)
- \( T_1 > T_2 \)

Thus, the correct option is:
**Option C: \( T_1 > T_2 \) and \( a_1 > a_2 \)**.