Asume that as narrow tunnel is dug between two diametricaly opposite points of the earth. Treat the earth as a solid sphere of uniform density. Show that ilf a particle is relesed in this tunel, ilt wil execute a simple harmonic motion. Calculate the time period of this motion.

Explanation

To solve the problem of a particle released in a tunnel dug through the Earth, we will follow these steps:

### Step 1: Understanding the Setup
We assume the Earth is a solid sphere of uniform density. If a particle is released in a tunnel that connects two diametrically opposite points, it will experience gravitational force directed towards the center of the Earth.

### Step 2: Gravitational Force Inside the Earth
According to Gauss's law for gravity, the gravitational force acting on a particle inside a uniform spherical shell is only due to the mass of the sphere that is at a radius less than the distance from the center to the particle. If the particle is at a distance \( x \) from the center, the gravitational force \( F \) acting on it can be expressed as:

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