A 2.0 cm ✕ 2.0 cm ✕ 6.0 cm block floats in water with its long axis vertical. The length of the block above water is 2.0 cm. What is the block's mass density?

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Hey, everyone in this problem, we have a cylindrical object of diameter 20 centimeters and height 50 centimeters that floats in oil with a length of 30 centimeters above the oil. We're told that the density of oil a row is 915 kg per meter cubed. And we're asked to determine the objects density. We have four answer choices here all in kilograms per meter cubed. Option A 549 option B 1530 option C 366 and option D 2209. We're gonna start by drawing a little diagram of what we have going on here. So we have our cylindrical object, OK? And that cylinder has a diameter of 20 centimeters in a height of 50 centimeters. We're drawing out our cylinder here, diameter 20 centimeters in height 57. Now we're told that 30 centimeters is above the oil. I'm gonna use blue here to just kind of draw this line separating the part of the cylinder above the oil and then under the oil. Let's leave the oil down below. And we know that 30 centimeters of our cylinder are above the oil. The total height of our cylinder is 50 centimeters, which means that we must have 20 centimeters of our cylinder underneath the oil. OK. So this is kind of what we have going on. We want to think about the forces acting on this cylinder as well. So if we go ahead and draw a free body diagram, and we're gonna have the weight of our cylinder acting downwards, we have a buoyant force which we're gonna call FB acting upwards. OK. Archimedes principle tells us that the fluid which in this case is oil exerts an upward force on the object. And, and Archimedes principle also tells us that that buoyant force is going to be equal to the weight of the fluid deiced and the weight of the oil displayed in this case. All right. So let's write that out in terms of the math. So we have that the buoyant force FB is going to be equal to the force of gravity or the weight, which is equal to MG. OK. So in this case, it's the mass of the oil displaced multiplied by the acceleration due to gravity. Now, we aren't given information about the mass of anything in this problem, but we do have the density of what. So let's recall the relationship between the mass and the density can recall that we can write the density row as a mass per volume. So the mass divided by the volume which tells us that the mass is going to be equal to the density row multiplied by the volume. OK. So we're gonna use that in this case and we can then therefore write the buoyant force FB as the density of oil, raw oil multiplied by the volume that gets displaced. And we're gonna call that the oil, but that's the volume that gets displaced by this cylinder multiplied by the acceleration due to gravity. All right. So we have information about this blank force now. But remember we're looking for the objects density, OK. We have the density of the oil in this equation, but we don't have the object density. So let's think about now, our forces and we talked about the point force. But what about the sum of force? We know that the cylinder is in an equilibrium situation. OK. And so the sum of the forces is going to be equal to zero. OK. So if we consider the sum of the forces in the y direction equal to zero, let's take up to be the positive direction that our summer forces is gonna be this buoyant force FB minus the weight W that's gonna be equal to zero. All right. Well, this tells us that our buoyant force FB is going to be equal to the weight. And what weight are we talking about here? We are talking about the weight of the cylinder and when we're talking about the weight in terms of our forces. In our free body diagram, we're talking about that entire weight from that entire cylinder, not just the part below or above the oil. All right. So let's start filling in what we know the buoyant force we worked out an equation for. And so we can substitute that in. We're gonna call this equation one. And that's that the buoyant force is equal to the density of the oil multiplied by the volume of oil dis sliced multiplied by the acceleration due to gravity. We're gonna substitute that into the left hand side of our force equation. We're gonna have the density of the oil multiplied by the volume of oil displaced, multiplied by the acceleration due to gravity is equal to the mass of the cylinder multiplied by the acceleration due to gravity. Now, we're back to this situation where we have the mass of the cylinder in our equation, we don't know anything about the mass and we wanna find the density. So we're gonna write the mass in terms of the density and the volume. We get that the density of oil multiplied by it, the volume displaced multiplied by the acceleration due to gravity is equal to the density of the cylinder multiplied by the volume of the cylinder multiplied by the acceleration to gravity. So this density of the cylinder, that's exactly what we're looking for. That's what the question is asking for. So we're gonna wanna solve for that, we can see that the acceleration due to gravity will divide out. We don't need to worry about that. We get the the density of the cylinder is going to be equal to the density of the oil multiplied by the volume of boiled sliced divided by the volume of the cylinder. Now the density of the oil we were given in the problem, the volume displaced and the volume of this cylinder weren't given, but we know the height, we know the radius. So we can go ahead and calculate these two things. So if we go up to our diagram, the volume displaced of the oil, OK, it's gonna be the bottom portion. It's gonna be the same as the volume of the cylinder that's below or in the oil. OK? So I've drawn that in green in her back. So they write that recall that the volume of a cylinder, it's going to be equal to pie R squared multiplied by the height H. And so for the volume of oil displaced, this is gonna be pie multiplied by R squared, multiplied by the height that's in the oil. All right. So if we get back to our equation for the density of our cylinder, we can now write this density of the cylinder as the density of oil multiplied by this volume. We just found IR squared multiplied by the height of oil. And all of this is gonna be divided by the total volume of the cylinder, there's just gonna be pi R squared multiplied by the height of the cylinder. Now again, we can see some things canceling out pi R squared is in the numerator and denominator. So we can divide both numerator and denominator by that term, we don't have to worry about it. And one thing that's really interesting here is when we do that, that means that the radius does not affect this die or this density that we're looking for. Ok. So we could have any radius here for our cylinder and we're gonna get the same result. OK. So it's just that amount or that height that's actually submerged, that matters in this case. All right. So now we're to a point, we can start plugging in numbers. We have that the density of the cylinder is going to be equal to the density of oil, which we're told is 915 kilograms per meter cubed multiplied by the height of our cylinder in the oil, which is 20 centimeters or 0.2 m divided by the total height of the cylinder. 0.5 m by 50 centimeters. All right. And when we work that out on our calculator, we get that the density of the cylinder, the density of this object we're looking for is 366 kg per meter cubed. And that is the final answer. If we go and compare this to the answer choices we were given, we can see that this corresponds with answer choice. Steve. Thanks everyone for watching. I hope this video helped see you in the next one.

A 2.0 cm ✕ 2.0 cm ✕ 6.0 cm block floats in water with its long axis vertical. The length of the block above water is 2.0 cm. What is the block's mass density?

Verified Solution

Key Concepts

Buoyancy

Density

Volume Displacement