A 10-cubic-cm cork floats in water with 1/2 of its volume submerged. Approximately what fraction of the cork’s volume will be submerged when in mercury (specific gravity = 13)?

a) 1/20

b) 1/2

c) The cork will completely submerge because of the mercury’s greater cohesiveness than water.

d) 5/13

**Explanation**

A floating object will displace a mass of fluid equal to its own mass. For example, if a 10-ton ship is floating in water, it must displace 10 tons of water. Or if a water strider weighing just 0.2 grams is skimming along the surface of the water, it must be displacing 0.2 grams of water.

Since the cork is floating, it must be displacing water (and mercury!) equal to its own mass. That lets us write the following:

m_{water} = m_{mercury}

Since ρ = m / V, we can rearrange the density equation to get:

m = ρV

Thus:

ρ_{water}V_{water }= ρ_{mercury}V_{mercury}

_{ }We’re told that the cork floats with half of its volume submerged. So the cork is displacing 5 cm^{3} of water.

(1 g/cm^{3})(5cm^{3}) = (13 g/cm^{3})(V_{mercury})

V_{mercury} = 5/13 cm^{3}

The cork is 10 cm^{3}, which means that 5/13 cm^{3} represents a tiny fraction of the cork’s volume. Among the answer choices, only (a) is anywhere close.

Alternatively, you can solve this question using the ratio of the specific gravity of the cork to the specific gravity of mercury. Since water has a specific gravity of 1 and half of the cork’s volume is submerged when in water, the cork must have a specific gravity of 0.5. The ratio of this value to the S.G. of mercury (13) gives us the fraction of the cork’s volume that is submerged: 0.5/13 = 1/26, closest to choice A.

a) 1/20th, correct

b) 1/2, incorrect, this answer reflects no change in submersion, even though mercury is far more dense than water

c) incorrect, although mercury has greater cohesiveness than water, this does not affect the buoyancy of objects floating in mercury

d) 5/13, incorrect, this answer reflects using 5/13 cm^{3}, 0.38, as the portion submerged rather than comparing to the total volume of 10 cm^{3}

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I don’t understand need help please

Hey Laila! Here’s a short way to get this answer without doing a ton of math:

Specific gravity represents the ratio of the density of an object or fluid to the density of water. Here, mercury is said to have a specific gravity of 13 (water has a S. G. of 1). This means that mercury has a density 13 times that of water! In other words, it is very dense for a fluid.

Now, our cork originally floated in water with half of its volume submerged, meaning that it has a specific gravity of 0.5 (half the density of water). Specific gravity is really awesome due to this exact trick: the fraction of an object that is submerged corresponds to its specific gravity divided by the specific gravity of the liquid it floats in. In water, 50% of the cork is submerged, so its specific gravity must be 0.5 because (0.5) / (1) = 50%.

But what happens in mercury, where the S. G. is 13? Well, the specific gravity of the cork is still 0.5, as its density isn’t going to change simply because we placed it in a different fluid. Now, the fraction submerged is (0.5) / (13) = 1/26, which is the exact answer to this question (the correct answer, 1/20, is rounded).

This question is also nice because only one answer is logical. If half of the cork already appeared above the surface in water, a LOT more of its volume will float in a dense fluid like mercury. (This is why people float better in very salty water – because it is more dense than fresh water, meaning that we are comparatively “lighter per unit volume.”) In other words, much less of the cork will be submerged. The only answer that is much less than 1/2 is 1/20.

Isn’t the statement that the MASS displaced by the cork in water and in mercury is the SAME, incorrect?

Archimedes principle states that a floating object will displace a VOLUME of fluid equal to the VOLUME of the object submerged.

BOTH the volume and mass displaced by the cork will be DIFFERENT for different fluids, right?

Hi there! You’re right that the volume of fluid displaced must be equal to the volume of the object that’s submerged, but Archimedes’ principle also deals with mass / weight. Consider this cork when it’s floating in water. We have a downward force that is equal to the cork’s weight, which is being balanced by an upward force. This upward force must also equal the cork’s weight – otherwise, the cork wouldn’t float. What is this force? The buoyant force, which is synonymous with the weight of the fluid displaced.

A good way to say this is that “for an object that’s floating, the weight of fluid displaced will be equal to the weight of the object.” Now, we can see that since the cork floats in both water and mercury, the mass (or weight) of water displaced must be equal to the mass (or weight) of mercury displaced. But great question!

I understand the explanation, but wouldn’t it be easier to ignore the 10cm^3 and just use the relationship of specific gravity? As in a SG of 1/2 indicates a density relationship of [500Kg/m^3(block)]:[1000Kg/m^3(water)] and an SG for Hg of [13000Kg/m^3(Hg):[1000Kg/m^3(water)], meaning a SG of 500:13000 for the block in Hg, or 5:130=1/27, which is the fraction of the volume submerged? Is this correct or am I assuming too much?

Hi Dylan, you are absolutely correct! You can certainly use the ratio of the relevant specific gravities to find the fraction of the object submerged. When we do this, we get 0.5:13, or 1:26, meaning that 1/26 of the volume will be submerged. With fluids, there are often shortcuts or reasoning tricks that can help us answer questions quickly. We have included a note at the bottom of the explanation for this question confirming that using your method works as well.