Physics General Question

Let’s say a stone is suspended in a round basin of water. I take a stick and swirl the water in a circular motion.

  1. Will the speed of the stone be equal to the speed of the water at any other points? Is there any way to measure the speed of the water or stone? Another question will be, is the speed of the water uniform throughout?

  2. The stone should experience a centripetal force. Will the magnitude of the centripetal force be reduced because of the viscous force of the water? If so, is it possible to calculate the magnitude of the resultant force?

  3. So can i assume that in this case, there will be four forces acting in four directions, each perpendicular to each other? The weight, upthrust (assuming the stone is light enough to float), the centripetal force and the drag force?

(P.S: Is viscous force equal to the drag force for the chapter on Physics of Fluid?)

  1. Can resonance be reached? i.e when the speed of the water reaches a maximum because the frequency at which i stir the water is equal to sort of a “natural frequency” of the water.

Let’s see; I might be able to help somewhat here. First, though, what do you mean by “suspended”? When I first read your OP, I assumed that you meant “hanging from a rope”, since otherwise the rock would sink. However, your later questions imply that you mean that the rock is neutrally buoyant; i.e., it has exactly the same density as the water does.

Anyway, first, let’s look at what the water in the basin does, without looking at the stone. I’m assuming that you’re swirling the water long enough to bring it up to speed. The velocity of the water at the edge of the basin, and on the bottom, is zero; this is because the unmoving walls of the basin impart viscous drag to the fluid. So, to answer one of your questions, the velocity of the water will NOT be the same throughout.

This observation has an interesting corollary: Since the water at the top of the basin is free to move, and the water at the bottom is not, swirling the water around horizontally sets up a VERTICAL swirl also, where water moves outward at the top, crawls down the outside walls, moves inward at the bottom, and upward in the center. You can see this in action in your favorite morning beverage: put a dollop of cream in your coffee and stir, watching where the cream cloud spreads. Also, if you drink loose-leaf tea, watch where the bits of tea leaves end up after stirring. The leaves are heavy, so they sink; after stirring, they will pile in a mound in the center of the cup. If your stone is heavy enough to sink but light enough to be pushed around by the water, it will also end up in the center of the bottom.

OK, how about the stone? We’ve covered what will happen if it’s heavy. If it’s light enough to float, it will go to the outside of the top of the basin and whirl around there. If it’s neutrally buoyant and small compared to the basin, it will do what any other blob of water will do: swirl around both vertically and horizontally. If it’s large compared to the basin, I’m not quite sure what will happen, but I suspect it will bob and whirl and twirl a lot depending on where in the velocity field it is.

Answers, as best I can, to questions:

  1. I’m sure the stone will move as fast as the water does at SOME point, given that there is a large difference in water velocities throughout the basin. Note that the (neutrally buoyant) stone will also twirl on its own axis. I believe there are a number of different ways to measure fluid velocity, but I’m not familiar enough with them to know how big the instrumentation is and how effective they are in a 3D flow field.

  2. If you were standing on this stone, you would feel a centripetal force because of the viscous force of the water, in the same way that a stone whirling around your head experiences centripetal force because of the rope. So it’s not really an additional force, if you are already calculating drag.

  3. Weight and buoyancy forces are constant; the stone experiences an additional drag force which is not, in general, perpendicular to the weight/buoyancy.

PS) I believe viscous force = drag. The caveat here is, for the interesting case where the stone is large in comparison to the basin, the velocity of the water is NOT constant at all points on the stone, which makes determining V in the drag equation tough.

  1. Resonances have mode shapes. The only applicable mode shape I can think of right now is is sort of a “slopping” back and forth of the water in the basin. Is this what you are thinking of? If so, then yes you can hit that natural frequency, although it would require stirring in a non-symmetrical way (not hard to do). If you are thinking of something else, please explain.