A heavy uniform chain lies on horizontal table top. If the coefficient of friction 0.25, then the maximum fraction of length of the chain that can hang over on one edge of the table is

Options

A. `20%`
B. `25%`
C. `35%`
D. `15%`

Answer

To solve the problem of determining the maximum fraction of length of a heavy uniform chain that can hang over the edge of a table given a coefficient of friction of 0.25, we can follow these steps:

### Step 1: Define Variables
Let:
- \( L \) = total length of the chain
- \( h \) = length of the chain on the table
- \( L - h \) = length of the chain hanging over the edge of the table

### Step 2: Determine Masses
The total mass \( M \) of the chain can be expressed in terms of its length:
- Mass of the part on the table: \( m_{\text{table}} = \frac{h}{L} \cdot M \)
- Mass of the part hanging: \( m_{\text{hanging}} = \frac{L - h}{L} \cdot M \)

### Step 3: Calculate Forces
The normal force \( N \) acting on the part of the chain on the table is equal to the weight of the chain on the table:
- \( N = m_{\text{table}} \cdot g = \frac{h}{L} \cdot M \cdot g \)

The maximum static friction force \( F_s \) can be calculated as:
- \( F_s = \mu_s \cdot N = 0.25 \cdot \frac{h}{L} \cdot M \cdot g \)

The weight of the hanging part of the chain is:
- \( W = m_{\text{hanging}} \cdot g = \frac{L - h}{L} \cdot M \cdot g \)

### Step 4: Set Up the Equation for Equilibrium
For the chain to be in equilibrium, the maximum static friction force must equal the weight of the hanging part:
\[ F_s = W \]
Substituting the expressions we derived:
\[ 0.25 \cdot \frac{h}{L} \cdot M \cdot g = \frac{L - h}{L} \cdot M \cdot g \]

### Step 5: Simplify the Equation
Cancel \( M \cdot g \) from both sides:
\[ 0.25 \cdot \frac{h}{L} = \frac{L - h}{L} \]
Multiplying through by \( L \):
\[ 0.25h = L - h \]
Rearranging gives:
\[ 1.25h = L \]
Thus,
\[ h = \frac{L}{1.25} = \frac{4L}{5} \]

### Step 6: Calculate the Length Hanging Over
The length of the chain hanging over the edge is:
\[ L - h = L - \frac{4L}{5} = \frac{L}{5} \]

### Step 7: Find the Fraction of the Length
To find the fraction of the total length of the chain that is hanging over the edge:
\[ \text{Fraction} = \frac{L - h}{L} = \frac{\frac{L}{5}}{L} = \frac{1}{5} \]
Converting this fraction to a percentage:
\[ \text{Percentage} = \frac{1}{5} \times 100\% = 20\% \]

### Final Answer
The maximum fraction of the length of the chain that can hang over one edge of the table is **20%**.

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