The Science of a Sinking Boat
A group of cadets is testing a model boat in a water tank. The boat is made of a hollow rectangular box of mass 200 g and external dimensions 20 cm × 10 cm × 8 cm (height). It is placed in fresh water (density 1 g/cm³). The boat floats upright. The cadets load the boat with small metal cubes, each of mass 50 g and volume 5 cm³. They want to know how many cubes can be added before the boat sinks (i.e., water enters the boat). They also observe that when the boat is empty, it displaces a certain volume of water, and they wonder about the buoyant force. Assume the boat's walls are thin and the inside is completely hollow. The boat is open at the top, so if the water level reaches the top edge, it will sink.
1. Calculate the maximum number of metal cubes that can be added without sinking the boat. Show your reasoning, including the condition for floating.
Read the worked answer
For the boat to float, the buoyant force must equal the total weight. The maximum buoyant force occurs when the boat is just submerged, i.e., displaced water volume equals the external volume of the boat: V_ext = 20×10×8 = 1600 cm³. The maximum mass that can be supported is the mass of displaced water: m_water = density × volume = 1 g/cm³ × 1600 cm³ = 1600 g. The boat's own mass is 200 g, so the maximum additional mass is 1600 − 200 = 1400 g. Each cube has mass 50 g, so number of cubes by mass = 1400/50 = 28 cubes. However, we must also consider the volume of the cubes: they displace water, but since they are inside the boat, they add to the total weight. The boat's external volume limits the displaced water. The cubes' volume does not directly reduce the boat's capacity because they are inside the hull; the boat sinks deeper as weight increases until the waterline reaches the top. The condition is total mass (boat + cubes) ≤ mass of displaced water at full submersion. So 28 cubes would give total mass 200 + 28×50 = 1600 g, exactly equal to the buoyant force at full submersion. At this point, the boat is on the verge of sinking. If we add one more cube, total mass = 1650 g > 1600 g, so it sinks. Therefore, maximum number of cubes is 28. But we must check if the cubes fit inside the boat without overflowing. The internal volume of the boat is approximately the external volume minus wall thickness, but walls are thin, so internal volume ≈ 1600 cm³. The volume of 28 cubes = 28×5 = 140 cm³, which is much less than 1600 cm³, so they fit. Also, the boat's height is 8 cm; when fully loaded to 1600 g, it will be submerged up to the top. So 28 cubes is the answer.
2. When the boat is empty, what fraction of its volume is submerged? And what is the buoyant force on the empty boat? (Use g = 10 m/s² for force in newtons, and density of water = 1000 kg/m³).
Read the worked answer
Empty boat mass = 200 g = 0.2 kg. For floating, buoyant force = weight = 0.2 kg × 10 m/s² = 2 N. Buoyant force also equals weight of displaced water: F_b = ρ_water × V_submerged × g. So 2 = 1000 × V_sub × 10 → V_sub = 2 / 10000 = 0.0002 m³ = 200 cm³. The boat's external volume = 1600 cm³. Fraction submerged = 200/1600 = 1/8 = 0.125 or 12.5%. So when empty, 12.5% of the boat's volume is submerged. The buoyant force is 2 N.