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. `(3)/(4)`
B. `(1)/(4)`
C. `(2)/(3)`
D. `(1)/(2)`

Explanation

To solve the problem step by step, we can follow this approach:

### Step 1: Understand the setup
We have a uniform metal chain placed on a rough table, with one-third of its length hanging over the edge. When this hanging part reaches a certain point, the chain starts to slide.

### Step 2: Define the variables
Let:
- \( L \) = total length of the chain
- \( M \) = total mass of the chain
- \( g \) = acceleration due to gravity
- \( \mu \) = coefficient of static friction

### Step 3: Determine the weight of the hanging part
When one-third of the chain is hanging, the length of the hanging part is:
\[ \text{Length of hanging part} = \frac{L}{3} \]

The mass of the hanging part is:
\[ \text{Mass of hanging part} = \frac{M}{L} \cdot \frac{L}{3} = \frac{M}{3} \]

The weight of the hanging part is:
\[ W_{\text{hanging}} = \frac{M}{3} \cdot g = \frac{Mg}{3} \]

### Step 4: Determine the weight of the part on the table
The length of the part on the table is:
\[ \text{Length on table} = L - \frac{L}{3} = \frac{2L}{3} \]

The mass of the part on the table is:
\[ \text{Mass on table} = \frac{M}{L} \cdot \frac{2L}{3} = \frac{2M}{3} \]

The weight of this part is:
\[ W_{\text{table}} = \frac{2M}{3} \cdot g = \frac{2Mg}{3} \]

### Step 5: Calculate the normal force
The normal force \( N \) acting on the part of the chain on the table is equal to its weight:
\[ N = W_{\text{table}} = \frac{2Mg}{3} \]

### Step 6: Write the equation for limiting friction
The limiting friction \( F_{\text{friction}} \) is given by:
\[ F_{\text{friction}} = \mu N = \mu \cdot \frac{2Mg}{3} \]

### Step 7: Set up the equilibrium condition
At the point of sliding, the weight of the hanging part equals the limiting friction:
\[ W_{\text{hanging}} = F_{\text{friction}} \]
Substituting the values:
\[ \frac{Mg}{3} = \mu \cdot \frac{2Mg}{3} \]

### Step 8: Solve for the coefficient of static friction
Cancelling \( Mg/3 \) from both sides (assuming \( M \) and \( g \) are not zero):
\[ 1 = 2\mu \]
Thus,
\[ \mu = \frac{1}{2} \]

### Conclusion
The coefficient of static friction between the metal chain and the table is:
\[ \mu = \frac{1}{2} \]