Archimedes' principle dictates that the buoyant force on a submerged object equals the weight of the fluid it displaces. The tension in the support cord is the difference between the weight of the object and the buoyant force. Both calculations of forces based on pressure and fluid displacement lead to the same buoyant force.
Understanding Buoyant Force via Archimedes' Principle
To solve for the tension in the cord, we must first understand Archimedes' principle, which states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. The density of the Styrofoam (p = 180 kg/m3) is less than the density of water (approximately 1000 kg/m3), so the Styrofoam will displace a volume of water with a weight equal to its own weight when fully submerged.
The tension in the cord is calculated by the difference between the weight of the Styrofoam and the buoyant force. Since the Styrofoam is fully submerged, the buoyant force is equal to the weight of the displaced water. The formula p = p0 + rgh can also be used to calculate the pressure at a given depth, and this pressure can help to determine the forces on the sides and bottom of the submerged object.
Summing up these forces will give us the same buoyant force that we can calculate using Archimedes' principle. This shows that the upward buoyant force is responsible for keeping the Styrofoam afloat, and it can be calculated both from the pressure difference on the object's surfaces as well as the weight of the displaced fluid.