Two concentric spherical shells have masses `m_(1)`, `m_(2)` and radit `R_(1)`,`R_(2) (R_(1) lt R_(2))`. Calculate the force by this system on a particle of mass `m`, if it is placed at a distance `((R_(1) + R_(2)))/(2)` from the centre.

Explanation

To solve the problem, we need to calculate the gravitational force exerted by two concentric spherical shells on a particle of mass \( m \) placed at a distance \( \frac{R_1 + R_2}{2} \) from the center. Here are the steps to find the solution:

### Step 1: Understand the Setup
We have two spherical shells:
- Inner shell with mass \( m_1 \) and radius \( R_1 \)
- Outer shell with mass \( m_2 \) and radius \( R_2 \) (where \( R_1 < R_2 \))

The particle of mass \( m \) is located at a distance \( r = \frac{R_1 + R_2}{2} \) from the center of the shells.

### Step 2: Determine the Position of the Particle
Since \( R_1 < R_2 \), the distance \( r = \frac{R_1 + R_2}{2} \) is greater than \( R_1 \) but less than \( R_2 \):
- This means the particle is outside the inner shell but inside the outer shell.

### Step 3: Gravitational Force Inside a Shell
According to the shell theorem:
- The gravitational force inside a spherical shell is zero.
- Therefore, the gravitational force due to the outer shell (mass \( m_2 \)) on the particle is zero since the particle is inside the outer shell.

### Step 4: Calculate the Gravitational Field Due to the Inner Shell
The gravitational field \( E_1 \) due to the inner shell at the position of the particle is given by the formula:
\[ E_1 = -\frac{G m_1}{r^2} \]
where \( r = \frac{R_1 + R_2}{2} \).

### Step 5: Calculate the Gravitational Field at the Position of the Particle
Substituting \( r \) into the equation:
\[ E_1 = -\frac{G m_1}{\left(\frac{R_1 + R_2}{2}\right)^2} \]
This simplifies to:
\[ E_1 = -\frac{4G m_1}{(R_1 + R_2)^2} \]

### Step 6: Calculate the Gravitational Force on the Particle
The gravitational force \( F \) on the particle of mass \( m \) due to the gravitational field \( E_1 \) is given by:
\[ F = m \cdot E_1 \]
Substituting \( E_1 \):
\[ F = m \left(-\frac{4G m_1}{(R_1 + R_2)^2}\right) \]
Thus, the force exerted on the particle is:
\[ F = -\frac{4G m_1 m}{(R_1 + R_2)^2} \]

### Final Answer
The force by the system on the particle is:
\[ F = -\frac{4G m_1 m}{(R_1 + R_2)^2} \]

---