Over the weekend, I finally watched Disney's Moana (it's been out for what, almost half a year?), and let me say I thoroughly enjoyed it. It was just the right combination of funny, dramatic, and the Rock singing to keep me in my seat for a solid hour and a half. Now, being Disney, I'm not even going to pretend that physics makes sense (how does the water move like it's alive? is it possible to have a giant air pocket directly underneath water? how is matter conserved when Maoi transforms?), but one part of the film particularly set off my physics sensors, and that was when Maoi was singing about his many accomplishments as a demigod. He stated that he "lassoed the sun," giving the people of Moana's earth longer days, and implying that he pulled the sun closer to the earth. Now, ignoring the fact that the sun is a giant fusion reactor and anything that came into contact with it would almost immediately burn up, I wanted to find out if pulling the sun closer to the earth would actually increase the length of the day.
Now, in order to make this simple, I'm going to make two assumptions. The first is that Moana's earth follows a geocentric model, that way the sun's movement will actually affect day length instead of year length, and the other is that the sun orbits Moana's earth in a perfect circle. Obviously this isn't true in reality, but it makes the math easier. So, being the sun follows uniform circular motion around the earth, Fc=Fg, meaning mv2/r=GMm/r2, where m is the mass of the sun, and M is the mass of the earth, and r is the distance between them. Simplifying and solving for v, we get v=(GM/r)1/2. Of course, this tells us nothing about the period of revolution. In uniform circular motion, the period T=2πr/v, and substituting in our previous equation, we find that the period of revolution, as a function of the radius (everything else is constant) T=2πr3/2/(GM)1/2. This means that as the radius increases, the period increases, and, more importantly, as the radius decreases, so does the period. Being our period of revolution in a geocentric model is equal to the day length, this means that Maoi pulling the sun closer should have decreased the length of a day, not increased.