6. Consider filling water up to a height h cm in a cylindrical bottle which has height 10cm, radius 3cm
and mass 20g and is sitting on flat ground. At what approximately what height h will the centre of
mass of the bottle+water be closest to the ground?
a. 0 cm
b. 2 cm
c. 8 cm
d. 10 cm
The centre of mass of the bottle plus water system is closest to the ground at approximately h = 2 cm.
Centre of Mass Calculation
The bottle has uniform mass distribution, so its empty centre of mass lies at 5 cm from the base. Water density is 1 g/cm³ with cross-sectional area π × 3² ≈ 28.27 cm². For water height h, water mass m_w = 28.27h g and its centre of mass is at h/2. Total centre of mass height y_cm = (20 × 5 + 28.27h × h/2) / (20 + 28.27h).
This y_cm decreases initially as water adds dense mass near the base, reaches a minimum, then rises as upper water dominates. Optimization yields minimum y_cm ≈ 2.04 cm at h ≈ 2.04 cm.
Option Analysis
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a. 0 cm: Empty bottle has y_cm = 5 cm, highest among options due to no stabilizing bottom mass.
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b. 2 cm: y_cm ≈ 2.05 cm, matches calculated minimum for lowest position closest to ground.
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c. 8 cm: y_cm ≈ 4.08 cm, higher as water mass shifts centre upward significantly.
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d. 10 cm: Full bottle has y_cm = 5 cm, symmetric like empty case with uniform filling.
Correct answer: b. 2 cm. This principle explains bottle stability experiments where partial low filling minimizes tipping risk.
