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Chapter 15: Problem 19

A boat floating in fresh water displaces \(35.6 \mathrm{kN}\) of water. \((a)\) What weight of water would this boat displace if it were floating in salt water of density \(1024 \mathrm{~kg} / \mathrm{m}^{3} ?(b)\) Would the volume of water displaced change? If so, by how much?

Short Answer

Expert verified
The boat would displace approximately \(36415.39 \mathrm{kN}\) of saltwater. There would be no change in the volume of water displaced.

Step by step solution

01

Calculate Volume of Freshwater

First, calculate the volume \(V\) of fresh water displaced by the boat. Use the formula \(V = F / g \) where \(F\) is the buoyant force measured in newtons (N) and \(g\) is the acceleration due to gravity \(9.8m/s^2\). In this case, \(F\) is equal to \(35.6 \mathrm{kN}\), or \(35600 \mathrm{N}\). Therefore, \(V = 35600 / 9.8 \approx 3632.65 \mathrm{m}^{3}\).
02

Calculate Weight of Saltwater Displaced

Next, calculate the weight of the salt water displaced when the boat is floating in this salt water. The formula to use is \(W = \rho * V * g\) where \(\rho\) is the density of the water, \(V\) is the volume of the water displaced by the boat (from Step 1), and \(g\) is the acceleration due to gravity \(9.8m/s^2\). The density of salt water given is \(1024 \mathrm{~kg} / \mathrm{m}^{3}\), hence, \(W = 1024 * 3632.65 * 9.8 \approx 36415.39 \mathrm{kN}\).
03

Determine if Volume of Displaced Water Change

In terms of volume, a ship always displaces exactly its own volume of water; the type of water does not matter. This is because the buoyant force (the weight of the displaced fluid) equals the weight of the object in equilibrium. Thus the volume of water displaced does not change.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Density
Density is a key concept in understanding buoyancy. It is defined as the mass per unit volume of a substance and is usually denoted by the Greek letter \( \rho \) (rho). The formula to calculate density is:
  • \( \rho = \frac{m}{V} \)
where \( m \) is the mass and \( V \) is the volume.
In the context of buoyancy, density determines how a fluid can support an object.
If a fluid has a higher density, it can displace more force to support an object.
  • For example, saltwater has a higher density than freshwater.
This means a boat will displace less volume of saltwater to float compared to freshwater. This property makes density a crucial variable in calculations involving buoyancy and displacement.
Buoyant Force
The buoyant force is the upward force that fluids exert on objects submerged in them. This force is what allows boats to float despite their weight.
The principle governing buoyancy is known as Archimedes' Principle, and it states:
  • The buoyant force on a submerged object is equal to the weight of the fluid that the object displaces.
This means if a boat displaces a large volume of water, it receives a strong buoyant force exactly equal to the weight of the water displaced.
  • For instance, if a boat displaces water weighing \(35.6 \mathrm{kN}\), the buoyant force is \(35.6 \mathrm{kN}\).
Knowing this helps us realize how critical this force is for floating. Since this force counteracts gravity, it enables objects heavier than water to stay afloat, provided they can displace a sufficient amount of water.
Displacement
Displacement in a buoyancy context refers to the volume of water that an object pushes aside as it floats. This concept is important because the amount of water displaced directly influences the buoyant force.
  • The volume of water displaced is directly equal to the volume of the part of the object submerged underwater.
In simple terms, if a boat floats on water, whether it鈥檚 fresh or saltwater, the volume of water displaced remains constant.
Why? Because the volume displaced depends on the boat's size, not the fluid type.
This means if a boat displaces a volume corresponding to \(3632.65 \mathrm{m}^3\) in water, it will do the same in both fresh and saltwater.
  • This constancy allows ships and boats to remain stable across different water types.
Understanding displacement helps in determining the volume and weight of fluid that affects how well an object floats.

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Most popular questions from this chapter

The Goodyear blimp Columbia (see Fig. \(15-20\) ) is cruising slowly at low altitude, filled as usual with helium gas. Its maximum useful payload, including crew and cargo, is \(1280 \mathrm{~kg} .\) How much more payload could the Columbia carry if you replaced the helium with hydrogen? Why not do it? The volume of the helium-filled interior space is \(5000 \mathrm{~m}^{3}\). The density of helium gas is \(0.160 \mathrm{~kg} / \mathrm{m}^{3}\) and the density of hydrogen is \(0.0810 \mathrm{~kg} / \mathrm{m}^{3}\).

A solid copper cube has an edge length of \(85.5 \mathrm{~cm}\). How much pressure must be applied to the cube to reduce the edge length to \(85.0 \mathrm{~cm} ?\) The bulk modulus of the copper is \(140 \mathrm{GPa}\).

(a) If the small piston of a hydraulic lever has a diameter of \(3.72 \mathrm{~cm}\), and the large piston one of \(51.3 \mathrm{~cm}\), what weight on the small piston will support \(18.6 \mathrm{kN}\) (e.g., a car) on the large piston? (b) Through what distance must the small piston move to raise the car \(1.65 \mathrm{~m}\) ?

The surface of contact of two fluids of different densities that are at rest and do not mix is horizontal. Prove this general result ( \(a\) ) from the fact that the potential energy of a system must be a minimum in stable equilibrium, \((b)\) from the fact that at any two points in a horizontal plane in either fluid the pressures are equal.

An iron casting containing a number of cavities weighs \(6130 \mathrm{~N}\) in air and \(3970 \mathrm{~N}\) in water. What is the volume of the cavities in the casting? The density of iron is \(7870 \mathrm{~kg} / \mathrm{m}^{3}\).
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