Question Video: Calculating Relative Movement of Objects Displaced by Fluid Pressure Physics

Video Transcript

A curved object that is underwater moves a distance capital 𝐷 toward a larger object of the same shape, as shown in the diagram. All the water that the smaller object displaces due to its motion impacts the entire curved surface of the larger object that is facing the smaller object. The area of the smaller object that displaces water is 0.25 meters squared, and the area of the larger object that the displaced water impacts is 1.5 meters squared. The larger object moves a distance lowercase 𝑑 due to the force of the water that impacts it. What is the ratio of lowercase 𝑑 to uppercase 𝐷?

Looking at our diagram, we see that initially this C-shaped object here, the smaller of the two, is moved a distance capital 𝐷 like this. This movement exerts a force on the water between these two objects. Pressure is transmitted through this water until it reaches the larger curved object. That pressure exerts a force on this larger object and causes it to move a distance lowercase 𝑑. What we want to solve for here is the ratio of these distances, lowercase 𝑑 to uppercase 𝐷.

As we get started, let’s give labels to these two curved objects. Let’s say the smaller of the two will have the label s for smaller and the larger of the two will have the label l. With that settled, let’s recall that any time an object moves some distance under the influence of a force, that means work is being done on the object. Both our small and our large curved objects then have work being done on them. Not only that, but we’re told that all of the water that’s displaced by the smaller curved object being set in motion impacts the entire curved interior surface of our larger object. In other words, as our smaller curved object moves, all of the energy it transfers to the water between these two objects is then transferred to the larger curved object without any loss. We can say then that the work done by our smaller curved object on the water equals the work done by the water on the larger curved object.

Noting that we can rearrange our work equation so it reads 𝑑 equals 𝑊 over 𝐹, we can therefore express the fraction we want to solve for this way. In the numerator, we have the work done on our larger object divided by the force on that larger object. And in the denominator, we have the work done by our smaller object on the water and the force that smaller object exerts on the water. We use the same symbol for the work done in each case because as we saw no energy is transferred out of this system. Therefore, these values are equal. If we multiply both numerator and denominator of this right-hand side by 𝐹 sub s divided by 𝑊, then the work 𝑊 entirely cancels out of this expression. Along with that, 𝐹 sub s cancels from the denominator. This leaves us with 𝐹 sub s divided by 𝐹 sub l. 𝐹 sub l, recall, is the force acting on the larger curved object, and 𝐹 sub s is the force with which the smaller curved object pushes against the water between the two objects.

We can now recall Pascal’s principle, which says that pressure in general is equal to a force divided over an area. We can rearrange this to read 𝐹 is equal to 𝑃 times 𝐴. And this means we can replace 𝐹 sub s with 𝑃 times 𝐴 sub s and 𝐹 sub l with 𝑃 times 𝐴 sub l. Notice that here we’re assuming the pressure in each case is the same. This follows from the fact that all of the water displaced by the moving smaller object impacts the surface of the larger object. Pressure, we could say, is conserved. Therefore, that factor divides out of this equation.

And now let’s recall that we’re given the two areas of these objects. We’ve called them 𝐴 sub s and 𝐴 sub l. The area of the smaller objects here, 𝐴 sub s, is 0.25 meters squared. At the same time, the inner surface area of the larger object, 𝐴 sub l, is 1.5 meters squared. So the ratio of lowercase 𝑑 to uppercase 𝐷 is 0.25 meters squared to 1.5 meters squared. And if we simplify this as far as we can, the units cancel, and this fraction equals one divided by six. This is the ratio of the distance moved by the larger curved object to that moved by the smaller object.