In this problem you should consider the earth as spherical with radius RE=6.374. 106 m and free fall acceleration g=9.8 m/s2 at the surface. For simplicity you can neglect the rotation of the earth when solving this problem. Imagine that an evacuated tube was built all the way around the earth at the equator in which capsules could travel at high speed.
(a) At what speed v relative to the surface of the earth would passengers experience weightlessness and what would be a practical advantage of operating at this particular speed?
Now imagine an equally outrageous transport system consisting of an evacuated tube passing straight through the center of the earth. You are to consider the motion of a capsule through this tube. At a distance x from the center of the earth it is only the mass within a sphere of radius x that contributes to the force of gravity acting on the capsule. This is a consequence of the 1/r 2 form of the force of gravity.
(b) Show this implies a force of gravity of the form F = -mg(x / RE ) acts on the capsule with mass m at a distance x from the center of the earth.
(c) Describe the type of capsule motion that this force implies and calculate the transit time (neglecting friction and any propulsion) from one side of the earth to the other. (Hint: Think harmonic oscillatory motion).
(d) Contrast the passenger experience in these two different transport systems including transit time and forces experienced.