Donald Duck and his nephews manage to sink Uncle Scrooge's yacht (m = 3527 kg), which is made of steel (ρ = 7850 kg/m^3). In typical comic-book fashion, they decide to raise the yacht (which is now on the bottom of a lake) by filling it with ping-pong balls. A ping-pong ball has a mass of 2.70 g and a volume of 3.35·10-5 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?
Mg = F – mg
Mg =N• •g - N•m•g <
where M = 3527 kg,
ρ = 1000 kg/m^3 – the density of water,
V =3.35•10^-5 m^3,
m =2.7•10^-3 kg
N =M/(ρ•V – m) =1.15•10^5
why is this a question
a) To find the buoyant force on one ping-pong ball in water, we can use Archimedes' principle, which states that the buoyant force on an object immersed in a fluid is equal to the weight of the fluid displaced by the object.
The weight of the fluid displaced by the ping-pong ball is equal to the weight of the water that occupies the volume of the ball. Given that the volume of the ping-pong ball is 3.35 * 10^(-5) m^3, we can calculate the weight of the water as follows:
Volume of water displaced = Volume of ping-pong ball = 3.35 * 10^(-5) m^3
Density of water = 1000 kg/m^3 (approximately)
Weight of water displaced = Volume of water displaced * Density of water
= (3.35 * 10^(-5) m^3) * (1000 kg/m^3)
= 0.0335 kg
Therefore, the buoyant force on one ping-pong ball in water is equal to the weight of the water displaced, which is 0.0335 kg multiplied by the acceleration due to gravity (approximately 9.8 m/s^2).
Buoyant force = Weight of water displaced * Acceleration due to gravity
= (0.0335 kg) * (9.8 m/s^2)
≈ 0.3283 N
b) To determine the number of ping-pong balls required to float the ship, we need to calculate the weight of the ship and compare it to the buoyant force provided by the ping-pong balls.
The total weight of the ship is equal to its mass (3527 kg) multiplied by the acceleration due to gravity (approximately 9.8 m/s^2):
Weight of the ship = Mass of the ship * Acceleration due to gravity
= (3527 kg) * (9.8 m/s^2)
≈ 34528.6 N
Now, we divide the weight of the ship by the buoyant force on one ping-pong ball to get the number of balls required:
Number of balls = Weight of the ship / Buoyant force per ball
= 34528.6 N / 0.3283 N
≈ 105171 balls
Therefore, approximately 105171 ping-pong balls are required to float the ship.