Chapter 13, Problem 14A

If you stepped into a hole that passed completely through Earth, you'd oscillate down and up. How long would a one-way trip take? How long would a round trip take?

To determine

Time taken for a one-way trip. Also, the timefor a round trip.

Answer to Problem 14A

For a one-way trip, $42.3\min$ would take. For a round trip, $84.6\min$ would take.

Explanation of Solution

Introduction: For a Simple Harmonic Motion, time period for the oscillation would be

$T=2\pi \sqrt{\frac{R}{g}}$

Let’s say mass $m$ travel inside the tunnel into the Earth. At the center of Earth, gravity is zero. The particle will move towards the center due to gravity. After reaching to center, it will move across due to the velocity acquired. But, moving away from the center will pull the mass and thus deaccelerate it, which leads to its zero speed after reaching the other end of the tunnel.

Calculation: Thus, Time period of oscillation $T=2\pi \sqrt{\frac{R}{g}}$

Where, $R=6.4\times {10}^{6}m$ and $g=9.8m/s^2$

Inserting the values, we will get

  $T=84.6\min$

Time duration to cover a one-way travel

  $t=84.6/2=42.3\min$

Conclusion: Thus, a one-way trip across the tunnel into the Earth will take $42.3\min$ and a round trip would take $84.6\min$ .