A rope of length 5 m has uniform mass per unit length. `lambda = 2 kg/m` The rope is pulled by a constant force of 10N on the smooth horizontal surface as shown in figure The tension in the rope at x = 2 m from polit A is

Options

A. 2 N
B. 8 N
C. 6 N
D. 5 N

Answer

To find the tension in the rope at a point 2 m from point A, we can follow these steps:

### Step 1: Identify the mass of the rope
The mass per unit length (λ) of the rope is given as 2 kg/m. The total length of the rope is 5 m. Therefore, the total mass (M) of the rope can be calculated as:
\[ M = \lambda \times \text{length} = 2 \, \text{kg/m} \times 5 \, \text{m} = 10 \, \text{kg} \]

### Step 2: Calculate the acceleration of the rope
The rope is subjected to a constant force (F) of 10 N. Using Newton's second law, we can find the acceleration (a) of the entire rope:
\[ F = M \cdot a \]
Rearranging gives:
\[ a = \frac{F}{M} = \frac{10 \, \text{N}}{10 \, \text{kg}} = 1 \, \text{m/s}^2 \]

### Step 3: Determine the mass of the segment of the rope
We need to find the tension at a point 2 m from point A. The mass of the segment of the rope from point A to this point (2 m) can be calculated as:
\[ m = \lambda \times \text{length of segment} = 2 \, \text{kg/m} \times 2 \, \text{m} = 4 \, \text{kg} \]

### Step 4: Calculate the net force acting on the segment
The net force (F_net) acting on the 2 m segment of the rope can be calculated using Newton's second law:
\[ F_{\text{net}} = m \cdot a = 4 \, \text{kg} \times 1 \, \text{m/s}^2 = 4 \, \text{N} \]

### Step 5: Calculate the tension in the rope at 2 m
The tension (T) at the point 2 m from point A can be found by considering the forces acting on the segment. The tension at this point must balance the force needed to accelerate the 2 m segment:
\[ T - F_{\text{pull}} = F_{\text{net}} \]
Where \( F_{\text{pull}} \) is the force pulling the entire rope, which is 10 N. Rearranging gives:
\[ T = F_{\text{net}} + F_{\text{pull}} = 4 \, \text{N} + 10 \, \text{N} = 14 \, \text{N} \]

### Final Answer
The tension in the rope at a point 2 m from point A is **14 N**.