Eureka!
Most people know that a boat floating on water displaces a volume of water that has the same weight as that of the boat. But, what about a rock that has a density greater than water and sinks? What happens as a result of the water displacement?
To explain this, we have to look at the forces in action that causes buoyancy. The primary force in all of this is the force of gravity. It appears as an attractive force between objects of mass. On the Earth, objects are attracted toward the center of the Earth with a force proportional to its mass. We commonly call this its weight at the Earth's surface at sea level.
The density of an object is defined as its mass to volume ratio, and since its weight is proportional to its mass, we could also say that density is proportional to the weight/volume ratio at sea level.
A glass filled with water exerts pressure on the walls of the glass due to the gravity force acting on the liquid. If we divided the water in the glass into layers of cubic centimeters (fig. 1), we see that the down force acting on the top layer is just the weight of the water. Now since 1 cm^3 of water weighs 1 gram, the pressure at that depth would be 1gm/cm^2. On the second layer we have the weight of the water on that layer plus the weight of the water from the first layer. And the third layer would have the additional weight of the first and second layers, and so on to the bottom layer. The result is an increasing water pressure as a function of depth in the glass. At sea level there is a pressure increase of 1 gm/cm^2 for every centimeter of depth in the water.
Now, if a buoyant object is placed into the water, its weight will force it down against the water until an equilibrium is reached in the downward and upward forces (fig 2). The upward force of the water against the object is equal to the water pressure against the object in the particular area it is in contact with. The area used for the force calculation is the effective area that is parallel with the surface at that point. The top has no additional force on it since it is above the water. A volume of water is displaced whose weight is the same as the weight of the object. The water level in the glass increases by the equivalent volume, and the weight of the glass of water has gone up by the weight of the buoyant object. Note that the liquid pressure varies with the depth, and the upward force per square centimeter is greater at deeper levels.
So what happens if a rock is submerged in the water but is still held suspended by a string? (fig 3)
In this case, the water pressure is applied over the entire surface area and the water volume goes up by the amount of the rock volume displacement. The forces up against the rock bottom are greater than the downward forces at the rock top but is not enough to make the rock float to the surface. The rock load on the suspension string is lightened by the weight of the volume of water that was displaced by the rock, and the extra weight is added to the glass of water. In the case where the rock isn't suspended and falls to the bottom of the glass, the total weight of the rock is added to the glass.
Now, you can figure out how much the rock weighs as measured with a scale at the bottom of the glass, and with how much momentum force it exerted when hitting the bottom of the glass when falling.
