# Why do pendulums swing more slowly at the equator?

1. Sep 11, 2008

### Blue_Jaunte

I was asked this recently and the only explanation I could come up with was that the Earth is oblate and the difference in R would account for the difference in the period. Is this wrong? Is this even a real phenomenon (the longer period at the equator than at other latitudes)?

2. Sep 11, 2008

### mgb_phys

The period of a pendulum is $$2\pi\sqrt{l/g}$$ so if g is less the period (time for a swing) becomes longer. g varies by a few percent around the world dependign on the density of local rocks.
Tt also depends on latititude because, as you said, the earth bulges at the equator and so you are futrther from the centre of the earth and the force of gravity is slightly lower.

3. Sep 11, 2008

### D H

Staff Emeritus
A pendulum swings slower at the equator because the Earth is rotating. The rotation acts to make the pendulum swing slower directly and indirectly. The direct effect is easiest to envision from the perspective of the rotating Earth-fixed frame. In this frame, a centrifugal force of $r\omega^2$ arises directly from the Earth's rotation. At the equator, this direct effect alone amounts to 0.034 m/s2. There is of course no centrifugal force at the poles.

For those who insist there is of course no such thing as centrifugal force anywhere, you will be forced to look at things from the perspective of an inertial frame. I leave this as an exercise to you curmudgeons. Keep in mind that the answer you get will be the same.

I also cited an indirect effect that results from rotation. The Earth's rotation makes the Earth bulge at the equator. As mgb_phs noted, this makes things at the equator further from the center of the Earth. Together, the direct and indirect effect act to make earth's gravitational acceleration 9.780 m/s2 at the equator. This is about 0.052 m/s2 less than it is at the poles, or a 0.53% reduction in the gravitational acceleration. The direct effect accounts for about 65% of the difference.

The next leading factor in variations in gravitational acceleration after Earth rotation and the J2 non-spherical harmonic term is altitude above the spheroid. Things weigh 0.28% less at the top of Everest than they do at sea level at the same latitude. Local variation in density is a distant fourth.