Video: Determining the Submergence Depth of a Buoyant Object

A cube-shaped object, with sides that are 130 cm long each, has a density of 950 kg/m³. The cube is placed into a body of water. The water has a density of 1000 kg/m³. What is the volume of the object in cubic meters? What is the mass of the object? Answer to the nearest kilogram. How many cubic meters of water have a mass equal to that of the object? How far below the water’s surface must the base of the object be in order to displace a mass of water equal to that of the object? Answer to the nearest centimeter.

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Video Transcript

A cube-shaped object with sides that are 130 centimeters long each has a density of 950 kilograms per meter cubed. The cube is placed into a body of water. The water has a density of 1000 kilograms per meter cubed. What is the volume of the object in cubic meters? What is the mass of the object? Answer to the nearest kilogram. How many cubic meters of water have a mass equal to that of the object? How far below the water surface must the base of the object be in order to displace a mass of water equal to that of the object? Answer to the nearest centimeter.

Alright, so in this question, we know that we’ve got a cube-shaped object. So let’s start by drawing a diagram. So here is our cube. And because it’s a cube, each one of the sides is going to have the same length. In this case, we’ve already been told the length of the sides — 130 centimeters long each.

Now, the first thing that we’ve been asked to do is to find the volume of the object in cubic meters. So we need to convert the side lengths into meters. To do this, we recall that one meter is equal to 100 centimeters. And if we multiply both sides of the equation by 1.3, then we find that on the right-hand side we have 130 centimeters and on the left we have 1.3 meters. Therefore, we can replace the side lengths in the diagram with 1.3 meters in each case.

Now, the other thing to remember is that the volume of a cube is equal to 𝑎 cubed, where 𝑎 is one of these side lengths here. And because all of these side lengths are the same, all of these are equal to 𝑎. Therefore, the volume of this cube is equal to 1.3 meters cubed. And when we plug this into our calculator, we find that the volume is 2.197 meters cubed. So this is our answer to the first part of the question. The volume of the cube-shaped object is 2.197 meters cubed.

Moving on to the next part of the question then, we are asked to work out the mass of the object. To do this, we can spot that in the question we’ve been given the density of the stuff that the object is made from. In other words, this object here has a density, which we’ll call 𝜚, of 950 kilograms per meter cubed. And the reason that this is useful is because we can recall that the mass of an object is given by multiplying the density of that object by the volume it occupies.

And hence, the mass of this cube-shaped object is given by the density of the object 950 kilograms per meter cubed times the volume of the object 2.197 meters cubed. And when we evaluate the right-hand side, we find that the mass is 2087.15 kilograms.

However, we need to give our answer to the nearest kilogram. So we need to round this value here. Now, this value is a seven. But depending on the value afterwards — that’s this one — it might either rounds up or stay the same. So this value after the decimal point is a one, which means that’s less than a five. And hence, our seven is going to stay the same. It’s not going to round up. So to the nearest kilogram, the mass of the cubic object is 2087 kilograms. And that’s our answer to the second part of the question.

So let’s move on to the third part: how many cubic meters of water have a mass equal to that of the object? In other words, how many meters cubed of water — how much volume of water — do we need to have so that it has the same mass as this object that we’ve drawn in the diagram?

Now of course, we can recall once again that the mass of any object is given by multiplying the density of that object by the volume it occupies. So if we’re looking at some water, we know that water has a density, which we’ll call 𝜚 sub 𝑤 of 1000 kilograms per meter cubed. We’ve been told this in the question. And we know that we want the mass of the water, which we’ll call 𝑚 sub 𝑤, to be equal to the mass of the cube-shaped object in the diagram and that masses 𝑚, which what we’ve calculated here in the second part of the question.

So in other words, for the water, we already know the mass that we want the water to have and the density of water and we’re trying to calculate what 𝑉 is. So we need to rearrange this equation. What we do is we divide both sides of the equation by 𝜚, the density. This way the density on the right-hand side cancels and we’re left with the mass divided by the density is equal to the volume.

So for the specific case of the water, the mass of the water 𝑚 sub 𝑤 divided by the density of the water 𝜚 sub 𝑤 is equal to the volume occupied by the water which we’ll call 𝑉 sub 𝑤. And at this point, we can sub values in because remember we want 𝑚 sub 𝑤 to be the same as the mass of the object. So this becomes 2087 kilograms. And the density of water is 1000 kilograms. This means that we find that the volume occupied by the water to have the same mass as the cube-shaped object is 2.087 meters cubed.

And that is our answer to the third part of the question, which means we can move on to the fourth part: how far below the water surface must the base of the object be in order to displace a mass of water equal to that of the object? And we need to answer this to the nearest centimeter. So in this case, what’s happening is that we’ve got the object partially submerged in the water. So here’s the surface of the water and all of this part of the object is the part that submerged in the water.

Now, what we’ve been asked to do is to find out how far below the water surface the base of the object must be. Okay, so let’s draw in the base of the object on our diagram. Now dotted lines are the way to go because of course we can’t see this part. But essentially, what we’ve been asked to find is this distance here. Because that distance is how far below the surface of the water the base of the object is.

Now the part that we’ve just colored in is the base of the object. And this distance which we’ll call distance 𝑑 is how far the base of the object is from this surface of the water. And the base of the object is a certain depth below the surface such that the water that the submerged part of the cube displaces has the same mass as the entire cube.

In other words, the submerged part of the cube displaces a certain volume of water. Now that volume of water is equal to the volume of the submerged part of the cube. And that volume of water corresponds to the water that has the same mass as the entire cube.

So we can write down an expression that gives us the volume of the submerged part of the cube. The volume of the submerged part of the cube is given by the width of the submerged part which is 𝑎. That’s this distance here multiplied by the length of the submerged part of the cube, which is another way. That’s this distance here multiplied by the height which is 𝑑. That’s this distance here or in other words the volume is given by 𝑎 squared times 𝑑.

Now, the water that occupied this volume has the same mass as the entire orange cube. And once again, we recall that mass is given by multiplying density by volume. So we want the mass of the water to be the same as the mass of the cube. That’s this mass here. And we know the density of water which is 1000 kilograms per meter cubed. And we have an expression for the volume of the submerged part of the cube. But it’s given in terms of a quantity that we don’t yet know. And that’s what we’re trying to find out.

In other words, the mass of the water is equal to 𝜚 sub 𝑤, the density of water, multiplied by 𝑎 squared 𝑑. And in order to find out what 𝑑 is, we can divide both sides of the equation by 𝜚𝑤 multiplied by 𝑎 squared so that 𝜚𝑤 on the right-hand side cancels and 𝑎 squared on the right-hand side cancels. What this leaves us with is that the mass divided by 𝜚𝑤 multiplied by 𝑎 squared is equal to 𝑑.

At this point, we simply need to plug in the values. We want the water to have the same mass as the cube. So the mass is 2087 kilograms, the density of water is 1000 kilograms per meter cubed, and 𝑎 squared is equal to 1.3 meters squared because 𝑎 is the length of the cube. Now when we evaluate the fraction on the left-hand side of the equation, we find that 𝑑 is equal to 1.2349 dot dot dot meters.

However, this is not our final answer because we need to give our answer to the nearest centimeter. So we first need to convert this quantity into centimeters. To do this, we multiply by 1000. And so we find that 𝑑 is equal to 123.49... centimeters and so we need to round this value to the nearest centimeter. Now once again, we look at the next value — this four — to tell us what happens when we’re rounding. Now, four is less than five. So this quantity is going to stay the same. It’s not going to round up.

And hence to the nearest centimeter, 𝑑 is equal to 123 centimeters. And so that is our answer to the final part of the question, at which point we’ve answered all of the parts of the question.

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