Serway & Jewett — Oscillatory Motion: Problem 50

Review. Why is the following situation impossible? You are in the high-speed package delivery business. Your competitor in the next building gains the right-of-way to build an evacuated tunnel just above the ground all the way around the Earth. By firing packages into this tunnel at just the right speed, your competitor is able to send the packages into orbit around the Earth in this tunnel so that they arrive on the exact opposite side of the Earth in a very short time interval. You come up with a competing idea. Figuring that the distance through the Earth is shorter than the distance around the Earth, you obtain permits to build an evacuated tunnel through the center of the Earth (Fig. P15.50). By simply dropping packages into this tunnel, they fall downward and arrive at the other end of your tunnel, which is in a building right next to the other end of your competitor’s tunnel. Because your packages arrive on the other side of the Earth in a shorter time interval, you win the competition and your business flourishes. Note: An object at a distance \(r\) from the center of the Earth is pulled toward the center of the Earth only by the mass within the sphere of radius \(r\) (the reddish region in Fig. P15.50). Assume the Earth has uniform density.