Trust Us: You Can’t Really Turn a Truck Into a Submarine

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A commercial comes on from time to time promoting a product that sprays some kind of plastic or rubber coating on things. I have no idea if it’s a great product, but the company uses the stuff to seal a truck and turn it into a submarine. If you don’t believe me, check out the video.

It’s a cool trick, but it’s not a serious submarine. Why not? Well, first of all, trucks aren’t submarines. But also because there are two kinds of submarines.

Constant Pressure Submarine

When most people think of a submarine, they probably think of this:

You get in and close the hatch. The submarine submerges and you can have a nice cup of tea–or do other things. Even if the submarine dives to 300 meters (I’m picking something at random), you’ll feel mostly the same. It will be just like enjoying a cup of tea in a windowless room at sea level. This is because the interior of the vessel is kept at an approximate pressure of 1 atmosphere, which is what you feel at the surface.

OK, a quick note on pressure. On the surface of the Earth, you are standing in all this air (mostly nitrogen and oxygen). Air particles move around and collide with things. When they collide with your body, these particles exert a tiny force. The larger the surface area of an object, the more collisions occur. I like to think of the collisions per unit area–this leads to force per unit area that is the definition of pressure.

The same is true underwater, except water molecules, not air molecules, collide with surfaces. Water has a much higher density than air, so there are many more particles. But what supports this water? More water. If you take a small imaginary cube of water in the ocean, the total forces on it must be zero. There is of course the gravitational force pulling down on this water, but there also is the force from the pressure of the rest of the water. All of the water above this cube of water pushes down on it even as the water below pushes up. The only way for the total force to be zero is if the force (and thus the pressure) is greater on the bottom of the cube. So, in the end water pressure increases with depth.

Because water is so dense compared to air, pressure increases rapidly. In seawater, you only need to descend about 10 meters (33 feet) to experience a significant increase. A human swimmer at at depth of 10 meters would have a total pressure of 2 atmospheres. Pressure increases by 1 atmosphere every 10 meters. So, yes–that submarine at 300 meters has a pressure on the hull of 31 atmospheres.

But don’t forget, there is someone inside that submarine drinking tea at 1 atmosphere. Here is a cross sectional diagram of a typical submarine:

Because the internal and external forces are not equal (but the surface areas are), there is a much greater force pushing on the outside of the hull than the inside. This means you need a sturdy wall to keep the submarine from collapsing.

Now I should say something about circles. Submarines are usually cylindrical or spherical. There is a good reason for this. When you press a circular shape with equal pressure from all sides, the stress on the material is uniformly distributed. This is not true of other shapes like a square. The circular shape means you get a stronger hull with less material.

Variable Pressure Submarine

Turning now to the video of a truck turned into a submarine. How about a quick calculation? Suppose the truckmarine (I just made that up. Maybe I should say submatruck) is 5 meters underwater. How much total force does the water exerted on the side window if the truck has an interior pressure of 1 atmosphere? Here is my diagram:

Yes, that is an underwater truck. Outside the window, the force is exerted by the atmosphere and 5 meters of water. Inside the truck, the force is due to the 1 atmosphere of air. So, in the end this is equivalent to only the force due to the pressure from the water. Fresh water has a density of around 1000 kg/m^{3} and the pressure dependence on depth would be:

In this expression, ? is the density of the water and g is the gravitational constant. Assuming the window is 20 cm by 20 cm and at a depth of 5 meters, the total force is 1960 Newtons (440 pounds). Yes, I made the approximation that the pressure on the window is constant but in fact it increases with depth. But trust me, this calculation will work.

A force of 1960 Newtons is rather large for a truck window. Just imagine 2 very large men standing on that glass. Do you think it would support them? I don’t. And this is at a depth of just 5 meters. What if the submatruck descended to 300 meters? Yes, the glass would clearly break. You can’t build a constant internal pressure submarine with these types of materials.

But you can still make a submatruck (I like that word) that can dive to 30 meters without breaking a window. The trick is to increase the pressure within the truck such that the interior pressure is the same as the exterior pressure. Now the net force on the window (and other parts of the truck) will be near zero. In fact, if you increase the interior pressure to just a bit higher than the external pressure you can prevent water leaks. Higher pressure interior air will push out through any weak spots and prevent water coming in. It’s a perfect submatruck.

But what about drinking tea? I guess you could still drink tea, but there are other problems. First, you will need to equalize your ears like you do when scuba diving. Second, you might need to ascend slowly to avoid decompression sickness–like you do when scuba diving. Yes, this is just like scuba diving except that instead of wearing a scuba mask, you are inside the scuba mask. You are inside a giant scuba mask that looks like a truck.

So, you can indeed make a submarine out of a truck. You probably can’t take it any deeper than you would scuba dive, and in fact you probably need some scuba tanks inside so you can add air to the interior as you go down. But it can be done. Here is another example of a variable pressure submarine from Top Gear. It’s cool.