A rope of length 5 m is kept on frictionless surface and a force of 5 N is applied to one of its end. Find the tension in the rope at 1 m from this end

Options

A. 1 N
B. 3 N
C. 4 N
D. 5 N

Explanation

To find the tension in the rope at a distance of 1 meter from the end where a force of 5 N is applied, we can follow these steps:

### Step 1: Understand the System
We have a rope of length 5 m placed on a frictionless surface, and a force of 5 N is applied at one end of the rope. We need to find the tension in the rope at a point 1 m from the end where the force is applied.

### Step 2: Define the Mass of the Rope
Let the mass of the rope be \( m \). The mass per unit length of the rope can be defined as:
\[ \text{Mass per unit length} = \frac{m}{5} \text{ kg/m} \]

### Step 3: Calculate the Mass of the Rope Segment
The segment of the rope from the end where the force is applied to the point where we want to find the tension (1 m from the end) has a length of 1 m. The mass of this segment is:
\[ \text{Mass of 1 m segment} = \frac{m}{5} \times 1 = \frac{m}{5} \text{ kg} \]

### Step 4: Set Up the Equation of Motion
Since the rope is accelerating due to the applied force, we can use Newton's second law. The net force acting on the 1 m segment of the rope is the applied force minus the tension \( T \) in the rope:
\[ F_{\text{net}} = F_{\text{applied}} - T = ma \]
Substituting the known values:
\[ 5 - T = \left(\frac{m}{5}\right)a \tag{1} \]

### Step 5: Calculate the Total Mass of the Rope
The total mass of the rope can be expressed as:
\[ m = \text{mass of the entire rope} = \frac{m}{5} \times 5 = m \]

### Step 6: Calculate the Acceleration of the Rope
The total force acting on the entire rope is equal to the applied force:
\[ F_{\text{net}} = 5 = ma \]
Thus, the acceleration \( a \) of the rope can be calculated as:
\[ a = \frac{5}{m} \tag{2} \]

### Step 7: Substitute Acceleration into the First Equation
Now, substitute the expression for acceleration from equation (2) into equation (1):
\[ 5 - T = \left(\frac{m}{5}\right) \left(\frac{5}{m}\right) \]
This simplifies to:
\[ 5 - T = 1 \]

### Step 8: Solve for Tension \( T \)
Rearranging the equation gives:
\[ T = 5 - 1 = 4 \text{ N} \]

### Conclusion
The tension in the rope at a distance of 1 m from the end where the force is applied is:
\[ \boxed{4 \text{ N}} \]