A rope of length L is pulled by a constant force F. What is the tension in the rope at a distance x from the end where the force is applied ?

Options

A. FL/x
B. `F(L-x)//L`
C. `FL//(L-x)`
D. `Fx//(L-x)`

Explanation

To find the tension in a rope at a distance \( x \) from the end where a constant force \( F \) is applied, we can follow these steps:

### Step 1: Understand the System
We have a rope of length \( L \) being pulled by a force \( F \) at one end (let's call it end B). We need to find the tension \( T \) at a point \( P \) which is at a distance \( x \) from end B.

### Step 2: Define Variables
- Let \( m \) be the total mass of the rope.
- The mass per unit length \( \lambda \) of the rope can be expressed as:
\[ \lambda = \frac{m}{L} \]
- The mass of the segment of the rope from point \( P \) to end B (length \( x \)) is:
\[ m_x = \lambda \cdot x = \frac{m}{L} \cdot x \]
- The mass of the segment from point \( P \) to end A (length \( L - x \)) is:
\[ m_{L-x} = \lambda \cdot (L - x) = \frac{m}{L} \cdot (L - x) \]

### Step 3: Calculate the Acceleration
The entire rope is being accelerated by the force \( F \). The acceleration \( a \) of the rope can be calculated using Newton's second law:
\[ a = \frac{F}{m} \]

### Step 4: Apply Newton's Second Law to the Segment
For the segment of the rope from point \( P \) to end B (length \( x \)), the tension \( T \) at point \( P \) must support the weight of this segment and provide the necessary force for its acceleration. Therefore, we can write:
\[ T = m_x \cdot a \]
Substituting for \( m_x \) and \( a \):
\[ T = \left(\frac{m}{L} \cdot x\right) \cdot \left(\frac{F}{m}\right) \]
The mass \( m \) cancels out:
\[ T = \frac{F \cdot x}{L} \]

### Step 5: Relate Tension to the Remaining Length
Now, we need to express the tension in terms of the remaining length of the rope \( L - x \):
\[ T = F \cdot \frac{(L - x)}{L} \]

### Final Result
Thus, the tension \( T \) in the rope at a distance \( x \) from the end where the force is applied is:
\[ T = F \cdot \frac{(L - x)}{L} \]