A baseball that lands in a lake (water) experiences a buoyant force of approximately 2N while submerged. A baseball’s mass is specified as 142-149g, or 5-5¼ oz., according to the official rules of baseball. What is the specific gravity of bromine if the same baseball experiences a buoyant force of 6.2N while submerged after landing in an open vat of liquid bromine at a nearby chemical factory?

a) 1.5

b) 3.1

c) 4

d) 40

**Explanation**

The buoyant force exerted on an object by a fluid is given by the equation F=ρVg, where ρ is the density of fluid displaced by an object, V is the volume of fluid displaced, and g is gravitational acceleration. Specific gravity is the ratio of fluid density to density of water, and the student should know the density of water as 1 g/cm^{3} or 1000kg/m^{3}.

The buoyant force in water is F_{water}=ρ_{water}V_{baseball}g, and the buoyant force in bromine is F_{bromine}=ρ_{bromine}V_{baseball}g. Since specific gravity is ρ_{bromine}/ρ_{water}, and all other variables are constant in the two buoyant force expressions, the specific gravity of bromine for this problem is equal to F_{bromine}/F_{water}. 6.2N/2N equals SG_{bromine}= 3.1.

a) 1.5, incorrect, This is the approximate weight of the baseball in newtons.

b) 3.1, correct.

c) 4, incorrect, This is the buoyant force in bromine divided by the ball’s weight in newtons.

d) 40, incorrect, This is the buoyant force in bromine divided by the ball’s mass in grams.

This is a great question! Challenged my knowledge of buoyancy for sure…

I have one question that needs clarification. In the explanation you’ve stated that the “volume of baseball” is equal for both cases, whether it is in water or bromine. This clearly simplifies the question to reach the density of bromine. However, in the expression for buoyant force, is it not supposed to be “volume of SUBMERGED baseball”? And would this not differ depending on whether the baseball is floating in water or bromine or any other differing density liquid?

I have understood the Archimedes principle as meaning that the buoyant force can be calculated from either the volume of fluid displaced (x density of fluid x g) or the volume of object submerged (x density of fluid x g), because these are equal. Does this volume not depend on the fluid’s properties?

Thanks in advance for your help!

Ah I think I figured it out…. The key point is that the ball is fully submerged for a moment, and we can actually consider densities and specific gravity from that scenario. When the buoyant force acts on the ball (immediately) it will rise up to the surface in both cases, because the buoyant force is greater than the weight of the ball. WHILE fully submerged, the volume of the ball = volume of water displaced = volume of bromine displaced. So we can take the buoyant force expressions for both cases, solve for volume, and equate the two expressions leading to the beautifully simplified calculation. =)