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Systematic Problem Solving: Part 3
Okay, so we've now covered the first 2 steps in our problem solving method. As a reminder, they were "determine the information" and "determine paths to solutions." Now we are up to our second example to help highlight step number two. It's a physics problem dealing with gravity, so we'll need just a little bit of background. Once we have that I think you will be well equipped to handle it, even if you've never taken a physics class before. The problem comes from the 9th edition of Fundamentals of Physics by Halliday, Resnick and Walker.
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Isaac Newton published his theory of gravitation in 1687 on the 5th of July [1]. Since then it has become part of the foundation of instruction in physics classes around the world. Relativity affects it's correctness, but in general it holds true. He gave us a particular constant, which we call G and an equation that says the force of gravity between two objects is given by the equation above. The F stands for the force due to gravity and the G is the constant we discussed and just gets multiplied with everything else. The m's are the mass of the two objects being considered and the r-squared is the radius of the distance times itself, or more easily, half the distance squared. The diagram with it shows us the same idea in a different way. There we are looking at the effects of the force of gravity on the point P that are caused by points S1 and S2. We're not worried about the forces acting them, our interest is only in point P. To see this we draw little arrows showing the directions of the forces. If all S1 and S2 have the same mass, the amount of force between each of them and the centre point P will be the same and cancel out! That's fantastic news. This is a very helpful tool that we'll wind up using a bit.
You probably won’t need to memorize any of that, just remember it's here to refer back to in a few minutes. So let's go ahead and take a look at our problem. Feel free to try tackling it on your own before we walk through it.
You probably won’t need to memorize any of that, just remember it's here to refer back to in a few minutes. So let's go ahead and take a look at our problem. Feel free to try tackling it on your own before we walk through it.
Let's start with step one, "determine the information." We are focusing on the point in the middle, this is like our point P in the smaller diagram back at the top. We know the little circle is at a distance r from that point and the bigger circle is a little further at R. All of the particles have the same mass, m. we want to find out how strong and in what direction the forces acting on the point in the middle are. We can call this two questions that we will work on at the same time for a bit. So for step two, "determine paths to solutions," what do we have? Well, we could apply the gravitation formula to each point and add up all of the combined forces like we did at the very beginning. We could also build a model and try to balance it to see which way it will tip. We could search for someone else's solution to the same problem, there's probably a solution manual for this book out there, or we could try to use the diagram given to try and achieve a similar method.
The first choice seems like a really hard way to solve this, and the second, while a very good way to help solve it, would be time consuming and probably costly too. The third option is to cheat, which if you want to do that just skip to the end and don't worry about learning anything. That leaves us with using the diagram.
The first choice seems like a really hard way to solve this, and the second, while a very good way to help solve it, would be time consuming and probably costly too. The third option is to cheat, which if you want to do that just skip to the end and don't worry about learning anything. That leaves us with using the diagram.
To start with we have a bunch of other points arranged around the one in the centre. We are going to want to consider the effects of each point in the diagram. If we try to find pairs maybe we can cancel out the forces like we did earlier.
If we draw some reference lines it will help us to more easily identify pairs. Also, we could quickly count how many points are on each side of each line to see in which ways it is balanced. This is the fastest way to figure it out, but we should probably still cancel opposite points just to make sure.
So with our lines in place, we can see that we get four "corners" we can cross out. After that we can match up three other "pairs" and we are left with just one other point besides the centre. So now we only have one thing to figure out since all the others cancelled. So if you check our equation, since they have the same mass, we're going to wind up with a very simple answer.