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•Pretend that the Earth is a perfect sphere of radiusRand that it has a constant massdensityr.•Pickanytwo distinct points on the surface of the Earth. Call themVandM.•Bore a tunnel straight through the Earth fromVtoM.18The 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. Agravity train was used in the 2012 movieTotal Recall.117
SURFACEINTEGRALS3.3 SURFACEINTEGRALS•Place a train in the tunnel atV. Assume that the only forces acting on the train aregravity,G, and a normal force,N, that the tunnel imposes on the train to keep itin the tunnel. In particular, there are no frictional forces, like air resistance, and thetrain does not have an engine. Release the train and assume that it does not melt asit passes through the centre of the Earth.VMOGNWhat 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 itO, is at theorigin,(0, 0, 0).˝Then rotate our coordinate system about the origin so that the origin,VandMalllie in thexz–plane.˝Then rotate our coordinate system about they–axis so thatVandMhave the samez–coordinateZ•0. So the coordinates ofVandMare(˘?R2´Z2, 0 ,Z). Let’s sup-