A monkey of mass $20 ext{ kg}$ is holding a vertical rope. The rope will not break when a mass of $25 ext{

Question

A monkey of mass $20 	ext{ kg}$ is holding a vertical rope. The rope will not break when a mass of $25 	ext{ kg}$ is suspended from it but will break if the mass exceeds $25 	ext{ kg}$. What is the maximum acceleration with which the monkey can climb up along the rope ($g = 10 	ext{ m/s}^2$)?

Accepted Answer

1. **Determine Maximum Tension ($T_{max}$):** The breaking strength of the rope corresponds to the weight of the maximum mass it can hold ($M = 25 	ext{ kg}$). Thus, the maximum tension is:
   $$ T_{max} = M \cdot g = 25 	ext{ kg} 	imes 10 	ext{ m/s}^2 = 250 	ext{ N} $$

2. **Apply Newton's Second Law:** For the monkey (mass $m = 20 	ext{ kg}$) climbing upwards with acceleration $a$, the forces are Tension ($T$) acting upwards and Weight ($mg$) acting downwards. The equation of motion is:
   $$ T - mg = ma $$
   [Source 118, 126]

3. **Solve for Maximum Acceleration ($a_{max}$):** Substitute $T_{max}$ into the equation:
   $$ 250 	ext{ N} - (20 	ext{ kg} 	imes 10 	ext{ m/s}^2) = 20 	ext{ kg} 	imes a_{max} $$
   $$ 250 - 200 = 20 \cdot a_{max} $$
   $$ 50 = 20 \cdot a_{max} $$
   $$ a_{max} = \frac{50}{20} = 2.5 	ext{ m/s}^2 $$

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