Potential energy of a particle at position x is given by `U = (x^2 - 4x) J`. Which of the following is equilibrium position of the particle?

Options

A. x=0
B. x=5m
C. x=2m
D. x=4m

Answer

To find the equilibrium position of a particle given its potential energy function \( U = x^2 - 4x \), we need to follow these steps:

### Step 1: Understand the relationship between force and potential energy
The force \( F \) acting on a particle is related to the potential energy \( U \) by the equation:
\[ F = -\frac{dU}{dx} \]
At equilibrium, the net force acting on the particle is zero, which means:
\[ F = 0 \implies -\frac{dU}{dx} = 0 \implies \frac{dU}{dx} = 0 \]

### Step 2: Differentiate the potential energy function
Given the potential energy function:
\[ U = x^2 - 4x \]
we need to differentiate \( U \) with respect to \( x \):
\[ \frac{dU}{dx} = \frac{d}{dx}(x^2) - \frac{d}{dx}(4x) \]
Calculating the derivatives:
\[ \frac{dU}{dx} = 2x - 4 \]

### Step 3: Set the derivative equal to zero
To find the equilibrium position, we set the derivative equal to zero:
\[ 2x - 4 = 0 \]

### Step 4: Solve for \( x \)
Now, we solve for \( x \):
\[ 2x = 4 \implies x = \frac{4}{2} = 2 \]

### Conclusion
The equilibrium position of the particle is at:
\[ x = 2 \text{ meters} \]