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 comic-book fashion, they decide to raise the yacht by filling it with ping-pong balls. A pingpong 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

Solution: a) The buoyant force on one ping-pong ball in water is approximately \(0.3286~\mathrm{N}\). b) Approximately 134,270 ping-pong balls are required to float the ship.

Step-by-step solution

Find the volume of displaced water

Use the volume of a ping-pong ball to determine the volume of the water displaced. We are given the volume of a ping-pong ball as \(3.35 \cdot 10^{-5} \mathrm{~m}^{3}\), so the volume of the displaced water is also \(3.35 \cdot 10^{-5} \mathrm{~m}^{3}\).

Calculate the mass of displaced water

To find the mass of the displaced water, we need to multiply the volume of the displaced water by the density of water (which is \(1000~\mathrm{kg/m^3}\)). Using the formula for mass, \(m = \rho \cdot V\), we have: \(m_{water} = 1000 \cdot 3.35 \cdot 10^{-5} = 0.0335 ~\mathrm{kg}\)

Calculate the buoyant force on one ping-pong ball

The buoyant force is equal to the weight of the displaced water. As weight equals mass times the acceleration due to gravity (\(g = 9.81~\mathrm{m/s^2}\)), we have: \(F_{B} = m_{water} \cdot g = 0.0335 \cdot 9.81 = 0.328635 ~\mathrm{N}\) Answer: The buoyant force on one ping-pong ball in water is approximately \(0.3286~\mathrm{N}\). #b) How many balls are required to float the ship?#

Determine the weight of the yacht

The weight of the yacht is given by the formula \(W = m \cdot g\): \(W = 4500 \cdot 9.81 = 44145 ~\mathrm{N}\)

Calculate the number of ping-pong balls required

In order to float the yacht, the combined buoyant force of all ping-pong balls should be equal to or greater than the weight of the yacht. Therefore, we need to divide the weight of the yacht by the buoyant force of one ping-pong ball to find the required number of balls. \(N_{balls} = \frac{W}{F_B} = \frac{44145}{0.328635} \approx 134270\) Answer: Approximately 134,270 ping-pong balls are required to float the ship.

Key concepts

Density and Buoyancy

Understanding density and buoyancy is crucial in solving the problem of how many ping-pong balls would be needed to float Uncle Scrooge's sunken yacht. Density is a measure of how much mass is contained in a given volume. Mathematically, it's expressed as \(\rho = \frac{m}{V}\), where \(\rho\) is density, \(m\) is mass, and \(V\) is volume. Substances with higher density will sink in fluids with lower density. In contrast, substances with lower density, like ping-pong balls, will float on or in the higher-density fluid.

Buoyancy refers to the upward force that a fluid exerts on an object that is submerged in it, which is crucial for floating objects. The amount of buoyancy depends on the volume of fluid displaced by the object. For example, even though ping-pong balls have a small mass, their larger volume compared to their weight allows them to displace enough water to overcome their own weight, thus floating.

In context of Uncle Scrooge's yacht, the steel yacht is denser than water, hence it sinks. But by adding ping-pong balls—which have a much lower density than water—we can increase the overall volume and displace enough water to generate the buoyant force required to lift the yacht.

Archimedes' Principle

Archimedes’ Principle is a law of physics fundamental to fluid mechanics that explains how the buoyant force on a submerged object is equal to the weight of the fluid displaced by the object. In formal terms, \(F_B = \rho_{fluid} \times V_{displaced} \times g\), where \(F_B\) is the buoyant force, \(\rho_{fluid}\) is the fluid’s density, \(V_{displaced}\) is the volume of the displaced fluid, and \(g\) is the acceleration due to gravity.

To apply this to our problem with the yacht, each ping-pong ball displaces a certain volume of water when submerged, thus experiencing an upward buoyant force equal to the weight of that volume of water. By calculating the buoyant force on a single ping-pong ball, we can use Archimedes’ Principle to estimate how many such balls would be needed to generate a buoyant force equal to the weight of the yacht—thus rendering it buoyant.

Fluid Mechanics

The study of fluids and how they behave under various forces and in different conditions is known as fluid mechanics. It covers both liquids and gases and is the basis for understanding phenomena like buoyancy and the principles described by Archimedes. Fluid mechanics is divided into statics, which studies fluids at rest, and dynamics, which involves the study of fluids in motion.

In this particular problem involving Donald Duck, his nephews, and Uncle Scrooge's yacht, fluid statics is at play as we are dealing with the water's ability to exert buoyant force on the submerged yacht and ping-pong balls. By manipulating the conditions – such as adding ping-pong balls to the submerged yacht – we apply principles of fluid mechanics to alter the yacht's buoyancy, aiming to offset the gravitational force pulling it downward and ultimately enable it to resurface.