19. Fluid Mechanics

Buoyancy & Buoyant Force

# Anderson Video - Density of an Object Underwater

Professor Anderson

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>> Hello, class. Professor Anderson here. Now let's take a look at a density problem. Let's say we have an object that is submerged under water and we want to figure out what the density is of that object. And let's do the following. Let's take a vat of water. And let's put a spherical object in there. And let's tie it to the bottom with a string. And now we want to determine what the density is of that spherical object if we know the tension in the string holding it to the bottom. All right. So, how do we attack this problem? Well, first off why don't we draw a free body diagram. Here's our object. We have a buoyant force B going up on the object. We of course have gravity going down, mg. And we have tension, T, in the wire or the string, also going down. Okay. If that thing is just stationary, then we can say the sum of the forces in the y direction is equal to zero. What do we have? We have B going up. We have mg going down. We have T going down. And all that's equal to zero. Now let's say we give you a little bit more information. Let's say you can measure the tension in that string and you measure it to be one-third the weight of the object, one-third mg. Let's plug this in now and see what we have. We have B minus mg minus one-third mg equals zero. And I can lump those two terms together and I get four-thirds and it's got a negative in front of it so I can add it to the other side and I get B is equal to four-thirds mg. But we also know that the buoyant force is the weight of the displaced fluid. So this thing is a sphere. It displaced water. And so the buoyant force we can write as the density of water times the volume of that sphere. That gets us a mass. And then we have to multiply by gravity. Now on the right side we have m. m is the mass of the sphere. So if we know the density of that thing, we can multiply by the volume and that gets us the mass of the sphere. And then again we have g hanging out. And now look what happens. A bunch of stuff cancels out, right. V sphere is on both sides. That cancels out. g is on both sides. And we can solve this thing for the density of the material that is in that sphere. And we got a four-thirds over here so if we multiply across by three-fourths, we get three-fourths the density of water. Okay. That's good. It's less than the density of water. The thing is trying to float but we're holding it down with the string. And if you know the density of water, you can calculate the density of the sphere. Good. Hopefully that's clear. If not, come see me in my office. Cheers.

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