I am looking to settle a bet. My friend and I were discussing truck bed pools, which are pick-up trucks with their beds tarped and filled with water converting them into small temporary swimming pools. Our conversation went as follows:

You would have trouble braking or going anywhere.

If you could seal the top of the pool with a cover and no air, you could do so.

But due to dynamic pressure caused by inertia, on the front of the pool the cover would be ripped off, at least at the front due to it not being designed to handle this amount of force.

There would be no difference in pressure front or back because there is no air to allow water to move.

At this impasse, we made a bet. Could someone clear this up for us.

We are both engineers: I am electronic and he is software, so we have backgrounds in science and mathematics and are looking to solve this with physics and math.

I am not looking to drive around with a truck full of water, which in the case of my Silverado truck would be 6272 lbs of water, well over the weight limit of my truck. I am looking for scientific principles to explain what is happening to the water, in the water, and why. For example, formulas to calculate forward forces in Newtons at a given point in the water.

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    $\begingroup$ Only in America! $\endgroup$ – innisfree Oct 5 '15 at 3:16
  • $\begingroup$ If the container is not completely full, then one could excite sloshing and in particular wave breaking. These breaking events can drive very large local accelerations, which could be very problematic for the integrity of the container. Note, a covered/non-covered bed doesn't really mater (constant pressure - the restoring force here is gravity), but rather the important parameter is the height over which the free surface can deform. $\endgroup$ – Nick P Oct 5 '15 at 3:28
  • $\begingroup$ Covered does matter because of the stated variables covered with no air to allow for sloshing so a completely enclosed rectangular cube filled with water (6297lbs worth in our pickup example) in motion when brought to an abrupt stop would have significantly higher pressure at the front of the water than the rear during the deceleration process $\endgroup$ – Rich Gant Oct 5 '15 at 3:55
  • $\begingroup$ Well, I agree that having water sloshing around in the pickup bed could be a big problem with the pickup's handling and braking. Also, I agree that, in principle, a possible solution to that is putting all the water into a container which has no air pockets in it because that would halt any large mass movements of the water. And I agree that there could potentially be large local stresses exerted on the container because the water will want to move around. To carry the solution beyond that requires the services of a mechanical engineer to do a detailed analysis of the stresses involved. $\endgroup$ – Samuel Weir Oct 5 '15 at 3:55
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    $\begingroup$ @NickP The simple static pressure the water develops while stopping will tear off any reasonable cover. See below. Certainly dynamics would make it worse, but it simply isn't needed. $\endgroup$ – dmckee Oct 5 '15 at 5:24

Before we talk about trucks, lets talk about merry-go-rounds. I'm sure that you've ridden on one and felt the relentless "force" trying to pull you off the edge. In your intro physics course they will tell you at some length that there is no such force and that the real force is the one you exert hanging on and that it points inward.

And they are right the experience of the centrifugal pseudo-force is an artifact of trying to do physics in a non-inertial frame., but ...

... with the right mathematical effort you can do physics in a non-inertial frame and the effort yields the set of pseudo-forces that you have to introduce.1

In the case of the truck the pseudo-force you are looking for is $-m\mathbf{\ddot{R}}$ (using the notation of Marion and Thornton). The interpertation of this is that from the point of view of the truck every object in it is subject to a "force" that has magnitude eqaul to its mass times the trucks acceleration (as measured from an inertial frame like, say, the ground) but pointing in the opposite direction from the acceleration.

So when you start moving from a stop the real acceleration points forward, so the pseudo-forces points backward. When you stop the real acceleration vector points backward, so you are "thrown against the seat belt".

Now the interesting thing about these pseudo-forces is that, like gravity, they are proportional to the mass of the object they act on and thus create the same acceleration in all free subjects. In fact we can treat them very much like an extra gravity.2

OK. Now lets talk about the water.

When you start the water is subject to a pseudo-gravity pointing back. I'll try to flow out the back.3 When you stop the water experienced a pseudo-gravity pointing forward and tries to pile up behind the cab.

The truck is badly overloaded, but let's say for arguments sake let's say that you can stop at two tenths of a $g$.5

Water is trying to pile up at the front and the cover is keeping it from doing that. The water develops a pressure just like the water in a pond. Assuming an 8-foot bed the pressure is about 1/8 of an atmosphere right by the cab fading off to zero at the back. Let's just say 1/14 atm averaged over the front half of the bed.4 The bed is also at least 4-feet wide. So 16 square feet at 1 psi. Hmmm ... 144 sq.in/sq.ft ... the force on the cover is more than a ton.

The cover comes off and the water spills around the cab.

1 We don't teach non-inertial frames in the intro course because the math is a pain and it can hide some very important principles we want the students to get.

2 Indeed, in general relativity the gravity you experience on a day to day basis is an inertia pseudo-force.

3 Really (that is as viewed from an inertial frame) it is mostly staying still while the tailgate plow through it, lifting it up and leaving it behind, but remember that we are now trying to do physics in the truck frame.

4 A slightly conservative choice and it makes an easy 1 pound per square inch, which is why I picked that particular figure.

5 For a typical 3/4 ton pickup you've overloaded by a factor somewhere between 3 and 4 so this might even be reasonable.

  • $\begingroup$ Interesting. How do tank trucks work then? Are they just really strong containers? $\endgroup$ – innisfree Oct 5 '15 at 7:33
  • $\begingroup$ and what if I put in a few barriers every so often into the water to stop more water moving to the front of the pool? I suppose they'd break too!? Or could it be useful? $\endgroup$ – innisfree Oct 5 '15 at 7:38
  • $\begingroup$ @innisfree Even so, these pressures are only several atmospheres. Pressure vessels like tanker trucks easily withstand them: steel tank and tarpaulins don't compare! AFAIK, the bigger problem is the impulsive forces arising from liquid flowing inside and stopping suddenly against a wall: if you google "Sloshing Pressure" you'll find a great deal on tank design.As for your second question: yes, putting walls in will spread the load, but again the weight from near-to-constant deceleration is not the problem. Tankers do have baffles and walls in them -but the main aim here is to prevent sloshing. $\endgroup$ – WetSavannaAnimal Oct 5 '15 at 10:56
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    $\begingroup$ Baffles won't affect the static pressure (which is what I calculated here). You could, I'm sure build a cover to withstand this load (a ton is not a lot in engineering terms), but the right-angle join is a bear, and the limited number of places to tie down is at least as bad. Tanks are built a heavy material, without corners and with attention to detail like 100% welds (that is all the way through). $\endgroup$ – dmckee Oct 5 '15 at 14:33
  • $\begingroup$ @dMcKee thank you for your well thought out and well written answer I do understand a bit of the math behind this I was however not able to convey the point to the others in a way they could grasp what I was trying to explane I am just not a teacher I guess thanks again for settling the dispute $\endgroup$ – Rich Gant Oct 5 '15 at 17:52

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