Static Electricity Legacy Problem #22 Guided Solution

Any two objects which are charged will exert electrical forces upon each other, and the two objects in this problem happen to be a balloon, which is charged negatively, and a plastic tube, which is charged negatively as well. Both having negative charge would mean that the two objects are going to repel each other. So if you've got a balloon charged negatively and a plastic tube charged negatively, and they're going to repel, it is possible to hold that tube underneath that balloon in order to levitate, so to speak, the balloon in mid-air. Levitate it because there's an upward force of repulsion upon the balloon, as well as a downward force of gravity. In this question, we're given some information about the balloon and the charges, and what we wish to understand, or wish to calculate, is the quantity of charge upon the balloon. I begin the problem by diagramming it, getting a middle picture of what's going on. The focus is upon the balloon, and so I draw a picture of a balloon. It can be just a circle, if you wish, and then I could put the tube underneath it, if I wish, and then I'm focusing on the balloon, and I'm looking at it in terms of forces. One of the forces is the obvious down force of gravity, so I draw an arrow down. I label it F-grab. The other force is the up force of electrical repulsion upon the balloon, as exerted by the tube. So I draw an up here. I label it F-electrical. Now, if the balloon is to levitate, these two forces must balance each other, so I start with the equation F-grab is equal to F-electrical. Now, the F-grab is simply going to be the m times the g, the mass of the balloon, times the value of the g, the gravitational field constant. So for the mass, I'm going to use 0.0014 kilograms. I've got to get it in kilograms, because my value of g is 9.8 newtons per kilogram. That allows me to calculate the F-grab. Now I set that numerical value equal to the k times the q times the q divided by the distance squared. Now the distance squared is 12 centimeters squared, but I've got to change that to meters, 0.12 meters. And one of the q values is the 4.3 times 10 to the negative 8 coulombs. The unknown q value is the q on the balloon. So I'm just going to do some algebra to solve for the q on the balloon. I have to use the k value of 8.99 times 10 to the 9th newtons times the meter squared per coulomb squared. And when I'm done, I end up solving for the q on the balloon. And I round it to two significant digits, and that becomes 5.1 times 10 to the negative 7th coulombs.