For stable equilibrium, which of the following is/are true ?

Options

A. `(dU)/(dx)=0` and `(d^(2)U)/(dx^(2)) gt 0`
B. `(dU)/(dx)=0` only
C. `(dU)/(dx)=0` and `(d^(2)U)/(dx^(2)) lt 0`
D. None of these

Answer

To determine the conditions for stable equilibrium, we can break down the problem into a series of logical steps:

### Step-by-Step Solution:

1. **Understanding Stable Equilibrium:**
- Stable equilibrium occurs when a system returns to its original position after being slightly disturbed. This implies that the net force acting on the body is zero.
- Mathematically, this can be expressed as:
\[ F_{\text{net}} = 0 \]

2. **Forces Acting on the Body:**
- Consider a body placed on a parabolic path. The forces acting on the body include:
- The gravitational force acting downward, \( mg \).
- The normal force acting upward, \( N \).
- For stable equilibrium, these forces must balance:
\[ N = mg \]

3. **Restoring Force Concept:**
- When the body is displaced from its equilibrium position, a restoring force acts to bring it back to equilibrium. This restoring force can be defined as:
\[ F = -\frac{du}{dx} \]
- For stable equilibrium, the condition for the net force to be zero leads us to:
\[ \frac{du}{dx} = 0 \]

4. **Second Derivative Condition:**
- To ensure that the equilibrium is stable, we need to check the concavity of the potential energy function \( u \). This is done by examining the second derivative:
\[ \frac{d^2u}{dx^2} < 0 \]
- This condition indicates that the potential energy function is at a minimum, which is characteristic of stable equilibrium.

5. **Conclusion:**
- Therefore, for stable equilibrium, the following conditions must be satisfied:
- The first derivative of potential energy with respect to position must be zero:
\[ \frac{du}{dx} = 0 \]
- The second derivative of potential energy with respect to position must be less than zero:
\[ \frac{d^2u}{dx^2} < 0 \]