The Overloaded Ferry and the Centre of Mass
A flat-bottomed ferry is a uniform rectangular deck 12 m long and 6 m wide, of mass 4,000 kg, floating in still water. A truck of mass 6,000 kg drives on and stops with its centre 2.0 m from the deck's geometric centre, measured along the length. The deck floats with its long axis horizontal. A student claims the ferry will tip because the truck is 'too far from the middle'. The ferry's freeboard (height of deck above water) is 0.40 m, and the deck is symmetric about its centre line. Assume the water is calm and the load is a point mass.
1. Determine whether the ferry tips, using the position of the combined centre of mass relative to the deck's geometric centre and the deck's footprint.
Read the worked answer
Take the deck centre as origin. Deck mass 4,000 kg at x = 0; truck 6,000 kg at x = 2.0 m.
x_cm = (4000·0 + 6000·2.0)/(4000+6000) = 12,000/10,000 = 1.2 m from the centre.
The deck's edge is 6.0 m from the centre along its length. Since 1.2 m < 6.0 m, the combined centre of mass lies well inside the footprint, so the ferry does not tip. The student's claim is wrong: what matters is not the truck's distance alone but the combined centre of mass relative to the support base.
Note also that tipping requires the centre of mass to pass beyond the edge; here there is a 4.8 m margin.
2. The truck now moves to the very edge of the deck (6.0 m from centre). Explain qualitatively what happens to the combined centre of mass and whether the ferry remains stable, then compute the new position.
Read the worked answer
New position: x_cm = (4000·0 + 6000·6.0)/10,000 = 36,000/10,000 = 3.6 m from centre.
Still inside the 6.0 m half-length, so the ferry does not tip in the idealised model. However, the centre of mass has moved from 1.2 m to 3.6 m, i.e. three times farther from the centre, so the restoring moment available is much reduced and any additional disturbance (waves, a second vehicle, off-centre loading) could push it over.
Judgement: the calculation shows stability is preserved only marginally in the ideal case; the real risk is that the model ignores free-surface effects, dynamic loads and the finite freeboard. A safe answer notes both the number (3.6 m) and the caveat that the ideal model overstates safety.