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A balloon of mass M with a light rope of length L and a man of mass m are at rest in equilibrium in air (buoyancy balances total weight). The man starts climbing up the rope and reaches the top. Which of the following statements is/are correct?

1. The balloon descends by a distance mL / (M + m)
2. The balloon descends by a distance ML / (M + m)
3. The center of mass of the system (balloon + man) remains stationary throughout
4. The displacement of the balloon equals the displacement of the man in magnitude

Correct answer: The balloon descends by a distance mL / (M + m)

Solution

Since the system is in equilibrium (buoyancy = total weight), the net external force remains zero as the man climbs. Thus the center of mass (CM) of the system stays fixed. Let the balloon descend by x (downward positive). The man moves from the bottom of the rope to the top, traveling a distance L relative to the balloon, so relative to ground the man moves up (L - x). CM fixed: M*(-x) + m*(L - x) = 0 → mL - mx - Mx = 0 → x = mL/(M + m). The CM remains stationary, confirming option C is also correct. The correct statements are A and C.

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