At what minimum acceleration should a monkey slide a rope whose breaking strength is `(2)/(3)`rd of its weight?

Options

A. `2//3`g
B. g
C. `1//3`g
D. zero

Answer

To solve the problem, we need to determine the minimum acceleration at which a monkey can slide down a rope without breaking it, given that the breaking strength of the rope is \( \frac{2}{3} \) of the monkey's weight.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Monkey**:
- Let the weight of the monkey be \( W = mg \), where \( m \) is the mass of the monkey and \( g \) is the acceleration due to gravity.
- The tension \( T \) in the rope acts upwards, while the weight \( mg \) acts downwards.

2. **Breaking Strength of the Rope**:
- According to the problem, the breaking strength of the rope is \( \frac{2}{3} \) of the monkey's weight. Therefore, we can express this as:
\[ T = \frac{2}{3} mg \]

3. **Apply Newton's Second Law**:
- When the monkey slides down with an acceleration \( a \), the net force acting on it can be expressed using Newton's second law:
\[ F_{\text{net}} = ma \]
- The net force acting on the monkey is the difference between its weight and the tension in the rope:
\[ ma = mg - T \]
- Substituting the expression for \( T \):
\[ ma = mg - \frac{2}{3}mg \]

4. **Simplify the Equation**:
- Factor out \( mg \) from the right side:
\[ ma = mg \left(1 - \frac{2}{3}\right) \]
- Simplifying the expression inside the parentheses:
\[ 1 - \frac{2}{3} = \frac{1}{3} \]
- Thus, we have:
\[ ma = mg \cdot \frac{1}{3} \]

5. **Cancel the Mass**:
- Since \( m \) is not zero, we can divide both sides by \( m \):
\[ a = g \cdot \frac{1}{3} \]

6. **Final Result**:
- Therefore, the minimum acceleration \( a \) at which the monkey can slide down the rope without breaking it is:
\[ a = \frac{g}{3} \]

### Conclusion:
The minimum acceleration at which the monkey can slide down the rope without breaking it is \( \frac{g}{3} \).