A uniform metal chain is placed on a rough table such that the one end of chain hangs down over the edge of the table, when one-third of its length hang over the edge, the chain starts sliding. Then the coefficient of static friction is

Options

A. (a)`3//4`
B. (b)`1//4`
C. (c)`2//3`
D. (d)`1//2`

Explanation

To solve the problem step by step, we will analyze the forces acting on the chain and apply the principles of static friction.

### Step-by-step Solution:

1. **Define the Length of the Chain**:
Let the total length of the chain be \( L \). When one-third of the chain hangs over the edge of the table, the length of the chain hanging down is:
\[ L_h = \frac{L}{3} \]
The length of the chain on the table is:
\[ L_t = L - L_h = L - \frac{L}{3} = \frac{2L}{3} \]

2. **Calculate the Weight of the Hanging Portion**:
The weight of the hanging portion of the chain can be calculated using the mass per unit length. Let \( m \) be the mass of the entire chain. The mass per unit length \( \lambda \) is:
\[ \lambda = \frac{m}{L} \]
Therefore, the weight of the hanging portion is:
\[ W_h = \lambda \cdot L_h = \frac{m}{L} \cdot \frac{L}{3} = \frac{m}{3} \]

3. **Calculate the Normal Force**:
The normal force \( N \) acting on the chain from the table is equal to the weight of the portion of the chain that is on the table:
\[ W_t = \lambda \cdot L_t = \frac{m}{L} \cdot \frac{2L}{3} = \frac{2m}{3} \]
Thus, the normal force \( N \) is:
\[ N = W_t = \frac{2m}{3} \]

4. **Apply Static Friction Condition**:
The maximum static friction force \( F_s \) that can act on the chain is given by:
\[ F_s = \mu_s \cdot N = \mu_s \cdot \frac{2m}{3} \]
where \( \mu_s \) is the coefficient of static friction.

5. **Set Up the Equation for Sliding**:
The chain starts sliding when the weight of the hanging portion equals the maximum static friction force:
\[ W_h = F_s \]
Substituting the values we have:
\[ \frac{m}{3} = \mu_s \cdot \frac{2m}{3} \]

6. **Solve for the Coefficient of Static Friction**:
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{1}{3} = \mu_s \cdot \frac{2}{3} \]
Multiplying both sides by \( 3 \):
\[ 1 = 2\mu_s \]
Dividing both sides by 2:
\[ \mu_s = \frac{1}{2} \]

### Final Answer:
The coefficient of static friction \( \mu_s \) is \( \frac{1}{2} \).

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