The minimum acceleration with which a fireman can slide down a rope of breaking strength two - third of his weight is

Explanation

To solve the problem of finding the minimum acceleration with which a fireman can slide down a rope of breaking strength two-thirds of his weight, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Fireman:**
- The fireman has a weight \( W = mg \) acting downwards (where \( m \) is the mass of the fireman and \( g \) is the acceleration due to gravity).
- The tension \( T \) in the rope acts upwards.

2. **Set Up the Equation of Motion:**
- When the fireman slides down with an acceleration \( a \), the net force acting on him can be expressed using Newton's second law:
\[ mg - T = ma \]

3. **Determine the Maximum Tension:**
- The breaking strength of the rope is given as two-thirds of the fireman's weight:
\[ T = \frac{2}{3} mg \]

4. **Substitute the Tension into the Equation:**
- Replace \( T \) in the equation of motion with \( \frac{2}{3} mg \):
\[ mg - \frac{2}{3}mg = ma \]

5. **Simplify the Equation:**
- Factor out \( mg \):
\[ mg \left(1 - \frac{2}{3}\right) = ma \]
- This simplifies to:
\[ mg \left(\frac{1}{3}\right) = ma \]

6. **Cancel the Mass \( m \) (assuming \( m \neq 0 \)):**
- Dividing both sides by \( m \):
\[ g \left(\frac{1}{3}\right) = a \]

7. **Final Expression for Acceleration:**
- Thus, the minimum acceleration \( a \) is:
\[ a = \frac{g}{3} \]

### Conclusion:
The minimum acceleration with which the fireman can slide down the rope is \( \frac{g}{3} \).