My sister is a teacher too. And she’s smart. And sometimes she poses questions which stump me. She posed a good physics problem on Facebook a while ago.
In case you can’t tell, the three fixed, point masses have masses 1, 4, and 9. She wants to know where you can place a mass so that it won’t move! So that the net gravitational force on it is nil.
Just in case you forgot Newton’s Law of Gravitation between two bodies:
Before starting, I thought this problem would be so easy. If the three masses were equal, we’d have a simple geometry problem. Since they aren’t, it turns out we have something more tricky. I thought the solution would come easy. For me, it didn’t. But I think I got an approximate solution.
Just so we can compare solutions, let’s put our masses on the cartesian plane as below:
As you can tell, I placed the three points on the unit circle.
I don’t want to give much away, so I’m just going to leave you to it. Throw your thoughts in the comments below. If you’re dying for the answer, I’ve hidden what I got on this site somewhere in some not-hard-to-find spot.
If you get stuck, look after the jump for some encouragement.
Just a note: I don’t know if I got the right answer… I think I did, when checking it, but I’m not totally sure. I got tired of working it. That’s why I wanted to throw this up there to see if anyone could corroborate, and also to see your approaches!
Okay, so you’re here because you’re stuck. Here’s some encouragement. I was able to write a couple of equations (net force in the x and y directions) and had to solve them simultaneously (set them both equal to 0). They were ugly. I used a computer. Yeah? Yeah? So what! So sue me! Things didn’t go elegantly for me. I went brute force. I never said this problem had a beautiful solution. Hey, I didn’t even come up with the blasted problem!
But I generated this really pretty image in the course of my investigation.
So if things look like they are ugly, they were for me too! Don’t let that discourage you! See if you can somehow use your computer or graphing calculator to get a solution…