Question

Donald Duck and his nephews manage to sink Uncle Scrooge's yacht \((m=4500 \mathrm{~kg}),\) which is made of steel \(\left(\rho=7800 \mathrm{~kg} / \mathrm{m}^{3}\right)\). In typical comicbook fashion, they decide to raise the yacht by filling it with ping-pong balls. A ping-pong ball has a mass of \(2.7 \mathrm{~g}\) and a volume of \(3.35 \cdot 10^{-5} \mathrm{~m}^{3}\) a) What is the buoyant force on one ping-pong ball in water? b) How many balls are required to float the ship?

Short answer

Answer: The buoyant force on one ping-pong ball is approximately 0.003285935 N, and it would require approximately 1,342,467 ping-pong balls to float the yacht.

Step-by-step solution

a) Buoyant force on one ping-pong ball

Archimedes' principle states that the upward buoyant force exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid the body displaces. The buoyant force can be calculated using the following formula: Buoyant Force (F_b) = ρ_fluid × g × V where ρ_fluid is the density of the fluid (in this case water, with a density of 1000 kg/m³), g is the acceleration due to gravity (approximately 9.81 m/s²), and V is the volume of the submerged object (in this case the volume of a ping-pong ball). Plugging in the values: F_b = (1000 kg/m³) × (9.81 m/s²) × (3.35 × 10^(-5) m³) F_b = 0.003285935 N The buoyant force on one ping-pong ball is approximately 0.003285935 N.

b) Number of ping-pong balls required to float the ship

To float the yacht, the total buoyant force exerted by the combined volume of the ping-pong balls must be equal to or greater than the weight of the yacht. The weight of the yacht, W_yacht, can be calculated using the formula: W_yacht = m × g where m is the mass of the yacht and g is the acceleration due to gravity. Plugging in the values: W_yacht = (4500 kg) × (9.81 m/s²) W_yacht = 44145 N In order to float the yacht, the buoyant force exerted by the combined volume of the ping-pong balls must be equal to or greater than the weight of the yacht: n × F_b >= W_yacht n = W_yacht / F_b n = (44145 N) / (0.003285935 N) n ≈ 1342466.18 Since we can't have a fraction of a ping-pong ball, we need to round up to the nearest whole number. n ≈ 1,342,467 It would require approximately 1,342,467 ping-pong balls to float the yacht.