A monkey of mass $$50 \text{ kg}$$ climbs on a rope which can withstand the tension ($$T$$) of $$350 \text{ N}$$. If monkey initially climbs down with an acceleration of $$4 \text{ m s}^{-2}$$ and then climbs up with an acceleration of $$5 \text{ m s}^{-2}$$. Choose the correct option ($$g = 10 \text{ m s}^{-2}$$)

A monkey of mass $$50 \text{ kg}$$ climbs on a rope that can withstand a maximum tension of $$T_{max} = 350 \text{ N}$$. We need to check if the rope breaks in either case.

We start by determining the tension when the monkey climbs down with acceleration $$a = 4 \text{ m/s}^2$$, taking downward as positive:

$$mg - T = ma$$

$$T = m(g - a) = 50(10 - 4) = 50 \times 6 = 300 \text{ N}$$

Since $$T = 300 \text{ N} \lt 350 \text{ N}$$, the rope does not break while climbing down.

Next, when the monkey climbs up with acceleration $$a = 5 \text{ m/s}^2$$, the net upward force must produce this acceleration:

$$T - mg = ma$$

$$T = m(g + a) = 50(10 + 5) = 50 \times 15 = 750 \text{ N}$$

Since $$T = 750 \text{ N} \gt 350 \text{ N}$$, the rope breaks while climbing upward.

Now we evaluate the options:

Option A: $$T = 700 \text{ N}$$ while climbing upward — Incorrect, $$T = 750 \text{ N}$$.

Option B: $$T = 350 \text{ N}$$ while going downward — Incorrect, $$T = 300 \text{ N}$$.

Option C: Rope will break while climbing upward — Correct, since $$750 \text{ N} \gt 350 \text{ N}$$.

Option D: Rope will break while going downward — Incorrect, since $$300 \text{ N} \lt 350 \text{ N}$$.

Therefore, the correct answer is Option C: Rope will break while climbing upward.