MCAT Physics Question — Floating Objects

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_waterV_water = ρ_mercuryV_mercury

 We’re told that the cork floats with half of its volume submerged. So the cork is displacing 5 cm³ of water.

(1 g/cm³)(5cm³) = (13 g/cm³)(V_mercury)

V_mercury = 5/13 cm³

The cork is 10 cm³, which means that 5/13 cm³ 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 cm³, 0.38, as the portion submerged rather than comparing to the total volume of 10 cm³