
Index
23a. The Centrifugal Force 23b. LooptheLoop 24a.The Rotating Earth 24b. Rotating Frames The Sun S1. Sunlight & Earth S1A. Weather S1B. Global Climate S2.Solar Layers S3.The Magnetic Sun S3A. Interplanetary Magnetic Fields S4. Colors of Sunlight 
The reduction of effective gravity by the Earth's rotationOne important rotating frame is the surface of the Earth, rotating with a period of about 24 hoursmore accurately, 23 hrs 56.07 min or 86164 seconds. If the equatorial radius of Earth is 6378 kilometers, the circumference comes to 40074 kilometersslightly more than the 40,000 kilometers supposed to be the poletopole circumference, implied by the definition of "one meter." The difference comes because the Earth bulges out at its equator. The velocity of the equator then is
That velocity is directed eastwards and is significantly faster than the speed of sound, which is about 335 m/s. Space rockets from Cape Canaveral need to attain about 8 km/s (in the frame of reference of the nonrotating center of the Earth ) to achieve orbit, so to give them a favorable starting velocity, such rockets are generally launched eastwards. To derive the centrifugal acceleration on the equator (i.e. the force in Newtons on one gram mass, rotating with the Earth), we calculate in meters and seconds 
Comparing this to the acceleration of gravitysay 9.81 m/s^{2}it is only 0.00346 or 0.346%. Effective gravity on the equator is reduced by the rotation, but only by about 1/3 of a percent The bulge of the Earth's equatorAssuming the Earth is exactly spherical, we expect gravity to always point towards the center of Earth. However, the centrifugal force is perpendicular to the axis of the Earth. Except on the equator, therefore, it is not exactly opposed to gravity, but adds a small horizontal vector component, pointing towards the equator (dashed arrow in the figure). As a result, not only is effective gravity weakened, but its direction is modifiedinstead of pointing to the center of the Earth, is slants (ever so slightly) towards the equator. Does this mean that if you placed a perfect ball on a very smooth horizontal surface, gravity would make it roll equatorward...? Suppose it was so. That same force would also act on the water of the ocean and make it flow equatorwards, and even the solid Earth might deform! How long would this go on? Well, until the equatorial pileup of material forms a "hill" around the Earth, rising slightly towards the equator, where its top would be. No more flow towards the equator would occur once the slope of the ground, as modified by the hill, would be exactly perpendicular to the effective direction of (modified) gravity. With such a slope, a perfect ball placed on a perfectly horizontal surface would no longer try to roll anywhere, and forces on oceans and on land would no longer try to move matter horizontally.
For more about the bulge of the Earth (including a more advanced mathematical treatment) see web site "The Bulging Earth". Jet planes above the equatorThe centrifugal force is a useful concept when figuring out equilibria in a rotating frame of reference, cases where forces balance exactly and no motion occurs. To calculate motion in a rotating frame is much more complicated, and in general is beyond the level of this presentation. All that will be done in this section and the one that follows is point out some complications. Let us assume the reduction of effective gravity is the same at jetplane altitudea small difference exists, but does not affect the argument below. Two jet airplanes fly along the equator at equal speedsone eastwards, the other westwards. Since they fly in circles around the center of Earth, the effective gravity aboard each of them is further changed. Do both sense the same change? You might think so, but it isn't true. It would be the same if the Earth did not rotate (neglect here the attraction of the Sun and the rotation of the Earth around it, two effects which about balance). Without Earth rotation, in the frame of the universe, the eastward and westward motions are completely symmetric. Actually, however, the speeds of the jets are measured relative to that atmosphere, which itself also rotates around the center of the Earth. Viewed from the frame of the universe, the speed of the Eastward jet is added to the rotation of the atmosphere, the one of the westward jet is subtracted. Suppose these are two supersonic Concorde airliners, flying at 465.1 meter/sec. Rather than study the motion in a rotating frame, which may or may not give correct answers, we can always go back to the nonrotating frame of the universe. In that frame, the rotational velocity of the westward jet is zero, and it senses no effective reduction of gravity. In the same frame, the eastward jet has twice the velocity, and since the centrifugal force is proportional to the velocity squared (v^{2}), it senses four times the centrifugal force. Effective gravity aboard that jet is reduced by four times the amount felt on the ground, or 1.384%. For airplanes flying east and west above the equator. this is essentially a modification of the centrifugal force. If the planes flew east and west at (say) latitude 35, the calculation would be more complicated (for the formula, see StarFAQ26.htm#492) What it meansThe above example suggests that we better watch out before applying Newton's laws in a rotating frame.Newton's first law certainly does not holda body not acted on by any force will not stay at rest but will fly off at a tangent. One can salvage this part of the law by adding a centrifugal force in any calculation of balanced forces. But if such forces are not balanced, if they cause motion in the rotating frame, then a modified form of Newton's laws must be used. They do exist, but are beyond the level of this course. The above example suggests some of the problems of rotating frames, and more examples are in the next section, which tells about the Coriolis effect. You and I, and anyone else on Earth, live in a rotating frame of reference. Should we therefore refrain from using Newton's equations in everyday life? It turns out that for motions on a scale much smaller than the size of Earth, at moderate velocities, the effects are negligible. One example concerns the famous question of whether draining water in kitchen sinks on opposite sides of the equator swirls in opposite directions, as is sometimes claimed. It turns out (see next section) that the effect in principle exists, but is much too small to affect observations. In the world wars, gunners firing big cannons corrected their aim (slightly) to account for the Earth's rotation, but you and I can usually ignore it, except for the modification of gravity by the centrifugal force. Questions from Users: What would happen if Earth rotated faster? And in the opposite direction: What if the rotation of Earth could be stopped?. Also: *** Acceleration due to gravity. *** Does the Earth's core share the Earth's rotation?. *** If the Earth's Rotation would Stop... *** If the Earth's Rotation would Change... *** What makes the Earth rotate? *** About Mountains *** The Earth's Spin reduced by Global Warming *** Why can't we feel the Earth's rotation? *** Does Earth rotation affect size of creatures? *** Weight on the Equator and at the Pole *** What if the Earth rotated in only 10 hours? *** On Earth's spin and Moon's appearance *** The centrifugal force from Earth's rotation *** Earth's rotation and human lifespan *** The Coriolis force on a moving air mass *** Change in Earth's gravity after tsunamis 
Next Stop: #24b Rotating Frames of Reference in Space and on Earth
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Author and Curator: Dr. David P. Stern
Mail to Dr.Stern: stargaze("at" symbol)phy6.org .
Last updated: 9222004
Reformatted 25 March 2006
Curators: Robert Candey, Alex Young, Tamara Kovalick
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