When there is only the brine (see Figure 1), the weight of the brine displaced (labelled A) is equal to the weight of the golf ball.
Let’s imagine the golf ball floating at the same level as before after oil has been added on top (see Figure 2). Since the golf ball now displaces oil as well, it must receive an additional upthrust that is equal to the weight of the displaced oil (labelled B). This means that at this level, there is a net upward force acting on the the golf ball. So surely the golf ball will float higher until the weight of the displaced brine plus oil equals the weight of the golf ball again (see Figure 3).
Wait, are we certain that the Archimedes Principle is applicable even when an object is submerged in two different fluids? Let’s use the “water banana” trick again.
Imagine a golf ball that is made of oil above the fluid boundary, and brine below. (Basically, this imaginary golf ball is the displaced fluids.) Since such a golf ball will be at neutral buoyancy, it must be experiencing an upthrust that is equal to its weight, which is the weight of the displaced fluids. So Archimedes Principle works even there are two (or more) layers of fluids.
Still, how does the oil, which clearly exerts only downward pressure forces on the golf ball, results in additional upthrust? This is easily explained if we use a cube instead.
When there is only brine (see Figure 4), the cube receives only upward pressure forces1. When oil is added (see Figure 5), besides resulting in downward pressure forces, it also results in an increase in the pressure in the brine. While the pressure at the top of the cube is increased by h1ρg, the pressure at the bottom of the cube is increased by h2ρg, resulting in a net increase in upthrust. Get it?
- Let’s ignore atmospheric pressure since it will be cancelled out when we take the difference of upward and downward pressure forces.