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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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Most popular questions from this chapter

Two identical cylindrical vessels with their bases at the same level each contain a liquid of density \(\rho .\) The area of either base is \(A\), but in one vessel the liquid height is \(h_{1}\) and in the other \(h_{2}\). Find the work done by gravity in equalizing the levels when the two vessels are connected.

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 \(140 \mathrm{GPa}\).

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 \(0.0810 \mathrm{~kg} / \mathrm{m}^{3}\).

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 \(7870 \mathrm{~kg} / \mathrm{m}^{3}\).

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.
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