Math Made Easy: Problem of the Day 26
In Star Wars: The Force Awakens, Starkiller base literally absorbs a star to power a superweapon. Now we all know the Star Wars movies require a substantial suspension of disbelief to be enjoyable (and they certainly are wonderful when taken as science fantasy), but for a moment, letís unsuspend our disbelief and examine this scenario. Just how much would it increase the gravitic pull on the surface of the planet? So hereís todayís problem:
There are two ways to solve this problem. Iíll do the harder one first, then the easier one. As I do this, remember weight is a force Ė the attraction of your mass to the Earthís (or whatever body youíre standing on). In metric, weight is measured in Newtons. In Imperial, pounds serves for weight (part of why the imperial system is so bad Ė it tries to make pounds serve for both mass and weight, when the two are completely different properties).
To do this the longer, harder way, we must use the the formula for gravitic force felt between two masses (where G represents the universal gravitational constant, m1 and m2 the two masses, and d the distance between the two objects centers of mass):
G is a known quantity, and is 6.673x10-11Nm2/kg2
Letís define m1 as the mass of the star. One solar mass is 1.989x1030kg. We actually arenít even going to add in the mass of the earth, as it is 6 orders of magnitude small than that of the sun. Which means in that scientific notation, its first digit doesnít even show up for 3 more digits. Itís negligible.
That leaves m2 as the mass of the person: 100kg. And the distance is the radius of the earth (the distance from the Earthís center of mass to yours is essentially one earth radius): 6.371x106 meters.
Note Iím working in meters and kg to stay consistent with the units of the universal gravitational constant. Also note that all units cancel except Newtons when we do this.
So letís just plug all of those in, and we get 3.270x108 Newtons:
Thatís not very intuitive, at least not for Americans. Since weíre used to working in pounds, and even in metric countries, weíre used to saying we ďweighĒ so-and-so many kilograms, what does that really mean? Well, letís divide it by the acceleration due to gravity at one earth mass. This will yield an answer in effective kilograms, which will give us a better idea of how much the personís weight has increased: Itís still in the order of magnitude of 107, which means he weighs over 3x105 times as much. Thatís three hundred thousand times more weight. I certainly doubt anyone would be able to stand on the surface of Starkiller Base after the star was absorbed if we didnít suspend our disbelief so we can enjoy the fantasy of the movie. Thank goodness we do.
Now that weíve done it that way, letís do the easier way. Just as the Earthís mass is insignificant to the Sunís, so yours is to the Earthís. By orders of magnitude, it doesnít even begin to compare. So when we use that formula for gravitational force, your mass practically doesnít even effect it, and the difference between your mass and anotherís doesnít effect it either. Thatís why your teachers confidently use the same figure for acceleration due to gravity on Earthís surface for pretty much everything, regardless of how much its mass is. So, knowing that, letís use a different formula for force, and just compare the suns mass to the Earthís. Well, the sun is 332,946 times the mass of the earth. Thatís 3.329x105. If we multiply that by the 9.8m/s2 figure we use for the acceleration due to gravity on the earthís surface, we have the new acceleration due to gravity. Now we can just use good olí Newtonís Second Law to calculate the fellowís weight after Starkiller Base has absorbed the star:
And lo and behold, we get essentially the same thing we did before. In fact, itís the difference in the mass of the Earth and Sun that is the difference in the poor individualís weight. So we didnít have to do much calculation to begin with. But, thank goodness Star Wars is fantasy, and we really donít have to worry about scientific accuracy to enjoy it. Because, man, was that freaking cool to see a star getting sucked into a giant planetary vacuum cleaner.