Pretend that the Earth is a perfect sphere of radius R and that it has a

Pretend that the earth is a perfect sphere of radius

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Pretend that the Earth is a perfect sphere of radius R and that it has a constant mass density r . Pick any two distinct points on the surface of the Earth. Call them V and M . Bore a tunnel straight through the Earth from V to M . 18 The British physicist and architect (he was Surveyor to the City of London and chief assistant to Christo- pher Wren) Robert Hooke (1635–1703) wrote about the gravity train idea in a letter to Isaac Newton. A gravity train was used in the 2012 movie Total Recall . 117
S URFACE I NTEGRALS 3.3 S URFACE I NTEGRALS Place a train in the tunnel at V . Assume that the only forces acting on the train are gravity, G , and a normal force, N , that the tunnel imposes on the train to keep it in the tunnel. In particular, there are no frictional forces, like air resistance, and the train does not have an engine. Release the train and assume that it does not melt as it passes through the centre of the Earth. V M O G N What happens? We’ll simplify our analysis of the motion of the train by picking a convenient coordi- nate system. ˝ First translate our coordinate system so that the centre of the Earth, call it O , is at the origin, ( 0, 0, 0 ) . ˝ Then rotate our coordinate system about the origin so that the origin, V and M all lie in the xz –plane. ˝ Then rotate our coordinate system about the y –axis so that V and M have the same z coordinate Z 0. So the coordinates ofVandMare(˘?R2´Z2, 0 ,Z). Let’s sup-

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