Classical Mechanics

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Unformatted text preview: 1 PHYS620 Fall 2009 Notes for Chapter 9 MECHANICS IN NON-INERTIAL FRAMES Accelerating reference frames with no rotation Suppose a particle P has position vector r pi in an inertial frame. Consider a second frame that is accelerating but not rotating relative to the inertial frame. Let the position vector of P in this frame be r pa . These position vectors are related by , pi pa ai = + r r R (9.1.1) where R ai is the position vector of the origin of the accelerating frame with respect to the inertial frame. The velocity and acceleration vectors are similarly related: , pi pa ai = + v v V (9.1.2) , pi pa ai = + a a A (9.1.3) where V ai and A ai are the velocity and acceleration of the non-inertial frame. If the particle P has mass m and is acted on by a force, F , then . pi pa ai m m m = = + F a a A (9.1.4) If the ‘force’ is determined by application of Newton’s second law to measurement of the acceleration in the non-inertial frame, then . a pa ai m m = =- F a F A (9.1.5) Hence there appears an additional force-like term that results from the acceleration of the frame. This is an example of a fictitious force, and in this case is called the inertial force, . inertial ai m = - F A (9.1.6) r pi r pa R ai P 2 Tides An application of this result is the explanation of tides. The dominant cause of tides is the gravitational pull of the Moon on the Earth. This gives the whole of the Earth, including its bodies of water, an acceleration towards the Moon. This acceleration is the centripetal acceleration of the Earth as the Earth and Moon orbit around their common center of mass. It corresponds to the gravitational pull an object would feel if it were at the center of the Earth. Objects that are on the side of the Earth nearer to the Moon experience a larger gravitational force than objects of the same mass that are on the other side of the Earth. Hence in the accelerating frame of the Earth, according to equation (9.1.5), objects on the Moon side of the Earth experience a force directed away from the Earth and towards the Moon. In particular the oceans on the Moon side of the Earth will bulge out towards the Moon. Objects on the opposite side of the Earth also experience a force directed away from the Earth, and the oceans on this side will bulge out away from the Moon. Hence, as the Earth rotates, there will be two high tides per day. Rotating reference frames Because of Earth’s rotation, reference frames fixed to the Earth’s surface are not truly inertial. As a consequence two new fictitious forces occur. These are the centrifugal force and the Coriolis force. To see how they arise, first consider the change in the position vector of an object fixed on the Earth’s surface over an infinitesimal interval of time as seen from an inertial frame. The rotation of the Earth is described by an angular velocity vector,...
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Chapter_9_Notes_new - 1 PHYS620 Fall 2009 Notes for Chapter...

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