Journey through the center of the Earth
Your initial acceleration would be the surface acceleration of gravity
but the acceleration would be progressively smaller as you approached the center. Your weight would be zero as you flew through the center of the Earth. For our hypothetical journey we will assume the Earth to be of uniform density and neglect air friction and the high temperature of this trip.
Taking positive r as outward from the center of the Earth:
This is the same form as Hooke's Law for a mass on a spring. It would cause the trans-Earth traveler to oscillate back and forth through the center of the Earth like a mass bobbing up and down on a spring. The angular frequency and period for this oscillation are
For this case the period of oscillation is
The traveler accelerates toward the center of the Earth and is momentarily weightless when passing through the geometric center at about 7900 m/s or almost 17,700 miles/hr. The traveler would pop up on the opposite side of the Earth after a little more than 42 minutes. But unless he or she grabs something to hold on, they will fall back for a return journey and continue to oscillate with a round-trip time of 84.5 minutes.
The period of the orbit is calculated from
which is the same as the period of the oscillating body.