Surely you have read The Hobbit (the book). It's a classic novel and the basis of the upcoming movie The Hobbit: The Desolation of Smaug. The book is old enough that I don't think I need to give any spoiler warnings. I want to consider the part where the dwarves escape from the elves by floating down a river in barrels.

In the book, Bilbo puts the dwarves in the barrels before they are pushed over into the river. This means that all of the dwarves travel in closed barrels while Bilbo rides on top. Of course some of the dwarves have a comfy trip down the river and some nearly drown. It is up to Bilbo to eventually release the dwarves.

The movie version of The Hobbit has some differences. The movie hasn't yet been released, but I can guess a few things from the trailer. In the image above, it looks like Thorin is in an open barrel and maybe fighting something or someone. Why? Who knows. Let's look at the physics involved in floating in a barrel.

Would the Barrel Float? ———————–

When an object is submersed in a fluid, the fluid pushes up on that object. We call this the buoyancy force. The magnitude of this buoyancy force is equal to the weight of the fluid that is displaced. If the density of the fluid is ρ and the displacement volume is V d , then the buoyancy force could be written as:

If you want more details about where this expression comes from, check out this older post about the famous water bridge. Before looking at the forces on a floating dwarf, we need some estimates. What is mass and height of a dwarf? What about the barrel? From my previous look at hobbits and dwarves, I am going to guess that Thorin (I think that's who's in the picture) has a height of about 1.4 meters with a mass of maybe 55 kg (with gear and stuff).

What about the barrel? I'll assume it is made of wood, something like oak. For the size, I can use the size of Thorin and the amount of Thorin that is sticking out of the barrel. From this, the barrel would have a height of about 0.94 meters with a radius of 0.3 meters. I can also estimate the thickness of the wood at about 2 cm.

Most barrels are barrel-shaped. They are a little wider in the middle than at the top and bottom. For this estimation, I will just pretend they are cylindrical barrels. This means that it would have an approximate mass of:

Just to be clear, r is the radius of the barrel, h is the barrel height and t is the thickness of the wood. For simplicity, I approximated the volume of the side as though it were just a large rectangular piece of wood with a length equal to the circumference of the barrel. Using a wood density of 750 kg/m3, I get a barrel mass of 30.8 kg.

Now I have the total mass of the dwarf plus the barrel. Here is a diagram that I can use to estimate the depth of the part of the barrel that is under water.

In the case of floating, the buoyancy and weight forces have the same magnitude. Since only part of the barrel is submerged, I can write:

Using my estimates for the mass and radius of the barrel I get a depth of 0.30 meters (oh, the density above is the density of water, not wood). But yes, that seems to not quite agree with the shot from the video. Clearly, MORE than 30 centimeters of the barrel are below the water level. How could this be? There must be some cargo in the barrel along with the dwarf.

Looking at the image, I can get a measurement of the amount of the barrel above the water at about 17 cm. This means that 77 cm of the barrel is below water. Let me use this value and solve for the mass of the payload (using the same expression as above).

With the masses of the dwarf and barrel from above, I get a payload mass of 132 kg. What could this payload consist of? It's probably not apples (like in the book version). How about I assume it doesn't take up more than half the volume of the barrel? If it took up more than that, a dwarf wouldn't fit in there. This means that this cargo has a volume of about 0.133 m3 and a density of 992 kg/m3. That's pretty close to the density of water (1000 kg/m3). Maybe the cargo is water. Or perhaps I should say that the dwarf is in a leaking barrel.

Stability of a Barrel ———————

There is still a problem with a dwarf in a barrel. This might not be so stable. Let's first look at a barrel sitting on the ground. Suppose you tip it a little bit and let go. Here is the tipped barrel along with the center of mass of the barrel (I'm not showing the dwarf).

Here the barrel has two forces acting on it. There is the gravitational force (weight). This force pulls on all parts of the barrel. However, it is convenient and equivalent to pretend that the gravitational force just acts at one point which we call the center of gravity. In a uniform gravitational field, the center of gravity is the same location as the center of mass. The other force is the force the floor pushes up on the barrel at the point of contact (since it's a contact force). These two forces mostly have the same magnitude. If you look at the torque about any point in this barrel, you will see that there is a non-zero net torque that causes the barrel to start rotating counter clockwise. The barrel will fall back to a position where it is no longer tipped (as long as it isn't tipped over too far).

Now, what if we do the same thing with a barrel in the water? Really, the only difference is that there is no longer a ground pushing up. There is water instead. Water is different than the ground (in case you weren't sure). The big difference is that water doesn't just push on the barrel at one point. I can still represent this water force (which is what causes the buoyancy force), but there are two important points. First, the deeper parts of the barrel have greater forces on them. Second, the water always pushes perpendicular to the surface of the barrel.

Ok, here is the same barrel tipped in water.

Remember, this is just a sketch. If you were to actually calculate these forces, the first thing you would see is that the total horizontal force from the water is equal to zero newtons (well, very close to zero). This means that the horizontal motion of the center of mass is mostly zero. Next, with an actual calculation you could find the "center of float". This is very similar to the center of gravity, but it is based on this differential force from the water. Then you could pretend like the buoyancy force acts as though it were just at this one point. Here is my guess for where this "center of floating" would be for the same barrel.

For the case of two forces like this, it would cause the barrel to tip over even further. This is bad. But what if you had some very heavy stuff in the bottom of the barrel? This would lower the center of gravity. It would change the diagram to something like this.

With a lower center of gravity, the combination of these two forces would cause the barrel to rotate back towards the upright position. This would be a stable case. And yes, this is why many ships have ballast - some type of heavy mass low in the bottom of the boat.

As a bonus, I made a video showing this exact thing. Here is used a dwarf made of rubber and cork stoppers in a floating beaker-barrel.

So dwarves probably shouldn't stand up in floating barrels. But wait. Where is the center of gravity for the barrel in the trailer? If it is half full with water, then I have three masses to consider: the water, the barrel and the dwarf. The water and the barrel have a center of mass in their center. For the dwarf, I will guess that the center of mass is just like a human - right about at the belly button. I wonder if dwarves have belly buttons. They probably do.

I have three different objects that all have their own center of mass with different values of mass. I can treat these just as if they were point masses. Now I can use the center of mass expression to find the combined center of mass.

For the heights (as measured from the bottom of the barrel), I get:

y w = 0.235 m.

= 0.235 m. y b = 0.47 m.

= 0.47 m. y d = 0.75 m.

Using these values, I get a center of mass at 0.398 meters above the bottom of the barrel. Is this too high? Well, I am pretty sure that the center of buoyancy can be calculated by finding the center of mass for the water that it displaces. I might be wrong, but this approach makes some sense. Suppose that I had a block of water floating in water. I know that seems silly, but just hang on. In that case, the water in water would obviously be stable. Wouldn't it make sense for the center of gravity and center of buoyancy to be in the same exact place? Now if you replace the water with some floating object, the center of buoyancy should still be in the same place.

For a barrel that is partially submerged, I can find the center of mass of the submerged part (assuming straight walls). This would just be half way down from the water level to the bottom of the barrel. Since the the water level is at 0.77 meters above the bottom, the center of buoyancy would be at 0.385 meters.

This is bad. If the center of gravity is higher than the center of buoyancy (which is just barely) then the barrel can tip over.

But Thorin is standing in that barrel. Is he really that foolish? I don't think so. What if Thorin's barrel has 132 kg of gold instead of water? Since gold has a much higher density, the center of mass for this gold would be very close to the bottom of the barrel. This should be enough to bring the center of gravity lower than the center of buoyancy.

I bet Thorin stole that gold from the wood elves. No wonder they dislike him.