Sometimes liking physics is a bad thing. I saw this post on Twitter and the picture really bugged me for some reason. BTW Be sure to Follow Universe and More. Matt does some great work!! His software looks so much more prettier than mine! 🙂
Here is the picture 
What irked me was where the ball landed in the picture. It shows the ball landing at his feet..but does it?
I talk about this case specifically in my classroom. A technique I often use in the physics classroom is looking at extreme cases. This situation is a great example of how to use this method. So lets start on how to solve it.
So here is the question
When you are in a spinning Space Station and you drop a ball, where does it land (or does it?)?
Obvious options: never will land, land in front of you, land at your feet, land behind you?
Lets look at extremes.
Case 1: When the height above the floor is zero.
When you let it go it stays on the floor. So it lands at same spot as you release it..
boring case…so lets look at the other extreme.
We can now discount the “never land” (or does it? Is it only landing because it is touching at the start?)
To look at a non-zero height solution, we need to introduce the idea of Inertial and non-inertial frames of reference. An inertial frame of reference is in an environment that isn’t accelerating. It makes observations “easier” to explain. You will see what I mean soon enough if unfamiliar with the types of frames. In an inertial frame of reference, Newtonian physics laws are followed. Most notably Newtons 1st law. Simplifying the law it states, An object will continue to travel in a straight line unless an external force acts on it. Sometimes a non-inertial frame can easily be explained and no need to go to an inertial frame. Case in point, we are living in a non-inertial frame of reference, since we are experiencing the acceleration of gravity, but we can easily describe the motions we see so no need to go to an Inertial frame. Coriolis force is an example of an exception to that case.
Lets explore why it will in fact land. Since the station is spinning, the people in the system are accelerating (their velocity is changing)
Go to the SIM and hit Start Spin. NO CHEATING, so DO NOT drop the ball yet! Alternate between the Inertial and non-inertial Frames of reference.
In the inertial frame, you see the stick figure spinning around holding a ball. (BTW that is a ball, not the stick figures head 🙂 ) We notice the ball is moving so a force must be being applied to it to get it change direction. So when you let it go it will keep going in a straight line in the direction that is was going when it was released, following Newtons 1st law. The linear path the ball follows ends up hitting the floor. We notice the straight line is shorter than the arc the foot follows so what gets to that spot 1st, the ball or the foot? You can see the path difference between the balls path and the path the foot follows in the above picture.
In the non-inertial frame we just see him standing there applying a force “up”. So “logically” when he lets go of the ball, it will fall down. But since we know that sometimes physics in non-inertial frames seems to break laws of physics, lets leave this part for now. Lets go back to discussing what happens in the inertial frame.
Now that we know the mass lands lets do the other extreme..REALLY high up
Case 2: High up.. lets take it right to the center of the circle.
At this spot we realize the ball isn’t moving since the center of the circle won’t move while it spins. so it obviously never moves to hit the floor. So lets lower it a bit. At this new position, just off center, we notice the ball is moving REALLY slowly compared to his feet so it will take a long time to move along the line of motion once “dropped”. Since it takes so long to “land”, the feet get to where the ball will land first, so the ball must land behind the person dropping it. The higher the drop, the slower it moves so the farther “behind” the person.
Lets now go and drop the ball in the inertial frame. (Drop the ball from a height about one third the radius. A much higher height is for later so don’t try it yet.) Remember..it isn’t the persons head falling off..it is the ball. 🙂 As it moves we notice our logic is confirmed!! I find using “extremes” useful in other situations. “Collision” questions are another example but I will leave that for another day.
Now run the SIM and drop the ball in the non-inertial frame. Is the motion easily explained? If you were standing in that frame and saw the ball move like that, you would realize that some weird unexplained force must be acting on it. You would need to make up some strange and exotic way to explain the motion. (imagine playing “space station basketball”!!) But why invent a force when you can change frames of reference!!
As a final question for the blog, drop a ball from a high height close to the center. Predict what you would see qualitatively in both frames before viewing the drop. Be sure to view in the Inertial frame 1st then the non-inertial frame since the physics makes more sense in that frame. There are some other questions posted on the bottom of the SIM page. Check them out.
For you “geeks” that like numbers, here is a question… What height must you drop the ball so it lands at the persons feet?
Remember, the trick to physics questions (or bane to all physics students) is figuring out what you need and interpret the question as you see fit.. One of my favourite physics lessons I give is “variables are your friend”. So use variables and most of all have fun!!! I have never done the calculations myself but I will post a solution if there is warrant for it! BTW How many solutions are there? 1? 2? mmmm
note:I can happily add to the sim if people have suggestions. Assuming they aren’t too time consuming I can do the mods. Be nice..1st blog post..so working on my writing style. 🙂