If the P.E. of a system of two atoms ( diatomic molecule) is given by `U(x) = -2 + 3( x-x_(0))^(2)`, where `x_(0)` is equilibium separation than

Options

A. The restoring force is`-6 ( x-x_(0))`
B. Acceleration of the one action with respect to other is other is `- ( 6 ( x-x_(0)))/(mu)[mu rarr` reduced of system ]
C. The restoring force is `( 3)/(2) ( x-x_(0))`
D. The acceleratoin of atom is ` - ( 3)/(2mu) ( x-x_(0))`

Answer

To solve the problem, we need to analyze the given potential energy function and derive the force acting on the system of two atoms. Let's go through the steps systematically.

### Step-by-Step Solution:

1. **Identify the Potential Energy Function**:
The potential energy \( U(x) \) of the system is given by:
\[ U(x) = -2 + 3(x - x_0)^2 \]
where \( x_0 \) is the equilibrium separation.

2. **Calculate the Force**:
The force \( F \) acting on the system is related to the potential energy by the formula:
\[ F = -\frac{dU}{dx} \]
We need to differentiate \( U(x) \) with respect to \( x \).

3. **Differentiate the Potential Energy**:
Differentiate \( U(x) \):
\[ \frac{dU}{dx} = \frac{d}{dx}(-2) + \frac{d}{dx}[3(x - x_0)^2] \]
The derivative of \(-2\) is \(0\), and using the chain rule for the second term:
\[ \frac{d}{dx}[3(x - x_0)^2] = 3 \cdot 2(x - x_0) \cdot \frac{d}{dx}(x - x_0) = 6(x - x_0) \]
Therefore,
\[ \frac{dU}{dx} = 6(x - x_0) \]

4. **Substitute into the Force Equation**:
Now substituting back into the force equation:
\[ F = -\frac{dU}{dx} = -6(x - x_0) \]

5. **Determine the Acceleration**:
The acceleration \( a \) of the system can be found using Newton's second law, where \( F = ma \). Here, \( m \) is the reduced mass \( \mu \) of the system:
\[ a = \frac{F}{\mu} = \frac{-6(x - x_0)}{\mu} \]

### Conclusion:
The force acting on the system is given by:
\[ F = -6(x - x_0) \]
And the acceleration of the system is:
\[ a = \frac{-6(x - x_0)}{\mu} \]