Potential energy of a particle is related to x coordinate by equation `x^2 - 2x` . Particle will be in stable equilibrium at :

Options

A. `x = 0.5`
B. `x = 1`
C. `x = 2`
D. `x = 4`

Explanation

To determine the position of stable equilibrium for a particle whose potential energy (U) is given by the equation \( U(x) = x^2 - 2x \), we will follow these steps:

### Step 1: Find the Force
The force acting on the particle can be derived from the potential energy using the relationship:
\[ F = -\frac{dU}{dx} \]
First, we need to differentiate the potential energy function \( U(x) \).

### Step 2: Differentiate the Potential Energy
We differentiate \( U(x) = x^2 - 2x \):
\[ \frac{dU}{dx} = 2x - 2 \]
Now, substituting this into the force equation:
\[ F = -\frac{dU}{dx} = -(2x - 2) = -2x + 2 \]

### Step 3: Set the Force to Zero
To find the equilibrium position, we set the force equal to zero:
\[ -2x + 2 = 0 \]
Solving for \( x \):
\[ 2x = 2 \implies x = 1 \]

### Step 4: Determine Stability of the Equilibrium
To determine whether this equilibrium is stable, we need to check the second derivative of the potential energy:
\[ \frac{d^2U}{dx^2} = \frac{d}{dx}(2x - 2) = 2 \]
Since \( \frac{d^2U}{dx^2} = 2 \) is positive, this indicates that the equilibrium at \( x = 1 \) is stable.

### Conclusion
The particle will be in stable equilibrium at \( x = 1 \).

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