Transportation to the other side of Earth It has been suggested that an efficient wa...

Transportation to the other side of Earth It has been suggested that an efficient way to travel from the United States to Australia would be to dig one or more holes through Earth and fall through one of the holes in a capsule. The gravitational force that Earth exerts on you and on the capsule when the capsule is at a distance r (r < R of Earth) equals that of the mass M inside a sphere of radius r with its center at Earth’s center. At Earth’s center, the force is zero. At some other distance r from the center of Earth, the force would be

where Minside r equals the volume of a sphere of radius r times the density of Earth. Substituting for Minside r, you find that the force that Earth exerts on you and on the capsule is proportional to r and provides a restoring force. You start at rest at Earth’s surface and accelerate as you get closer to the center. At the center you are moving at maximum speed and Earth exerts no force on you. After you pass the center, Earth starts pulling back on you, causing your speed to decrease. With no friction or air resistance, you should stop on the other side of Earth. If you can find an expression for the restoring force that Earth exerts on you, you can substitute that in place of k in the expression for period T = 2π(m/k)^1/2. The travel time will be half the period.

Which expression below represents the mass m of Earth inside a sphere of radius r smaller than the radius R of Earth? Note that r is the density of Earth, assumed uniform.