Static Electricity Legacy Problem #29 Guided Solution

Here is a very difficult problem which involves a good deal of thinking as well as some computational skills. What we wish to determine is the charge on a balloon that is repelling another balloon of identical charge. So you need to kind of picture as you read this problem what's going on. You've got two balloons with like charge and identical charge that are hung by threads to common point on a ceiling. Because they're charged with like charge, they repel each other. And instead of hanging straight down, they hang out at an angle. And the angle between the two threads ends up being 13.1 degrees. The distance from the point on the ceiling where they're hung to the center of either one of the balloons is 155 centimeters. That's the length of the thread plus a little bit more given the diameter of the balloon. Now what you have to do is you have to use Coulomb's Law and trigonometry as well as a Neff-Graff equation in order to calculate the charge on the balloon. You'll notice that I've included a free-body diagram showing the three forces acting up on the balloon on the left, let's say. It doesn't matter which balloon I pick, but I just happen to pick the balloon that is on the left as I look at these two repelling balloons and there's three forces acting up on it. The usual F-Graff directed down. The tension in the string directed up towards the pivot point that would be mainly upwards, but also pulling a little bit rightwards towards the other balloon. And finally, the repulsive force directed leftward up on the leftward balloon. I've labeled these three forces F-Graff, epileptical, and tension, and I should be able to easily calculate F-Graff by using the F-Graff equals mg equation. I need to make sure that my value for m is in units of kilograms since my value for g is 9.8 newtons per kilogram. So when I do this calculation, I end up getting 1.7444 times 10 to the negative second newtons is the force of gravity on this left balloon. Now this downward force of gravity must be balanced by some other force since these two balloons are in equilibrium. And so the only force that's pulling up is the vertical force in the thread. So you'll notice that I've drawn a diagram and I've taken the tensional force going up and to the right and I've broken it up into two components, the vertical F-Y and the horizontal F-X. And this F-Y component has got to balance the F-gravity and as such, F-Y's also got to be the same value, 1.7444 times 10 to the negative second newtons. Now what I can do is use this little force triangle that I have here and the idea that the angle theta in the diagram is one half of the 13.1 degrees. After all, 13.1 degrees is the angle between the two strings deflected from the vertical. So if you slice that in half, you get 6.55 degrees. That's the angle theta and I can relate that angle theta to the side opposite F-X and the side adjacent F-Y using the tangent function. I would say the tangent is 6.55 degrees is equal to F-X divided by F-Y where the F-Y is my force of gravity. I can do some math on this and I can end up finding that F-X is 2.0029 times 10 to the negative third newtons and this is the horizontal component in the thread. This has to balance the only other horizontal force on the left balloon and that's the F-Electrical. So not only have I calculated F-X, I've calculated F-Electrical. I'm going to write that down, 2.0029 times 10 to the negative third newtons, that's F-Electrical, and I have to equate that with K times Q1 times Q2 divided by D-squared. Now the Q1 and the Q2 are the same Q values that charge on both the balloons so Q1 times Q2 becomes Q-squared. So I have this F-Electrical as K times Q-squared divided by the distance between the two balloons squared. Now the next big challenge, and I have to find this distance between the two balloons. So I go back to that little triangle I've drawn, it's a force triangle, but if I pretend for a moment that it's a distance triangle, the diagonal, the red diagonal in there is 155 centimeters or 1.55 meters and that's the side hypotenuse in that right triangle that has an angle of theta of 6.55 degrees and the side opposite that is simply the distance from the vertical to the center of the balloon. Now that is one half of this distance of separation we're talking about in the Coulomb's law equation. So what I can say is that tangent of theta is equal to that X distance divided by the 1.55 meters and I can solve for X using, I said tangent, that's actually the sine of theta and doing so yields an answer for X of 0.1768 meters, that's the distance from the center of the balloon to the vertical line and if you double it you get 0.3536 meters and that's the separation distance. I need to plug that into the Coulomb's law equation as D as in K, Q1 times Q2 divided by D-squared. Now the only unknown left in that equation is the Q. So what I do is I regroup my terms and get Q-squared by itself and then I find out what Q-squared is, 2.7859 times 10 to the negative 14th, I take the square root of it and I get 1.6691 times 10 to the negative 7th Coulomb's and that's the value, the quantity of charge on the left balloon as well as the quantity of charge on the right balloon and I simply round it to three digits.