Your significant other gave you a watch for Valentine ’s Day, and tells you it’s made of “pure gold” (density of gold = 19,000 kg/m3). Being a mistrusting boy/girl friend, you decide to find out whether this is true. Later that day, you place the watch on a spring scale and lower it into water. The weight on the scale reads 10N out of water and 6N when completely submerged. Is the watch really made out of pure gold?

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Correct Answer: **B.** *No. The watch’s density is only 2,500 kg/m3. *

We should start by finding the buoyant force. In this case, it is easy, because we know the mass of the watch out of water and the mass of the watch when completely submerged. The buoyant force is of course the difference between the two (Buoyant force = “mass lost” * acceleration due to gravity). Hence, the Buoyant Force = 10N – 6N = 4N.

From this, because the buoyant force = “mass lost” *acceleration due to gravity, we can find the “mass lost” (or rather, the mass of the displaced water):

4N = mass * 10m/s^{2}. And so the mass of displaced water = 0.4kg

We are also told the mass of the watch outside of water (10N), so we also know the mass of the watch using the same logic:

10N = mass * 10m/s^{2}, and so mass watch = 1 kg

Next, in order to determine whether the watch is pure gold, we would have to find the density of the watch (so we can compare it to the density of pure gold). We know the mass of the watch (1kg), so we must now find its volume so we can solve for density using Density (p) = mass/volume. The key to solving this problem is recognizing that the displaced water and the watch both have the same volume. The density of water is common knowledge (1000 kg/m3). (If you don’t know this, commit it to memory now! It’s important for the MCAT). Thus, using the density formula (p = m/V) we can find the Volume of water displaced:

1000 kg/m3 = 0.4 kg/Volume. Solving for Volume gives 0.4/1000 L. This is also the Volume of the watch.

SO now we have both the mass of the watch AND the volume, and can solve for the density of the watch:

Density of watch = 1kg/(0.4/1000L) = 2,500 kg/m3… which leads us to choose answer B. And now when we compare this density to the density of pure gold (19,000 kg/m3), we realize your watch is NOT made of pure gold (and neither is your romance)!

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Photo attributed to ScientificRelevance.