angular velocitycentrifugal forceearthhomework-and-exercisesnewtonian-gravity
The question I am working on is, "Consider the following.
(a) Find the angular speed of Earth's rotation about its axis. rad/s
(b) How does this rotation affect the shape of Earth?"
I am fully capable of solving part (a); however, I am not sure how to describe the effect earth's rotation on its shape. I tried to search my textbook for the answer, but could not find anything. Is there an actual effect on the shape?
How fast would a sphere need to rotate for a dust speck at its equator to achieve balance between gravitational attraction and centrifugal force?
If you do the math (equating $G M m / R^2$ to $m \omega^2 R$ and using $M = \frac{4\pi}{3} \rho R^3$ as well as $\omega = 2\pi f$), it follows that the size of the sphere is entirely irrelevant and that only the density $\rho$ of the sphere enters into the equation for $f$, the number of revolutions per unit time: $$f^2 = \frac{1}{3\pi}G\rho$$ For $\rho = 5.5 \times 10^3$ kilogram per cubic meter (the density of planet earth) it follows that $f=0.197 \times 10^{-3}$ revolutions per second, corresponding to a revolution period of $5070$ seconds (1 hour and 24 minutes).
- Are there any other rivers in the world for which this is true?
The Mekong, at least after it has left the Tibetan plateau.
Location Latitude Elevation (m) Radial distance (km)
Source 33° 42' 30" 5224 6376.8
Manwan Reservoir 24° 45' 15.5" 997 6375.4
Ruak River mouth 20° 21' 16" 336 6375.9
Mekong delta 9° 27' 30" 0 6377.6
While the Mekong does flow "downhill" (toward the center of the Earth) at the start thanks to that four kilometer drop in altitude across the Tibetan plateau, it's pretty much all uphill from the Manwan Reservoir on.
- If one uses the actual gravitational potential for a non-rotating oblate spheroid, is the Gulf of Mexico at a higher gravitational potential than Minnesota?
There's a nice what-if scenario at esri, the Environmental Systems Research Institute, that asks and answers this question: If the Earth Stood Still: Modeling the absence of centrifugal force. The scenario investigates what would happen to the Earth's waters if the Earth somehow stopped rotating over the course of a few decades. That's a long enough span of time that the waters would have a chance to adjust, but far too short a span of time for isostasy to readjust the shape of the Earth.
Not only would the Mississippi stop flowing southward, the waters of the oceans would flow away from the equator. This would leave a globe-spanning equatorial bulge of land surrounded by two polar oceans. The end result:
Best Answer
The Earth is mostly fluid. This may seem a strange claim but the rock in the mantle behaves like an extremely viscous fluid, which is why continental drift can happen.
Anyhow, if you imagine a stationary drop of liquid it will form a sphere. This is a bit of a cheat because small drops form spheres due to surface tension not gravity, but the end results are similar. If you start the drop rotating the water at the "equator" is going to feel an outwards force due to the rotation, so the drop will change shape and get bigger around the equator while the poles flatten. This shape is known as an oblate spheroid, and indeed it's the shape of the Earth because the Earth behaves like a rotating fluid drop.
To try and calculate the change of shape is a little messy, but luckily someone has done all the hard work for you and you can find the results: Thayer Watkins: The shape of a rotating fluid mass.