Could you do this? Could a spinning human slow down the Earth? Theoretically, yes.
It's All About Angular Momentum
In an introductory physics course, there are three big ideas. There is the work-energy principle, the momentum principle and then the angular momentum principle. I'll skip the work-energy principle since it doesn't matter too much here. You might be familiar with the momentum principle. Basically, it says that the net force on an object changes its momentum. I can write it like this:
Yes, this isn't always the expression for momentum - but it's a good starting place. What about the angular momentum principle? It essentially says that there is a property of objects called the angular momentum. You can change this angular momentum by applying a torque. For this spinning Earth problem, we don't have to worry about torque (no net torque) so I'll just say that it's like a rotational force. Now I can write the angular momentum principle:
τ is the torque, but what about the "o" subscript? When we are talking about rotations we have to have some point about which we calculate the torque and the angular momentum. I am referring to this point as "o". L is the angular momentum and ω is the angular velocity. The I is called the moment of inertia, but I prefer to call it "rotational mass". This is a property of an object that makes the connection between angular momentum and angular velocity just like mass does for linear momentum. Now, there is one small point here. In the above expression, I is a scalar value. This is only true if the object is rotating about a fixed axis. This won't be true with the Earth, but I will use it anyway. Just trust me.
Now back to linear momentum. Suppose that I am riding on a friction free train car that is moving at some constant velocity (without an engine). What would happen if I ran towards the front of the car while it's moving? Since there are no net external forces on the system (car plus me), it will have a constant total momentum. As I run backwards, I will have a momentum in the forward direction. The only way for the total momentum to remain constant is for the car to slow down just a little bit.
The same thing is true with angular momentum.
Angular Momentum Example
Here is a quick demo I made to show this conservation of angular momentum.
It's not the best demo, but I put it together rather quickly. Let's see how this works. In the first example, the platform and the disk are both stationary. This means that the total angular momentum is zero. Since there are no torques on the system, the total angular momentum must stay zero. I can represent this with a drawing, but there is something you need to know first. We represent the angular momentum as a vector (I already said that). This vector is parallel to the axis of rotation. If you let the fingers of your right hand curl in the direction of rotation, then your thumb will point int he direction of the angular momentum.
After the small disk starts spinning, the big platform must spin in the opposite direction such that the two angular momentum vectors add up to zero (vector).
When I turn the small disk off, it slows down. This decrease in angular momentum of the small disk should decrease the angular momentum of the big wheel. The reverse is true also. If the big wheel starts spinning and the small disk is turned on, it can slow down the rotation of the big thing.
But wait. What if I turn the small disk 90 degrees (like I did in the video)? In this case, the disk increases in angular momentum. However, the big platform doesn't rotate. Why? Torque is the answer. Here is a drawing of the disk in the second orientation.
If it could, the platform would rotate the opposite direction. But it can't. The floor pushes against the platform and exerts a torque to counteract the change in angular momentum. But what if the small disk was at some angle? In this case, only the vector component of angular momentum in the vertical direction would matter.
Slowing Down a Day
Now for the xkcd question. Could I slow down a day? Yes. How much? That is the fun part. If a person increase in angular momentum, the Earth must also change in angular momentum such that the sum of Earth plus person angular momentum is constant.
Let me start with some assumptions. First, the Earth. I am going to approximate the Earth as a solid and uniform density sphere (which it isn't - see this example). Second, I will pretend that the Earth is on a fixed and non-wobbling axis (which it isn't). Oh, the angular speed of the Earth is about (1/24) revolutions per hour. I guess I can also ignore the angular momentum of the Earth as it moves around the Sun. Yes, there are lots of assumptions here. I can calculate the moment of inertia for a spinning solid sphere as:
But what about the spinning person? Let's say that the person is a cylinder - why not? How about this person-cylinder has a mass of 70 kg and a radius of 0.15 meters (which is probably too high, but it's just an estimate). Now, how fast can this person spin? According to this video, an ice skater can spin up to 400 rpm (41.9 radians/second).
There's one last thing to consider. Where on the Earth is this spinning person? If they are on the equator (and standing up straight), the spin will have no effect on the length of the day. Technically, it will do something - it will change the axis of rotation since the Earth is a zero-torque system - but I'm just considering the length of the day so I will ignore that. Only the component of the spinning person's angular momentum in the same direction as the Earth's angular momentum maters. If the person is in New Orleans, the latitude is about 30 degrees. If I call Earth's rotation axis the z-axis then I can write:
Where θ is the latitude angle. Now I can write the angular momentum of the Earth plus person as (just the z-component):
Since there isn't any torque, the angular momentum of the Earth before the spin is equal to the new angular momentum of the Earth plus the angular momentum of the person.
That's about it. I essentially know all the values to put into that equation. Notice that the z-component of the person's angular velocity will have to be positive in order to decrease the angular speed of the Earth. Are you ready for the bad news? Even if I put this spinning human at the North pole and even if the human spins at 400,000 rpm, I essentially get a zero change in angular speed. Well, at least in python the difference in angular speeds is less than 10-19 rad/s.
I'll go ahead and say it. You can't slow down the day. Sorry. Live for the moment.
Homework
We can't slow down a day, but we can enrich our lives with physics homework. Here are some questions for you.
- How fast would a person have to spin in order to increase the day by 1 second? Ignore relativistic effects at first just to see what kind of answer you get.
- What if all the people on Earth moved as far North as possible and then they all spun? How much longer would a day be?
- What if everyone got in their car and drove East at about 70 mph? How would this effect the length of the day?
- What is the highest angular velocity a human can have without falling apart?
- How many songs can you name that talk about slowing down time?
- In the 1978 movie, Superman flies so fast around the Earth that he reverses time. Forget for a second that changing the rotational direction of the Earth isn't the same as reversing time, estimate the change in angular speed of the Earth if Superman flies 0.5 times the speed of light around the Earth and angular momentum is conserved.
