Today’s repeat of the fuel tank problem had an incorrect answer. The 1/4 line is NOT at the center of mass of the semicircular pizza cardboard. The calculus boys can prove that. The basic issue is that the 1/4 line depends only upon mass, and the balance point depends on both the mass and the distance (distribution) of the mass. Just like an adult and a kid on a teeter-totter, there is a balance point even when the masses are not equal. Click and Clack should never listen to an anonymous physicist!
oldnotdead is correct. The balance point does not mean equal mass on both sides, but equal MOMENT (mass times distance). To use the pizza circle trick, he should have sliced thin sections parallel to the table until the weight of the sections on both sides were the same.
Click and Clack SHOULD listen to anonymous engineers!
The answer was wrong. It’s easier to calculate that the center of mass is not the correct spot if you make the the tank diamond shaped with a distance 1 to each corner from the center. The 1/4 area point is a distance 1/sqrt(2) up from the bottom point. The center of mass of the triangular bottom half of the diamond is a distance 2/3 up from the bottom.
If we are going to morph the tank, I’ll morph it into two identical rectangles placed into a TEE shape. Then the center-of-mass of each rectangle is obvious, and it clear that they are different distances from the intersection line of the two. For a rectangle of size LxW, one moment arm is L/2 and the other is W/2. This “proves” that the line of equal masses is not the balance point. There is nothing “special” about a circle that changes this principle for the fuel tank.
I was wondering about this answer and didn’t get a chance to look at it until now. This center of mass idea would yield a top ?half? of the half-tank that was nearly 10% larger than the bottom. If the tank were diamond-shaped, as Dekabg suggested, the top portion would be 25% larger.
You?d think that Click and Clack would have an assistant on the show that could check these puzzlers and answers for accuracy.
Congratulations Denise from Hawaii for correctly guessing the incorrect puzzler answer.
Yep, it’s an obvious, but wrong, answer to the problem
just as you say.
A hypothetical mismatched pair of brothers on the playground,
(call 'em Maximus Tappet and Minimus Tappet) would have
first found out this principle on some Boston area see-saw.
They had to discover the distance-from-pivot rule in order
to properly balance.
I guess they’ve forgotten the playground wisdom of
their long-departed youth.
Again, your’re over-analyzing the problem. The solution assumes uniform density of the fuel. If the density of the mass resting on a lever such as a teeter-toter was uniform, then it would balance. If a fat kid on one side was to spread his mass over his entire side of the lever and a child of lesser mass did the same on the opposite side, the teeter-toter would fall in favor of the fat kid every time. Think of the fuel in the tank as a mass resting on the fulcrum that is equal at every point. The cardboard “slice” is a brilliant example of using assumptions to simplify a problem.
The true way to avoid over-analysis is to check every idea by experiment. If you are one of those who believe the balance point is the right answer, I have two suggestions:
- Find the balance point of the semicircle pizza board, and cut the board into two pieces along this line. Then weigh each piece and you will find they are not identical.
- If you don’t have scales this sensitive, tape a coarse graph paper over each portion, and count the squares. You will find the two portions do not have the same area.
I’ll repeat: the calculus of finding two equal areas is entirely different from the calculus of finding two equal torques (balance). The 1/4 line is not the balance point.
Again, your're over-analyzing the problem...You're under-analyzing it, and not understanding the problem, or the answer.
Oldnotdead suggested an experiment to prove this. While I’m not going to take the time to try to cut out some cardboard and balance things, etc. Here is a graphical representation of the problem that I posted in the other thread on this matter. I created a semicircle in AutoCad the same dimensions required by the problem. I let the program tell me where the centroid, or center of mass, was. I then created two regions based on that centroid. They are most definitely not the same area, which would equate to mass in a uniformly dense material, of course. I also created a diamond-shaped tank (as suggested earlier) with corners 10 inches from center. Indeed the difference becomes even more apparent in that shape.
Great drawings, jt1979. Now can you show the location of the COM of the TEE that I suggested above? That shape shows the equal masses clearly, and the fact the COM is not at the “expected” location.
And, for those who don’t have a scales, but like “balance”: take the two pieces, and tie thread to them and hang them from each end of a ruler. That is, use the ruler as a balance to see which piece weighs more.
Oldnothead, it sounded like this is what you were talking about.
LMAO, I just realized I wrote “old-not-head” instead of “old-not-dead” in that response. A simple typing error, I assure you; not trying to call you a knot head!
The only thing I would add is the measurement showing the balance point. However, given the data you do show, I can see that it is 8.50.
So the COM is at 8.50 and the “1/4 line” is at 10.00. Of course, they are different numbers for every different tank shape. I don’t think you have given these two numbers for the cylindrical tank.
I’m slowly approaching the “oldknothead” time of life.
“oldnotdead” is definitely correct. As Tom and Ray mentioned the week after the original call, the integral calculus is a little ugly, so I just created a simple Excel sheet (attached) and divided up the lower half of the tank into 1000 slices. Imagine each slice as adding a teaspoon of fuel to the tank. Actually, we’re filling it from the mid-point down to the bottom instead of bottom-up, but mathematically it’s all the same.
I can explain the various columns in detail if anyone’s interested, but the bottom line is this: Counting up the area alone, the 1/4 mark is at 0.404Radius (0.202Diameter) below the center of the tank. Using the center of mass approach, you get 0.424Radius (0.212Diameter), which is a 5% error. Pretty close, but definitely wrong.
The reason that it’s as close as it is is because the half circle is sort-of close to a rectangular shape, not super-imbalanced top-to-bottom. As some of the other posters have noted, as the shape gets more and more imbalanced (e.g. triangle, T shape), the error becomes larger and larger.
Fascinating analysis! I too had jumped to the conclusion that balancing a cardboard cut-out supplied the correct answer.
But I also got hung up on an earlier step in the solution. In their answer, Tom & Ray said “Using 10th grade geometry, he drew two chords on the circular piece of cardboard to find the center of the circle.” It sounds like they’re talking about this technique:
http://www.mathopenref.com/constcirclecenter.html
But that requires improvising something that you can use as a compass. There’s a better way, using only the right-angle corners of the pizza box:
I don't think you have given these two numbers for the cylindrical tank.The center of mass on the cylindrical tank is shown in my first drawing to be at 5.7359 inches above the bottom. In the earlier threads relating to the puzzler that started this whole mess, I, and many others, calculated the correct answer for the quarter tank mark to be at approx. 5.96 inches above the bottom if I recall correctly.