Veritasium – The SAT Question Everyone Got Wrong
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That was great! Thank you, Chris McLaughlin.
I would have said 3, too, so, glad I took the SAT prior to 1982.
When I first saw this problem (I saw this video a few months back), my intuitive answer was “4,” because it is the same problem as the number of times the Earth rotates on its axis in the year that it goes around the Sun. The answer was obvious by inspection.
It is an old concept, and it is a tricky one. As our presenter said, it is a concept that is vitally important for astronomers to understand when figuring out where to point their telescopes each night. There are 365¼ days in a year, but the Earth rotates 366¼ times in that year. This is why the Zodiac constellations change positions relative to midnight. The Moon rotates once a month (actually, a little more), but from our point of view it does not rotate at all, and if we do think of it as rotating, we would usually think it rotates 12 times a year, but it rotates once more than that (actually, closer to 13½ times).
It just goes to show that the people who put together the SAT tests (and similar tests) are not rocket scientists, brain surgeons, nuclear engineers, or astronomers.
The SAT people also have not played much with Spirograph sets, because that is another experience that can show the same results.
It is also a concept that is important for orbital mechanics, so I will add them to the list of people not writing these tests. Orbital mechanics (the engineers doing the calculations) need to understand this concept in order to calculate where their spacecraft is over the planet. If they don’t understand this one, they don’t take pictures of Mars in the right place, and Robert would be giving us the wrong locations for some of the Cool Image Time! pictures.
The concept appears in a few unexpected areas of life. There are gears called “orbital gears,” or “planetary gears,” that do this same thing, and their shafts turn more than the number of times they go around the center (or less, if they are on the inside).
The presenter gave us examples from different reference frames, showing how important frame of reference can be. From our frame of reference on the Earth, the Sun rises in the east, but from a reference frame fixed to the Sun, it doesn’t rise at all; the Earth rotates as it revolves around the Sun. It is a completely different experience, giving completely different realities. Frame of reference is one reason why Lagrange Points are so difficult to understand, because they are presented in a rotating frame of reference, but they only work because of the revolutions of the secondary body. For instance, the L1 point is not a balance of gravity, as the usual presentation of Lagrange points suggests, but is the balance of gravity and the momentum of a body at the revolving L1 point.
It’s interesting that if both sprockets are turning, the ratio (the pivot points being fixed) in a 3 to 1 diameters equal “three” revolutions. equivalent of his example of the ribbon when laid flat. For instance a four cycle engine turns at a ratio of 2 to 1 against the camshaft. With both gears turning in a fixed location, the extra revolution is canceled out.
The mention the spirograph triggered a memory from grade school, trying to make a three lobe flower using a cog three times smaller, resulted and four petals and not three… now I know why. The rotating pivot adding distance from the surface increasing the diameter of the circle that the cog must travel adding an extra turn.
Now that I think about it, from mechanical theory over 40 years ago, explained this is how the sun, planetary, and ring gear work together (as Edward explained) has a “torque multiplying affect” that’s used in heavy equipment everywhere.
A mechanical equivalent of orbital mechanics comes to mind, when I think about a driveline on a vehicle. The acute angle (inner) of the driveline as it exits the transmission travels slower then the outside of the constant velocity joint. Opposite but similar to how a satellite speeds up in an elliptical orbit as it gets closer to the planet.
(The speed difference in the driveline is offset and balanced by the same angle in the opposite direction going into the rear end. If the driveline splines ever come apart then reassembled with the caps “not aligned”, that results in a catastrophic vibration leading to failure. The angle of the transmission and rear end also must be within one degree of each other, body lifts and modifications must take this into consideration or drive lines will continuously fail)
360° in the circle… 365 days in a year. A little bit more than 1° per day, the equivalent of two full moons difference in the position of the stars every night.