Sunday, September 11, 2011

Why Walking South Would Feel Like Walking Downhill if the Earth was a Perfect Sphere

As Treebeard in Lord of the Rings says, "I always like going south; somehow, it feels like going downhill."

Well I decided to put this to test and actually figure out if walking south actually feels like going downhill and that it's not completely psychological. I'm going to solve this problem assuming the earth is a perfect sphere. This is not a true assumption, so make sure you read the Edit section at the bottom of this page.

If you're going to solve this problem, you should draw a picture. In physics, we like to draw free-body-diagrams that show all the forces acting on the thing that matters. In our case, the thing that matters is a man standing on the surface of the earth. Pointing outward from his body are two arrows that show the forces acting on him (at least the only ones I care about).

So gravity is pulling down on the man and I called it Fg. There's another arrow called Fc. This represents centrifugal force. Centrifugal force is only an apparent force. Believe it or not, your body doesn't want to spin around in circles all day, every day. It wants to keep moving in a straight line. But gravity is strong enough that it keeps you on the ground, moving in circles (much to your body's chagrin) for all eternity. So because the earth is spinning you experience an apparent centrifugal force, think Gravitron.

Well we just might be in luck, because look at the direction the centrifugal force is pulling. It's not pulling straight up or straight down. It's pulling you both upward and south, towards the equator!

So the feeling of walking downhill while you're walking south is not 100% psychological. It might be 99.999999% psychological, but by golly, it's not 100% (at least in 99.99999999% of the cases).

The next question to ask is, How strong is this "Downhill Force" at different latitudes? Is it stronger at the poles or at the equator or somewhere in between?

Well I did some calculations and drew this picture which shows the total force on someone at various latitudes in the northern hemisphere, taking into account both gravity and the centrifugal force.
It pretty much just looks like you'd be pulled towards the center of the earth at any latitude. But instead of looking at the total force, let's look at the tangential force. The tangential force is the force pushing you either north or south (not up or down). Here is a similar picture plotting only the tangential force and ignoring the vertical forces.

If you look closely, you'll see points at 0, 18, 36, 54, 72, and 90 degrees. But there are no arrows coming out of the points at 0 and 90 degrees and the forces in between are the strongest. This makes sense. If you're on the north pole, there is very little centrifugal force since you're close to the axis of rotation. If you're on the equator, you have the most centrifugal force, but unfortunately, it's all pointing straight up into the air, which doesn't help push you forward. In the next figure, I've plotted the tangential force as a function of latitude, including the southern hemisphere.

So the places with the most tangential force are at + and - 45 degrees latitude. And it turns out that if you're in the southern hemisphere, walking north feels like walking downhill. For reference points, I've plotted where New Orleans, Denver, Portland, and Anchorage lie. Lucky Portlanders get maximum downhill force. And my parents who recently moved from Anchorage area to New Orleans can happily say they at least didn't forfeit any downhill force by their recent move.

Although, probably the most important thing to look at on that graph is the vertical axis, which is measured in G's. So that means that even in Portland, OR, you only get a maximum downhill force of about 2/1000 of your body weight. Sad day.

Below, Wintergreen pointed out that the shape of the earth is not exactly spherical. The shape of the earth is determined both by gravity and by the centrifugal force. So that means that the earth is somewhat flattened. The equilibrium shape of the earth would be one where the surface is always perpendicular to the combined force of gravity and the centrifugal force. That means that if the earth has reached that equilibrium shape, then there is no "Downhill Force." The downhill force would only exist before the earth has reached that equilibrium shape. 


  1. ... You sir are most excellent. MOST excellent.

  2. Hi,

    I think that Earth's surface is shaped by the overall force (gravity + centrifugal) in a way that the surface is always perpendicular to the overall force. Tangential force is therefore zero everywhere... CMIIAW

  3. I hadn't thought about that. I'd heard that the earth is somewhat flattened because of the centrifugal force. You're definitely right if the earth has reached that shape. I guess it just depends on whether or not the earth is morphing toward that shape or if it has already reached it. Thanks.

  4. Well, maybe Treebeard was old enough to remember times when the Earth was perfectly spherical:)

    But when I think more about it there are two scenarios:
    A) Plastic Earth: The Earth's is morphing quickly enough to reach the shape with no tangential forces. Then there is no difference between walking south or north.
    B) Rigid Earth: The Earth's is not morphing quickly enough. In this case the Earth has (more or less) shape which was formed when the Earth was hot and plastic enough, i.e. few billions years ago. But at that time, the Earth was rotating somewhat faster (it is decelerating, mainly due to the tidal force of the Moon) therefore now the surface is deformed "too much" and tangential forces have now opposite direction as they would have, if the Earth was perfectly spherical (deformed "not enough"). In other words, walking NORTH feels like walking downhill:)

    But thanks for the post, it was nice to spend a while thinking about this stuff.

    PS: Excuse my English, please:)