Application of Archimedes’ Law Consider the situation in Fig. E1.7(a), in which a barrel rests on a raft that floats in a swimming pool. The barrel is then pushed off the raft and may either float or sink, depending on its contents and hence its mass. The cross-hatching shows the volumes of water t

Question

Application of Archimedes’ Law

Consider the situation in Fig. E1.7(a), in which a barrel rests on a raft that floats in a swimming pool. The barrel is then pushed off the raft and may either float or sink, depending on its contents and hence its mass. The cross-hatching shows the volumes of water that are displaced. For each of the cases shown in Fig. E1.7 (b) and (c), determine whether the water level in the pool will rise, fall, or remain constant, relative to the initial level in (a).

Accepted Answer

Initial state. Let the masses of the raft and barrel be [latex]M_{r}[/latex] and [latex]M_{b}[/latex], respectively. If the volume of displaced water is initially [latex]V[/latex] in (a), Archimedes' law requires that the total weight of the raft and barrel equals the weight of the displaced water, whose density is [latex] ho:[/latex]

[latex]\left(M_{r}+M_{b} ight) g=V ho g.[/latex] (E1.7.1)

Barrel floats. If the barrel floats, as in (b), with submerged volumes of [latex]V_{r}[/latex] and [latex]V_{b}[/latex] for the raft and barrel, respectively, Archimedes' law may be applied to the raft and barrel separately:

[latex] ext { Raft : } M_{r} g=V_{r} ho g, \quad ext { Barrel : } M_{b} g=V_{b} ho g.[/latex] (E1.7.2)

Addition of the two equations (E1.7.2) and comparison with Eqn. (E1.7.1) shows that:

[latex]V_{r}+V_{b}=V ext {. }[/latex] (E1.7.3)

Therefore, since the volume of the water is constant, and the total displaced volume does not change, the level of the surface also remains unchanged.

Barrel sinks. Archimedes' law may still be applied to the raft, but the weight of the water displaced by the barrel no longer suffices to support the weight of the barrel, so that:

[latex] ext { Raft : } M_{r} g=V_{r} ho g, \quad ext { Barrel : } M_{b} g>V_{b} ho g.[/latex] (E1.7.4)

Addition of the two relations in (E1.7.4) and comparison with Eqn. (E1.7.1) shows that:

[latex]V_{r}+V_{b}

Therefore, since the volume of the water in the pool is constant, and the total displaced volume is reduced, the level of the surface falls. This result is perhaps contrary to intuition: since the whole volume of the barrel is submerged in (c), it might be thought that the water level will rise above that in (b). However, because the barrel must be heavy in order to sink, the load on the raft and hence [latex]V_{r}[/latex] are substantially reduced, so that the total displaced volume is also reduced.

This problem illustrates the need for a complete analysis rather than jumping to a possibly erroneous conclusion.

Question 1.7: Application of Archimedes’ Law Consider the situation in Fig...

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