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The "Gravity Train" refers to the following curious yet impractical experiment, suggested by Robert Hooke (1635- 1703) in a letter to Isaac Newton. The idea was used in the 2012 movie Total Recall. Imagine that the Earth is a perfect sphere of radius R and that it has a constant mass density p. Pick any two points V and M in the same latitude (z coordinate; see Figure 1 below). Bore a tunnel straight through the Earth from V to M. 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. There are no frictional forces and the train does not have an engine. Release the train from point V and assume that it does not melt as it passes through the center of the earth. M (-√R²Z2,0, Z), NA G (0,0,0) (√R²-Z², 0, Z) Recall Newton's second law of motion and Newton's law of gravity. mā = G + N Mm G=-G -T ||7||³ -x (1) (2) Here, M is the mass of earth, m is the mass of the train, G is universal gravitational constant, and 7 is the vector from O to the train. Solve for the following. 1. What is the equation of the motion of the train? Hint: Recall that the solution to the ordinary differential equation x" (t)+ ax(t) = 0 is x(t) = C₁ cos(√at) + C₂ sin(√āt) 2. How long does it take for the Gravity Train to go from V to M? What is its relationship with |7|?

The "Gravity Train" refers to the following curious yet impractical experiment, suggested by Robert Hooke (1635- 1703) in a letter to Isaac Newton. The idea was used in the 2012 movie Total Recall. Imagine that the Earth is a perfect sphere of radius R and that it has a constant mass density p. Pick any two points V and M in the same latitude (z coordinate; see Figure 1 below). Bore a tunnel straight through the Earth from V to M. 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. There are no frictional forces and the train does not have an engine. Release the train from point V and assume that it does not melt as it passes through the center of the earth. M (-√R²Z2,0, Z), NA G (0,0,0) (√R²-Z², 0, Z) Recall Newton's second law of motion and Newton's law of gravity. mā = G + N Mm G=-G -T ||7||³ -x (1) (2) Here, M is the mass of earth, m is the mass of the train, G is universal gravitational constant, and 7 is the vector from O to the train. Solve for the following. 1. What is the equation of the motion of the train? Hint: Recall that the solution to the ordinary differential equation x" (t)+ ax(t) = 0 is x(t) = C₁ cos(√at) + C₂ sin(√āt) 2. How long does it take for the Gravity Train to go from V to M? What is its relationship with |7|?

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The "Gravity Train" refers to the following curious yet impractical experiment, suggested by Robert Hooke (1635- 1703) in a letter to Isaac Newton. The idea was used in the 2012 movie Total Recall. Imagine that the Earth is a perfect sphere of radius R and that it has a constant mass density p. Pick any two points V and M in the same latitude (z coordinate; see Figure 1 below). Bore a tunnel straight through the Earth from V to M. 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. There are no frictional forces and the train does not have an engine. Release the train from point V and assume that it does not melt as it passes through the center of the earth. M (-√R²Z², 0, Z), NA G (0,0,0) V (√R²-Z², 0, Z) Recall Newton's second law of motion and Newton's law of gravity. mā= G + N Mm G=-G- G||F7||37 (1) (2) Here, M is the mass of earth, m is the mass of the train, G is universal gravitational constant, and is the vector from O to the train. Solve for the following. 1. What is the equation of the motion of the train? Hint: Recall that the solution to the ordinary differential equation x"(t) + ax(t) = 0 is x(t) = C₁ cos(√at) + C₂ sin(√at) 2. How long does it take for the Gravity Train to go from V to M? What is its relationship with 7*?
Transcribed Image Text:The "Gravity Train" refers to the following curious yet impractical experiment, suggested by Robert Hooke (1635- 1703) in a letter to Isaac Newton. The idea was used in the 2012 movie Total Recall. Imagine that the Earth is a perfect sphere of radius R and that it has a constant mass density p. Pick any two points V and M in the same latitude (z coordinate; see Figure 1 below). Bore a tunnel straight through the Earth from V to M. 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. There are no frictional forces and the train does not have an engine. Release the train from point V and assume that it does not melt as it passes through the center of the earth. M (-√R²Z², 0, Z), NA G (0,0,0) V (√R²-Z², 0, Z) Recall Newton's second law of motion and Newton's law of gravity. mā= G + N Mm G=-G- G||F7||37 (1) (2) Here, M is the mass of earth, m is the mass of the train, G is universal gravitational constant, and is the vector from O to the train. Solve for the following. 1. What is the equation of the motion of the train? Hint: Recall that the solution to the ordinary differential equation x"(t) + ax(t) = 0 is x(t) = C₁ cos(√at) + C₂ sin(√at) 2. How long does it take for the Gravity Train to go from V to M? What is its relationship with 7*?
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