Rotating Earth

Spinning Sphere  A point fixed to the Earth is a non-inertial system.  The surface is nearly spherical. Radius R E = 6.37 x 10 6 m  The axis is fixed except for very long intervals. Angular velocity  = 7.27 x rad/s RERE

Surface Point  A point on the surface is in the body frame of the Earth. Define by radius vector  A point rotating with the Earth can be defined from the center or surface. Primed from center Unprimed from surface RERE r’ r

Rotating Equation of Motion  The equation of motion in the rotating Earth’s frame includes external forces. Gravity mgGravity mg Other forces FOther forces F  This is transformed to the surface frame. Earth is large compared to displacementsEarth is large compared to displacements

Plumb Bob  A weight hanging at rest is only subject to gravity and tension. Effective force due to rotation correctionEffective force due to rotation correction RERE g r g eff  The angle is the colatitude. Magnitude: 0.03 sin  m/s 2 About 0.3% of g 

Centripetal Correction  The spin of the earth creates a centripetal acceleration toward the axis of the earth.  The material of the Earth is pulled less at the equator.  Items weigh less at the equator. a =  2 r

Latitude Changes  As the Earth spins the equator slightly bulges.  The radius is about 22 km bigger compared to the pole.  Expect a few Gal difference from equator to pole. Equator: m/s 2 North Pole: m/s 2 Colorado School of Mines next