Phys 325 Spring 2011 Lecture 24 - Physics 325 Lecture 24...

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Physics 325 Lecture 24 Motion near the Earth’s surface In this lecture, we describe the motion of particles near the surface of the earth. The dominant non-inertial effect is due to the rotation of the earth about its axis, as the effects of other motion (e.g. revolution of the earth around the sun and the motion of the solar system in our galaxy) are comparatively negligible. Consider the coordinate system shown below used to describe the motion of a particle near the surface of the earth: We have a fixed inertial frame ,, x y z    with origin at the center of the earth and the moving frame x y z on the Earth’s surface. The effective force eff F as measured in the moving system becomes (recall Equation 23.3) 0 ( ) 2 eff f r F mg mR m r m r m v   (24.1) where 0 g is the Earth’s gravitational field vector given by 0 2 E R M g G e R  (24.2) where E M is the Earth’s mass and R is the Earth’s radius. We choose Earth’s angular velocity to lie along the inertia system’s z -direction and recall that 5 7.3 10  rad/sec which is a pretty slow rotation (why the Earth’s frame is a decent approximation
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of an inertial frame). Since is very nearly constant in time, we can safely ignore the mr term in Equation (24.1). According to Equation 22.10, we have () ff RR R   (24.3) Equation (24.1) then becomes 0 ( ) 2 eff r F mg m r R m v   (24.4) The first and second terms in Equation (24.4) are what we experience (and measure) on the surface of Earth as the effective g , and we will henceforth denote it as g . Due to the rotation of the earth, the acceleration due to gravity (i.e. the acceleration of a freely falling body in the rotating frame of Earth) appears to be modified by the centrifugal acceleration and the magnitude is reduced, as shown in below
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Phys 325 Spring 2011 Lecture 24 - Physics 325 Lecture 24...

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