This in't new, but it is popular. Balancing a broom on its brushes. Cool trick, but the big problem is what people say.
"Hey, today is special because the planets are aligned and you can balance a broom!"
Well, today may indeed be special (it could be your birthday or something) but the position of the planets really have no effect on stuff. One more note. I am almost certain that others have shown very similar calculations to what I will show. However, I just can't remember where. If I had to guess, I would say it was Ethan at Starts With a Bang. But all of this has happened before and all of it will happen again.
Let me start with gravity. Not your dad's "mass times g" gravity, no the REAL stuff. Newton's gravity (unless your dad was Newton, then these two are the same thing). Gravity is an interaction between objects with the property mass. It is not just an interaction between things and the Earth. That just happens to be the thing with the most obvious interaction. Suppose I have two objects, mass 1 and mass 2 that are separated by a distance r(as measured from the centers of the objects).
The magnitude of the gravitational force between these two would be:
Where M1 and m2 are the masses of the two objects and G is the gravitational constant with a value of 6.67 x 10-11 N*m2/kg2. Yes, both masses have the same forces on them because forces are an interaction between two objects.
Some Sample Calculations
Let me look at the broom and estimate its mass at around 1 kg. What objects could be interacting with this broom? Well, obviously the Earth. The Earth has a mass of 5.97 x 1024 kg and the broom is 6.38 x 106meters from the center (the radius of the Earth). Using these values, the gravitational force on the broom from the Earth is:
You know why that looks the same as your "mass times g" formula? Because it is. Where do you think g = 9.8 N/kg comes from? Now, how about a couple of planets? Right now, Venus is fairly bright in the night sky. But how far away is it? This is a perfect job for WolframAlpha. It says the distance to Venus is 1.292 x 1011 meters. Since Venus has a mass of 4.87 x 1024, this means the magnitude of the gravitational force on the broom will be 1.94 x 10-8 Newtons. This force is tiny compared to the gravitational force from the Earth. Why? Because the mass of Venus is around the same mass of the Earth, but its center is WAY farther away. Ok, how about a planet with a little more mass. What about Jupiter? It has a mass of 1.90 x 1027 kg and is currently 118.29 x 1011 meters away. This will create a gravitational force of 1.8 x 10-7 Newtons - still tiny. One more object. What is the gravitational force between YOU and the broom? Let's say you have a mass of 65 kg with a distance of maybe 0.3 meters between your centers. This would create a gravitational force of 4.8 x 10-8Newtons. Yes, this is also tiny. But look, the gravitational force from you is greater than the gravitational force from Venus. So, here is your answer. How could the alignment of the planets matter when there are people around the broom that could matter almost as much (or maybe more)?
Then How Do You Balance a Broom?
It isn't difficult. Really, there are two important things. First, the center of mass of a broom is quite low. Much closer to the ground than many people would estimate. Since the "brushes" part is at the bottom and bigger than the handle, the center of mass is low. Here is a picture of me with my hands at the center of mass for a broom.
As a quick note, finding the center of mass for objects like broom is quite fun and simple. Here is a demo of how you can do that. What does the center of mass have to do with balancing a broom? Well, if the center of mass for the broom is not directly over some part of the support for the broom, it will fall over. In this case, the support area of the broom is covered by the brushes. There is another thing that is probably important. The brushes bend and act like a springy-type restoring force. This means that you don't EXACTLY have to get the thing balanced before you let go. You just have to be close. Let describe a similar situation. Suppose you have a completely spherical bowl turned upside down. Try to balance a marble on the top of this inverted bowl and you will find it quite difficult. I guess theoretically, it is possible - but it will be tough. Now imagine a marble on top of an inverted bowl that looks like this:
I know, not my best drawing. Sorry, I will try better in the future. But here you can see that there are several places you can put this marble such that it will stay near the top. Of course, you can't put it just anywhere. The broom is sort of like this. That is why it can stay up. I guess the next thing would be for me to plot the restoring force on the broom as a function of angle. Maybe someday.