A uniform rope of length l lies on a table if the coefficient of friction is , then the maximum length l, of the part of this rope which can over hang from the edge of the table without sliding down is -

Options

A. `l/(mu)`
B. `l/(mu)+1`
C. `(mul)/(mu+1)`
D. `(mul)/(mu-1)`

Answer

To solve the problem of determining the maximum length \( X \) of the part of a uniform rope that can overhang from the edge of a table without sliding down, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables**:
- Let the total length of the rope be \( L \).
- Let the length of the rope hanging over the edge be \( X \).
- The length of the rope on the table will then be \( L - X \).
- Let \( \lambda \) be the mass per unit length of the rope.

2. **Calculate the Mass**:
- The mass of the hanging part of the rope is given by:
\[ m_{\text{hanging}} = \lambda X \]
- The mass of the part of the rope that is on the table is:
\[ m_{\text{on table}} = \lambda (L - X) \]

3. **Determine Forces**:
- The gravitational force acting on the hanging part of the rope is:
\[ F_{\text{gravity}} = m_{\text{hanging}} \cdot g = \lambda X g \]
- The maximum static friction force that can act on the rope to prevent it from sliding is:
\[ F_{\text{friction}} = \mu \cdot m_{\text{on table}} \cdot g = \mu \cdot \lambda (L - X) g \]

4. **Set Up the Equation**:
- For the rope to remain at rest, the friction force must be equal to the gravitational force acting on the hanging part:
\[ \lambda X g = \mu \cdot \lambda (L - X) g \]

5. **Cancel Common Terms**:
- We can cancel \( \lambda g \) from both sides (assuming \( \lambda \neq 0 \) and \( g \neq 0 \)):
\[ X = \mu (L - X) \]

6. **Rearrange the Equation**:
- Rearranging gives:
\[ X + \mu X = \mu L \]
- This simplifies to:
\[ X (1 + \mu) = \mu L \]

7. **Solve for \( X \)**:
- Finally, we can solve for \( X \):
\[ X = \frac{\mu L}{1 + \mu} \]

### Conclusion:
The maximum length \( X \) of the part of the rope that can overhang the table without sliding down is:
\[ X = \frac{\mu L}{1 + \mu} \]