A monkey of mass 20 kg is holding a vertical rope. The rope will not break when a mass of 25 kg is suspended from it but will break if the mass exceeds 25 kg. What is the maximum acceleration with which the monkey can climb up along the rope ( g= 10 m / s²)

The correct option is 2.5 m / s².

Step-by-Step Explanation:

1. Identify the given data:
- Mass of the monkey, $m=20\text{ kg}$
- Maximum mass the rope can support without breaking, $M=25\text{ kg}$
- Acceleration due to gravity, $g=10{\text{ m/s}}^2$

2. Calculate the maximum tension the rope can withstand:
The maximum tension ($T_{\text{max}}$) the rope can support is equal to the weight of the maximum suspended mass:
$T_{\text{max}}=M·g$
Substituting the values:
$T_{\text{max}}=25\text{ kg}\times 10{\text{ m/s}}^2=250\text{ N}$

3. Determine the equation of motion for the monkey climbing up:
When a monkey of mass $m$ climbs up the rope with an acceleration $a$, the tension $T$ in the rope is given by:
$T=m(g+a)$

4. Find the maximum acceleration:
To find the maximum acceleration ($a_{\text{max}}$), we set the tension in the rope equal to the maximum allowable tension ($T_{\text{max}}$):
$T_{\text{max}}=m(g+a_{\text{max}})$
Substituting the values we have:
$250=20(10+a_{\text{max}})$
Divide both sides by 20:
$12.5=10+a_{\text{max}}$
Subtract 10 from both sides:
$a_{\text{max}}=12.5-10=2.5{\text{ m/s}}^2$

Therefore, the maximum acceleration with which the monkey can climb up along the rope is 2.5 m / s².