A hollow plastic sphere is held below the surface of a freshwater lake by a cord anchored to the bottom of the lake. The sphere has a volume of 0.650 m^3 and the tension in the cord is 1120 N. (c) The cord breaks and the sphere rises to the surface. When the sphere comes to rest, what fraction of its volume will be submerged?

Video transcript

Hey everyone. So this problem is working with buoyancy force. Let's see what it's asking us. We know a ball is tied to the bottom of a tank containing fresh water using a string. The balls volume is 5.56 times 10 to the negative third years cubed. And the tension in the string is 50.1 newtons. The thread is cut, letting the ball float to the surface at the moment the ball becomes stationary. They're asking us to determine how much volume of this ball is submerged. So this is a tricky question. The key to this is to recognize that there are two different scenarios. We're going to have to do free body diagrams. Some of the forces for both of those scenarios to figure out all of our unknowns and use both scenarios to solve equations to find our this percent of volume, which is what we're in the end looking for. So I'm going to draw the two different scenarios, the two different free body diagrams first and then we'll go from there. So the first scenario, we know the block or the ball is fully submerged. So that's important because we have the tension of the string pulling down keeping this ball submerged. We also have the weight of the ball as a downward acting force and then our upward force is our buoyancy force. The second scenario there, the way the problem states it is the moment the ball becomes stationary. What that really means is the moment where we are floating on the surface. That means our weight, which is our only downward force now is equal to our buoyancy force. So let's recall that our buoyancy force is given by rho G. B where rho is the density of the fluid in question. Gs or gravitational force and the is the volume. So this problem gives us the volume as 5.56 times 10 to the negative third meters cubed. It also gives us the tension which we'll use in this first scenario As 50.1 Newtons. So let's recall Newton's second law and as at people's M. A. Will take the sum of our forces. We are not moving. There's no acceleration this question. Um So they are going to equal zero. So we have the positive buoyancy force is equal to the sum of tension plus the weight. Our buoyancy force, we know we're given that it was in water. So let's recall that the density of water is kg per meter cubed. So the density of water times gravity times the volume. Those are all knowns equals the tension also unknown. Plus the weight. So everything except for the weight here is no they're not asking us for weight. They're asking us for a percent of volume. But let's look at the second question and see if we can kind of figure out how we're gonna use one to solve for the other. So in this scenario we have the same sum of forces equals zero and some of our forces here are just the buoyancy force equals the weight buoyancy force grow G. V. Equals w Now we do know the volume but we know the volume of the fully submerged um ball. What we don't know is what the volume is when it's not fully submerged and that's actually what we're going to be going after this partially submerged volume. So the density is still the same is the density of water. The gravitational force is the same and the weight doesn't change. So we can use the weight from this first equation, plug it into the second equation because in the second equation we have two unknowns and then solve for this second volume. Okay, so let's do that. It's kind of a plug and shut from here, density of water. 1000 kg per meter cubed times 9.8 years per second squared times the volume that was given to us. And the problem 5.56 times 10 to the minus Turn to the nice 3rd meters cubed. We're going to subtract tension onto the other side. So that was 50. newtons. And that equals our way. So you plug that in. Our weight comes out to 4.4 mutants down in our us down into our second equation. We're gonna solve for volume. So that's gonna be 4.4 mutants over 1000 kg per meter cubed times 9.8 m per second squared. And that is going to give us the volume when it is floating Is 4.48 times 10 to the -4. It was cute. And so the question is actually asking us for a percent of what percent of the total volume. And so to do that, we are going to take we'll call this volume floating. We're gonna take volume floating over the volume total times is going to give us our percent volume. So we have 4.48 times 10 to the - meters cubed over 5.56 times 10 to the minus third meters cubed Equals 8.05. Right? Change that into a percent. Multiply that by 100 gives us 8.5%. That's our final answer here. Look at our options are choices a is 8.05%. So that is the correct answer for this problem. That's all for this one. We'll see you in the next video.