A monkey is descending from the branch of a tree with a constant acceleration. If the breaking strength of the branch is 75% of the weight of the monkey, the minimum acceleration with which the monkey can slide down without breaking the branch is

Options

A. g
B. `(3g)/(4)`
C. `(g)/(4)`
D. `(g)/(2)`

Answer

To solve the problem, we need to analyze the forces acting on the monkey as it descends from the branch of the tree with a constant acceleration.

### Step-by-Step Solution:

1. **Identify the Weight of the Monkey:**
The weight (W) of the monkey can be expressed as:
\[ W = mg \]
where \( m \) is the mass of the monkey and \( g \) is the acceleration due to gravity.

2. **Determine the Breaking Strength of the Branch:**
The breaking strength of the branch is given as 75% of the weight of the monkey. Therefore, we can write:
\[ \text{Breaking Strength} = 0.75 \times W = 0.75 \times mg = \frac{3}{4} mg \]

3. **Set Up the Equation of Motion:**
When the monkey is descending with a constant acceleration \( a \), the net force acting on the monkey can be expressed using Newton's second law:
\[ F_{\text{net}} = W - F_{\text{tension}} = ma \]
Here, \( F_{\text{tension}} \) is the tension in the branch, which is equal to the breaking strength when the branch is about to break.

4. **Relate Forces:**
Since the tension in the branch must not exceed its breaking strength, we can set up the following inequality:
\[ mg - T = ma \]
where \( T \) is the tension in the branch. When the branch is at its breaking point:
\[ T = \frac{3}{4} mg \]
Substituting this into the equation gives us:
\[ mg - \frac{3}{4}mg = ma \]

5. **Simplify the Equation:**
Simplifying the left side:
\[ \frac{1}{4}mg = ma \]

6. **Solve for Acceleration \( a \):**
Now, we can solve for \( a \):
\[ a = \frac{1}{4}g \]

### Final Result:
Thus, the minimum acceleration with which the monkey can slide down without breaking the branch is:
\[ a = \frac{1}{4}g \]