Assume that as narrow tunnel is dug between two diametrically opposite points of the earth. Treat the earth as a solid sphere of uniform density. Show that if a particle is released in this tunnel, it will execute a simple harmonic motion. Calculate the time period of this motion.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Setup
We have a narrow tunnel dug between two diametrically opposite points of the Earth. We will treat the Earth as a solid sphere with uniform density.

### Step 2: Define the Variables
Let:
- \( R \) = radius of the Earth
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