A block of mass 10 kg lying on a smooth horizontal surface is being pulled by means of-a rope of mass 2 kg. If a force of 36N is applied at the end of the rope, the tension at the mid point of the rope is,

Options

A. 33 N
B. 30 N
C. 24 N
D. 12 N

Answer

To solve the problem, we need to find the tension at the midpoint of the rope when a force of 36 N is applied at the end of the rope. The block has a mass of 10 kg and the rope has a mass of 2 kg.

### Step-by-Step Solution:

1. **Identify the total mass being accelerated**:
The total mass includes the mass of the block and the mass of the rope.
\[ \text{Total mass} = \text{mass of block} + \text{mass of rope} = 10 \, \text{kg} + 2 \, \text{kg} = 12 \, \text{kg} \]

2. **Calculate the acceleration**:
Using Newton's second law, \( F = ma \), where \( F \) is the total force applied (36 N) and \( m \) is the total mass (12 kg).
\[ a = \frac{F}{m} = \frac{36 \, \text{N}}{12 \, \text{kg}} = 3 \, \text{m/s}^2 \]

3. **Determine the tension at the midpoint of the rope**:
Let \( T_a \) be the tension at the midpoint of the rope. The mass of the rope that is being pulled by this tension is half of the rope's total mass, which is \( 1 \, \text{kg} \) (since the rope has a total mass of 2 kg).

The force acting on the half of the rope can be calculated using:
\[ F = T_a - m_{half} \cdot a \]
where \( m_{half} = 1 \, \text{kg} \).

Substituting the values:
\[ T_a - (1 \, \text{kg} \cdot 3 \, \text{m/s}^2) = 0 \]
\[ T_a - 3 = 0 \implies T_a = 3 \, \text{N} \]

4. **Calculate the tension at the midpoint**:
Now, we need to find the tension at the midpoint of the rope considering the entire system. The tension at the midpoint can be determined by considering the force acting on the entire system minus the tension acting on the half of the rope.
\[ T_a = 36 \, \text{N} - (2 \, \text{kg} \cdot 3 \, \text{m/s}^2) = 36 - 6 = 30 \, \text{N} \]

5. **Final calculation for tension at midpoint**:
The tension at the midpoint of the rope is:
\[ T_a = 36 \, \text{N} - 3 \, \text{N} = 33 \, \text{N} \]

### Conclusion:
The tension at the midpoint of the rope is **33 N**.