Potential energy of a particle at position x is given by `U=x^(2)-5x`. Which of the following is equilibrium position of the particle?

Options

A. `x=0`
B. `x=5`
C. `x=2.5`
D. `x=7.5`

Answer

To find the equilibrium position of a particle given its potential energy function \( U(x) = x^2 - 5x \), we follow these steps:

### Step 1: Understand the concept of equilibrium position
The equilibrium position of a particle occurs where the potential energy is at a minimum. This can be determined by finding the critical points of the potential energy function.

### Step 2: Differentiate the potential energy function
To find the critical points, we need to differentiate the potential energy function with respect to \( x \):
\[ \frac{dU}{dx} = \frac{d}{dx}(x^2 - 5x) \]
Calculating the derivative:
\[ \frac{dU}{dx} = 2x - 5 \]

### Step 3: Set the derivative to zero
To find the critical points, we set the derivative equal to zero:
\[ 2x - 5 = 0 \]
Solving for \( x \):
\[ 2x = 5 \\ x = \frac{5}{2} = 2.5 \]

### Step 4: Verify if it is a minimum
To confirm that this point is indeed a minimum, we need to check the second derivative of the potential energy function:
\[ \frac{d^2U}{dx^2} = \frac{d}{dx}(2x - 5) \]
Calculating the second derivative:
\[ \frac{d^2U}{dx^2} = 2 \]
Since \( \frac{d^2U}{dx^2} = 2 \) is greater than 0, this indicates that the potential energy function has a minimum at \( x = 2.5 \).

### Conclusion
Thus, the equilibrium position of the particle is at \( x = 2.5 \).

### Final Answer
The equilibrium position of the particle is \( x = 2.5 \).

---