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http://www.newton.dep.anl.gov/askasci/phy99/phy99x82.htm


Gravity and Spin

Name: Meredith
Status: student
Age: 30s
Location: N/A
Country: N/A
Date: 1999

Question: I am embarrassed to ask this, but I cannot think of one physics equation that would say that gravity is anyway related to the rotation of the earth. Yet all my science students think that the earth has gravity because it spins. I want to correct them, but then in the back of my mind, I get nervous, and think maybe I'm wrong. It's driving me crazy. Would the earth's gravity still be 9.8 m/s per second if the earth kept revolving around the sun, but stopped spinning?

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Replies: Why are you embarassed? You are correct. You can't think of any physics experiment that would demonstrate that gravity is related to rotational motion because it isn't. If the Earth were not rotating relative to the stellar background, its gravitational field would be unchanged. Mind you, it IS true that the acceleration ``due to gravity'' would be very slightly more than 9.8 m/s^2 anywhere but at the poles because there would no longer be any centrifugal force reducing the effect of gravity. But that isn't what your students are saying anyway. (And we are ignoring any issues of general relativity here, as they are weensy corrections.)

You probably want a counter-argument to throw their way. Let's see. You need first of all to convince them that the gravitational attraction between two bodies depends on the properties of both (e.g. the mass of both bodies appears in in Newton's law of gravitation). Point out to them that the weight of any small body, which is the gravitational attraction between it and the Earth, varies with its size. Having two students step on the nearest scale will verify this.

Now note that they have assumed that the gravitational attraction between a large body (the Earth) and a small body (a person or object on the Earth's surface) depends on the spin of the LARGE body. (Because if the Earth were not spinning, there would be no such force, according to them.)

And ask them why, in that case, the gravitational force between the small body and the large body does NOT depend on the spin of the SMALL body. (You can easily demonstrate that the time it takes a spinning object to fall is identical to the time it takes a nonspinning object to fall.)

You can also perform the thought experiment of considering two bodies near each other, one spinning planet, one nonspinning small object. Gradually you transfer mass from one to the other, keeping the angular momentum constant (which means the planet starts spinning faster and faster, incidentally, but perhaps irrelevantly). After a while the object will become as large as the planet was, and the planet as small as the object was. Is the gravitational attraction still due to the (very quickly) spinning ``micro''planet left over? If so, then we should see that two spinning objects -- if necessary very quickly spinning objects -- should attract each other more strongly then they do when not spinning. And this is not, of course, observed.

If their arguments are vaguely based on centrifugal forces, ask them why gravity does not vary near the poles, where you are going around the Sun once a year, but not being flung around in a circle every 24 hours.

I hazard that your students are suffering from a peculiarly American malady -- the tendency to confuse proximity to causation. The Earth does two interesting things as far as human beings are concerned: it spins, giving us day and night, and it exerts gravity, giving us up and down. (I suppose a case can be made for a third: it goes around the Sun tilted, which gives us summer and winter.) It is human, or perhaps more correctly American human nature, to assume that because the two phenomena occur on the same object one must cause the other. Gravity is the more mysterious fact, more in need of explanation, so it is assigned to be the effect, and spin assigned to be the cause.


The root cause is that American students, in particular, find it difficult to cope with the state of ``not knowing''. They need to feel they know, even if they can't possibly given the evidence available to them. They are happier feeling they know, and later finding themselves wrong, than in feeling ignorant, and later being enlightened. It's an unfortunate tendency, which tends to hamper their scientific abilities, since a good scientist needs to cherish the feeling of ``not knowing'' as long as possible. It is only in this mental silence, free of unjustifiable hypothesizing, that the truth, which always starts out as a tiny voice indeed, can be heard.

Hence you might do them a favor if you explore with them first WHY they feel there is a connection between spin and gravity. If the answer is ``just because'' or ``it seems sensible'' or something equally data- and logic-free, then you can do them some good by discussing how very dangerous trusting that kind of unsupported evidence-free ``hunch'' can be, not only scientifically, but in other areas. The tendency to judge too hastily and on too little actual data is a particularly unfortunate American vice. You can point out to them that the very best and most successful scientists and inventors are those who are the least inclined to think they know the answer before they are absolutely, positively, convinced-by-mountains- of-data sure they know. In science plausible hypotheses are a dime a dozen, but good, fact-supported theories are rare gems indeed.

Grayce

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Dear Meredith,

You are correct. Gravity is a universal property of all objects and is not considered in ordinary

Newtonian mechanics to have anything to do with the state of motion. So the number 9.8 applies whether the earth is spinning or standing still.

Best regards,
prof. topper

The acceleration due to gravity is indeed 9.8 m/s/s (g) as a result of the size of the mass of the earth pulling equally on every kg of mass on its surface. This number becomes smaller as the distance increases between the center of the earth and an object. It actually decreases by the square of the increase in distance meaning that the effect of gravity falls quickly. Even Mars does indeed pull on you and me - gravity never drops to zero because you can never reach zero by dividing a number - but the force is so small as to never be measure.

Back to the earth... Although I have never seen the calculation, I expect we could prove (and maybe another reply will do it for you) that the effect of the rotation of the earth lessens g at the equator and causes it to be greater at the poles.

If anything, the spin of the earth lessens the effect of gravity and it definitely does not cause it. Perhaps using the pull of the moon as an example would help make this clear. It is removed from the surface of the earth and yet has a considerable effect on the oceans. You might also look up the work of Henry Cavendish and Philipp von Jolly, both of whom used a very sensitive balance and a large lead mass to measure gravity.

Good luck with your young and impressionable minds.

Larry Krengel
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A common misconception is that anything spinning has gravity due to its spin. Newtonian Gravitation has to do with a field that is created by the mass that is in it. The greater the mass, the greater the attractive force. Sir Isaac Newton found:

F = G(m1 * m2)/(r^2)

to be the gravitation force of attraction between two bodies. He used calculus to find his results for a uniform spheroid. The earth and moon are good approximations of sphereoids. The constant of proportionality in the equation, G, was measured years later by Lord Cavendish. Cavendish, in an ingenious experiment, found:

G = 6.67 * 10^-11 N m^2/s^2

This is a very tiny number, but is very important in astronomical calculations where the masses of the bodies are very large. You may wish to tell your class about this experiment. The experiment does not involve any spinning of bodies.

The gravitational field intesity at any point in space is found by the slope of a Force versus Mass graph. Using a spring scale, measure the amount of force different masses exert at a given point in space. At the earth's surface, the gravitational field intensity is about 9.8 N/kg. By using Newton's second law, we find that the downward acceleration due to gravity near the earth's surface is about 9.8 m/s^2. Go elsewhere in space and this number will differ. If you go to a region of lower mass, say the moon, the gravitational field intensity is lower (1.6 m/s^2). Measuring the local gravitational field intensity is easy to do in the classroom with spring scales (callibrated in Newtons) and a set of hooked masses.

Another common misconception about gravity is that there is no gravity in space. They see that the astronauts are weightless and think that gravity is not present. The Space Shuttle would continue in a straight line forever if it were not for the gravitational field of the Earth pulling the craft into orbit. (see Newton's First Law of Motion) Your weight is the force of support pushing up on you. If you are not supported by anything, you will be accelerated in the gravitational field. Stand on a table with a spring scale hand. Hang a mass from it. Note that there is a measureable support force on the object at rest. Now, while holding this system, jump off the table. While you are falling, you are weightless (and so is the mass on the scale). Clearly, the force due to gravity was working on you (you accelerated), but you were weightless due to the lack of a support force. Please note that your mass never changed.

Throw a ball, lightly, horizontally. It falls a meter away. Throw the ball a bit harder, and it strikes the ground a bit farther away. Is the ball weightless while in flight? (Yes.) Is the ball under the influence of the Earth's gravitational field? (Yes, it is falling) What would happen if you could throw the ball fast enough that its path would match the curvature of the earth? The ball would be constantly falling . . . around the earth. That is what the astronauts are doing: falling around the earth. They are high enough in the atmosphere where air friction is so extremely small that they do not burn up. They are weightless since they have no support force. They are in the earth's gravitational field since they are always falling.

Good question! Thanks for using NEWTON BBS.

---Nathan A. Unterman
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The force of gravity is related to the earth's MASS, not its rotation. Also, the value of the force of gravity is related to the separation between the two masses used to measure the force (the earth is usually one of the masses). The gravitational acceleration constant depends on where on the earth's surface it is measured but the average value of 9.8 m/(second squared) is accurate enough for most work.

In addition, the force of gravity is slightly offset by the 'centripetal' force or the inertial effect of the rotation. This effect would be greatest at the equator and zero at the poles. If the earth were not rotating the measured gravitational acceleration constant would be slightly larger because this compensating inertial action would not be present.

Bradburn
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Your students have part of a right idea, but it is leading them in entirely the wrong direction. They are probably thinking about science fiction books and movies in which space stations generate "artificial gravity" by spinning. This is due to the commonly misunderstood centripetal acceleration, which is perceived as a "centrifugal force." The centripetal acceleration is what makes objects move in curved paths, instead of in straight lines. when an object moves in a cirle, the centripetal acceleration is constantly pulling it toward the center of the circle. This is misinterpreted as a force pulling away from the center of the circle, but that "force" is actually just the tendency of an object to keep moving in a straight line.

If it weren't for gravity, the spinning of the earth would cause us all to fly off into space, not stick to the earth. This is just like swinging a weight on a string in a circle; if you let go, it sails off in the direction it was moving when you let go. In fact, the spinning of the earth works against gravity. This effect is small everywhere, greatest at the equator, and zero at the poles. So if the earth stopped spinning, gravitational acceleration would increase everywhere on earth except at the poles.

So, go ahead and correct your students. Maybe you should even have them do the experiment of swinging a weight on a string. (outside.) Let them see for themselves that the spinning of the earth does not pull us inward. Then maybe they'll be convinced.

Richard E. Barrans Jr.
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The Earth does not have gravity because it spins. The gravitational attraction between the Earth and a person on the Earth depends only on the mass of the Earth, the mass of the person, and the distance between the Earth's center of mass and the person's center of mass.

However, the force a person's feet exert on the ground (i.e., the person's weight) does depend on the Earth's spin. What if the Earth started spinning faster and faster? At some speed, the Earth's gravitational force would not be enough to keep us on the surface. We'd fly off into space because the acceleration (v^2/r) required to keep us moving in a circle of the Earth's radius at the rotational speed of the Earth would be greater than the acceleration of Earth's gravity.

The figure 9.8 m/s/s is a measured value, and it includes the effect of the Earth's spin as a small correction. If the Earth were not spinning, the measured value would be .034 m/s/s larger. If the Earth were spinning so fast that a day took only five minutes, then the gravitational acceleration would be spent entirely to keep us moving in a circle, and we'd all feel weightless.

Here are the numbers and equation I used:

radius of Earth: 6.4x10^6 meters

speed of object at the equator: 465 m/s

acceleration required to keep an object moving at speed v in a circle of radius r: v^2/r

Tim Mooney
Beamline Controls & Data Acquisition Group
Advanced Photon Source, Argonne National Lab

 

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