When I was a young child, my uncle Fred used to try to convince me of some pretty outlandish things. His favorites were to say that the earth was flat and that England was a hoax. I was old enough to know better and I tried to argue as best I could, though with limited success. Looking back, I think uncle Fred was trying to stretch my mind a bit so that I could use critical thinking to defend true propositions in the face of opposition.
In trying to dismantle the idea that the earth was round, he suggested that the people in the southern hemisphere would fall off. I could dispatch that well enough - gravity comes from the earth and there’s nothing “below” it to pull people off. Then he argued that if it were spinning, people would be flung into space, like water droplets from a spinning basketball. I wasn’t sure what to say about that. It took a few days of thinking before I realized the answer: the earth spins pretty dang slowly. It takes a full 24 hours just to make one rotation. That wouldn’t throw water off a basketball, and it probably wouldn’t throw people off the earth.
But it is spinning, so even if the effect is small people will still feel a little less force than they would have if the earth were stationary. The details of this are a classic problem which you can find in many texts and websites, but it’s fun so let’s give it a try. The principle is fairly simple. Circular motion requires an acceleration because the direction of travel and thus the velocity is constantly changing. In this particular case that force is provided by gravity. The force that keeps you from falling to the center of the earth is just the pressure the ground exerts on your feet. Your feet exert an equal and opposite pressure on the earth and so you experience that as your weight. But if you’re in circular motion by virtue of the earth’s rotation there has to be a net force downward, and so you experience this as a decrease in the force from the ground and thus your weight decreases. The acceleration required for uniform circular motion is
Therefore a person on the surface will perceive that much less acceleration, since it’s no longer being matched by upward force of the earth’s surface. Plug in the maximum value for v (at the equator) and the radius of the earth r, and you’ll get a value of about
a = 0.0339 m/s2
This is pretty small compared to the 9.8 m/s2 or so that is the standard downward acceleration at the earth’s surface, and so fortunately no one gets thrown off by the rotation. But measuring it would be possible with adequately sensitive equipment. Since F = ma, I can multiply my own mass (about 170 pounds*) and see that I should weigh less on the equator by about 0.94 9.4 ounces (Thanks to Dr. Pion in the comments for catching my careless calculator mistake!) than I would be on a stationary earth or the north pole. Not much of a weight loss solution, but at least I won’t be thrown off the earth.
That’s all we need if you’re standing still, and that’s where most published solutions stop. But if you’re moving - for instance in a boat or airplane - additional complications arise. Your motion in the east-west direction changes v. If you’re traveling with the earth’s rotation, v will be larger. Vice versa for traveling against the earth’s rotation. This produces an additional term that must be taken into account. Finally there’s yet another (generally smaller) term that takes into account the fact that following the curvature of the earth itself requires a centripetal acceleration. These refinements are called the Eötvös effect and are necessary for precision measurements of the earth’s gravitational field.
*The pound is a little ambiguous as to whether it’s a unit of mass or force. Here I take 1 pound as a mass equal to the usual ~0.45359 kg. Why not just use metric and make the units a lot easier to deal with? Because having been raised on them, I find English units more intuitive. Metric readers are invited to take advantage of the much easier math and recalculate for their own weights!
2 responses so far ↓
1 CCPhysicist // Jun 10, 2008 at 4:40 pm
Nitpick: 170*(0.0339/9.8) = 0.59 lb = 9.4 oz.
You should be comparing your answer to the acceleration calculated directly from Newton’s formula, not “g”. The value of “g” includes a multitude of factors, one of which is the effect you calculated. That is why GM/R^2 = 9.83 m/s^2.
[The problem is actually simplest in the rotating laboratory frame where one employs a fictitious force when calculating the downward acceleration of gravity, in effect "using up" some gravity because we are in a circular orbit of sorts, but can also be cast in terms of the normal force on a body at rest in the lab, as you did.]
The standard value of g = 9.80665 m/s^2 is an exact value used globally to convert a mass in kg or lb to a weight* in N or lbf** or poundals. At one time it was probably the acceleration around Paris, France. [The back of my envelope says that GM/R^2 - 0.0339*cos(lat) is about 9.808 m/s^2, which is pretty close.]
The actual value of g is significantly smaller in Florida than in New York. As you calculated, the rotational effect is on the order of 0.03 from the pole to the equator, but there is also a large effect because the earth is oblate and a bit pear shaped, as well as effects due to your elevation above sea level.
*Legally, worldwide, weight refers to a mass - but physics and engineering use that term for m*g as a matter of convenience. That is why the Net Weight is given in kg on packages, where the legal terms of weights and measures are used. The ounce or pound on a package is also a mass, in the Avoirdupois system, and is defined in terms of the standard kg. (This is one detail that Wiki gets right but most physics books get wrong.) The Troy pound is smaller, by the way.
**A one pound force, 1 lbf, will accelerate a one pound mass at 9.80665 m/s^2. That makes the lbf the same as what most people mean by lb. We adjust scales and load cells so the force they measure is converted into the correct mass for you local value of g. The poundal is more interesting, in that it takes about 32 poundals to give 1 lb that same acceleration.
2 Carl Brannen // Jun 10, 2008 at 5:21 pm
To a very large degree, people believe pretty much what they want to believe and they will cling to those beliefs in the face of whatever arguements they come across to the contrary. This is a universal feature of all humankind, physicists exhibit it just like anybody else.
Whenever a new physics theory comes around, the majority of the old physicists reject it, and a good portion of them argue against it for the rest of their lives. I started college as a mining engineering major. The introductory geology class was taught by a really old guy who knew his rocks but didn’t believe in plate techtonics. This was unsatisfactory in the department so they brought in another instructor for 2 weeks to cover it.
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