Consider the earth as a uniform sphere if mass `M` and radius `R`. Imagine a straight smooth tunnel made through the earth which connects any two points on its surface. Show that the motion of a particle of mass m along this tunnel under the action of gravitation would be simple harmonic. Hence, determine the time that a particle would take to go from one end to the other through the tunnel.

Explanation

To solve the problem, we need to show that the motion of a particle of mass \( m \) along a straight tunnel through the Earth is simple harmonic motion (SHM) and then determine the time taken for the particle to travel from one end of the tunnel to the other.

### Step 1: Understanding the gravitational force inside the Earth

When a mass \( m \) is at a distance \( r \) from the center of the Earth, only the mass of the Earth that is enclosed within that radius contributes to the gravitational force acting on \( m \). The mass \( M' \) of the Earth within radius \( r \) can be expressed as:

\[ M' = M \left( \frac{r^3}{R^3} \right) \]

where \( M \) is the total mass of the Earth and \( R \) is the radius of the Earth.

### Step 2: Applying Newton's law of gravitation

The gravitational force \( F \) acting on the mass \( m \) at distance \( r \) from the center of the Earth is given by:

\[ F = \frac{G M' m}{r^2} = \frac{G \left(M \frac{r^3}{R^3}\right) m}{r^2} = \frac{G M m r}{R^3} \]

### Step 3: Expressing the force in terms of displacement

If we let \( x \) be the displacement from the center of the Earth, where \( x = r \), then the force can be rewritten as:

\[ F = \frac{G M m}{R^3} x \]

Since the force is directed towards the center, we can express it as:

\[ F = -\frac{G M m}{R^3} x \]

### Step 4: Identifying the motion as simple harmonic motion

The equation \( F = -k x \) (where \( k = \frac{G M m}{R^3} \)) is characteristic of simple harmonic motion. Thus, we can identify that the motion of the particle is indeed SHM.

### Step 5: Determining the angular frequency

From the SHM relation, we know that:

\[ F = m a = -k x \]

This gives us the angular frequency \( \omega \):

\[ \omega^2 = \frac{G M}{R^3} \]

### Step 6: Finding the time period of the oscillation

The time period \( T \) of the oscillation is given by:

\[ T = 2\pi \sqrt{\frac{R^3}{G M}} \]

### Step 7: Calculating the time taken to travel from one end to the other

The time taken to travel from one end of the tunnel to the other is half the time period:

\[ \text{Time taken} = \frac{T}{2} = \pi \sqrt{\frac{R^3}{G M}} \]

### Final Answer

Thus, the time taken for the particle to go from one end of the tunnel to the other is:

\[ \text{Time taken} = \pi \sqrt{\frac{R^3}{G M}} \]