Question

Ricardo, mass 80 kg, and Carmelita, who is lighter, are enjoying Lake Merced at dusk in a 20 kg canoe. When the canoe is at rest in the placid water, they exchange seats, which are 3.0 m apart and symmetrically located with respect to the canoe's center. Ricardo notices that the canoe moved 50 cm relative to a submerged log during the exchange and calculates Carmelita's mass, which she has not told him. What is it?

Answer 1

Since there are no external forces acting on the canoe at rest, the center of mass of the system is stationary. It does not change relative to a fixed coordinate system. Consider a coordinate system in which the x-axis is along the boat and x=0 is the position of the submerged log. The two scenarios are shown in the following figure

The two scenarios are separated by the blue line. In the first scenario, Ricardo is at the left end of the canoe while Carmelita is at the right end. The center of mass of the canoe is at the center of the canoe (that is 1.5m from the left end of the canoe). The center of mass of the system is at a distance away from the submerged log. In the second scenario, the positions of Ricardo and Carmelita are reversed. But the center of mass of the system is still at a distance from the submerged log. The canoe is shifted to the left by 50cm or 0.5m.

The center of mass in the first case is

Ricardo's position is at , the center of mass of the canoe is at and Carmelita's position is at . We get

The center of mass in the second case is

Ricardo's position is at x=2.5m. The center of mass of the canoe is 1.5m from the left end of the canoe which is at x=1m from the submerged log. Carmelita's position is at x=-0.5m. We get

We know that the center of mass remains stationary. Using equation (1)

Substituting we get