A uniform rope so lies on a table that part of it lays over. The rope begins to slide when the length of hanging part is 25% of entire length. The co-efficient of friction between rope and table is

Options

A. 0.33
B. 0.25
C. 0.5
D. 0.2

Explanation

To solve the problem step by step, we will analyze the forces acting on the rope and use the concept of friction to find the coefficient of friction between the rope and the table.

### Step-by-Step Solution:

1. **Understand the Setup**:
- A uniform rope is lying on a table, and part of it is hanging off the edge.
- The length of the hanging part is 25% of the total length of the rope.

2. **Define Variables**:
- Let the total mass of the rope be \( m \).
- The mass of the hanging part (25% of the rope) is \( \frac{m}{4} \).
- The mass of the part lying on the table (75% of the rope) is \( \frac{3m}{4} \).

3. **Identify Forces**:
- The hanging part of the rope experiences a gravitational force pulling it down, which is \( F_{\text{hanging}} = \frac{m}{4} g \).
- The part of the rope lying on the table experiences a normal force due to its weight, which is \( F_{\text{normal}} = \frac{3m}{4} g \).

4. **Frictional Force**:
- The frictional force \( F_{\text{friction}} \) that opposes the motion of the rope is given by:
\[ F_{\text{friction}} = \mu F_{\text{normal}} \]
- Here, \( \mu \) is the coefficient of friction.

5. **Set Up the Equation**:
- For the rope to begin sliding, the frictional force must equal the weight of the hanging part:
\[ F_{\text{friction}} = F_{\text{hanging}} \]
- Substituting the expressions we have:
\[ \mu \left(\frac{3m}{4} g\right) = \frac{m}{4} g \]

6. **Simplify the Equation**:
- We can cancel \( g \) and \( m \) from both sides (assuming \( g \) and \( m \) are not zero):
\[ \mu \left(\frac{3}{4}\right) = \frac{1}{4} \]

7. **Solve for \( \mu \)**:
- Rearranging gives:
\[ \mu = \frac{1/4}{3/4} = \frac{1}{3} \]

8. **Final Answer**:
- The coefficient of friction \( \mu \) is \( \frac{1}{3} \) or approximately \( 0.33 \).

### Summary:
The coefficient of friction between the rope and the table is \( \frac{1}{3} \) or \( 0.33 \).