The key to many buoyancy problems is to treat the buoyant force like all the other forces we've dealt with so far. What's the first step? Draw a free-body diagram.
A basketball floats in a bathtub of water. The ball has a mass of 0.5 kg and a diameter of 22 cm.
(a) What is the buoyant force?
(b) What is the volume of water displaced by the ball?
(c) What is the average density of the basketball?
(a) To find the buoyant force, simply draw a free-body diagram. The force of gravity is balanced by the buoyant force:
ΣF = ma
Fb - mg = 0
Fb = mg = 4.9 N
(b) By Archimedes' principle, the buoyant force is equal to the weight of fluid displaced.
Fb = ρVdispg
Vdisp= Fb/ρg = 4.9/(1000*9.8) = 5 x 10-4 m3
(c) To find the density of the ball, we need to determine its volume. The volume of a sphere is:
V = (4/3)πr3
With r = 0.11 m, we get:
volume of basketball = V = 5.58 x 10-3 m3
The density is mass divided by volume:
ρ = m/V = 0.5 / 5.58 x 10-3 = 90 kg/m3
Another way to find density is to use the volume of displaced fluid. For a floating object, the weight of the object equals the buoyant force, which equals the weight of the displaced fluid.
mg = Fb = ρfluid Vdisp g
m = ρobject V, so:
ρobject V = ρfluid Vdisp
Factors of g cancelled. Re-arranging this gives, for a floating object:
ρobject / ρfluid = Vdisp/V