A uniform rope of length l lies on a table . If the coefficient of friction is `mu` , then the maximum length `l_(1)` of the part of this rope which can overhang from the edge of the table without sliding down is :
`(1)/(mu)`
`(l)/(mu+1)`
`(mul)/(1+mu)`
`(mul)/(1-mu)`

Options

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

Answer

To solve the problem of determining the maximum length \( l_1 \) of the part of the rope that can overhang from the edge of the table without sliding down, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( l \) = total length of the rope
- \( l_1 \) = length of the rope hanging off the table
- \( \mu \) = coefficient of friction
- \( \lambda \) = linear mass density of the rope (mass per unit length)

### Step 2: Determine the Weight of the Hanging Part
The weight of the hanging part of the rope can be expressed as:
\[ W_{\text{hanging}} = \lambda l_1 g \]
where \( g \) is the acceleration due to gravity.

### Step 3: Determine the Weight of the Part on the Table
The length of the rope lying on the table is \( l - l_1 \). Therefore, the weight of the part of the rope lying on the table is:
\[ W_{\text{table}} = \lambda (l - l_1) g \]

### Step 4: Calculate the Friction Force
The frictional force \( F_{\text{friction}} \) that prevents the rope from sliding is given by:
\[ F_{\text{friction}} = \mu \cdot N \]
where \( N \) is the normal force. The normal force is equal to the weight of the part of the rope lying on the table:
\[ N = W_{\text{table}} = \lambda (l - l_1) g \]
Thus, the frictional force can be expressed as:
\[ F_{\text{friction}} = \mu \cdot \lambda (l - l_1) g \]

### Step 5: Set Up the Equation for Equilibrium
For the rope to not slide down, the frictional force must equal the weight of the hanging part:
\[ \lambda l_1 g = \mu \cdot \lambda (l - l_1) g \]

### Step 6: Simplify the Equation
We can cancel \( g \) and \( \lambda \) from both sides (assuming \( \lambda \neq 0 \)):
\[ l_1 = \mu (l - l_1) \]

### Step 7: Solve for \( l_1 \)
Rearranging the equation gives:
\[ l_1 + \mu l_1 = \mu l \]
\[ l_1 (1 + \mu) = \mu l \]
\[ l_1 = \frac{\mu l}{1 + \mu} \]

### Conclusion
The maximum length \( l_1 \) of the part of the rope that can overhang without sliding down is:
\[ l_1 = \frac{\mu l}{1 + \mu} \]

### Final Answer
Thus, the correct option is:
\[ \frac{\mu l}{1 + \mu} \]