Imagine a straight tunnel through the centre of a uniform Mars (g = 3.7 m/s² at the surface, R =…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
Imagine a straight tunnel through the centre of a uniform Mars ( at the surface, ). A stone is dropped in at one end.
- Show that its motion is simple harmonic, and find the time it takes to reach the other end.
- How fast does it pass the centre?
Need a reminder? The key ideaShow
A uniform spherical shell pulls an outside body as if its mass were at its centre. Inside it, the field is zero and the potential is the same everywhere, . Inside a uniform solid sphere, .
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The force. At distance from the centre: , in proportion to the displacement and toward the centre: SHM.
Step 2. Period. , so ; one way is half of that, .
Step 3. (b) Largest speed .
Answer: (a) About minutes. (b) .
Check. The one-way time equals half the period of a satellite skimming the surface: both have .
Watch out. is not constant in the tunnel, so the equations of motion with steady acceleration do not apply.
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