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

### Step 1: Understand the concept of equilibrium
In mechanics, a particle is in equilibrium when the net force acting on it is zero. The net force can be derived from the potential energy function.

### Step 2: Relate force to potential energy
The force \( F \) acting on the particle is related to the potential energy \( U \) by the equation:
\[ F = -\frac{dU}{dx} \]
This means we need to find the derivative of the potential energy function with respect to \( x \).

### Step 3: Differentiate the potential energy function
Given:
\[ U(x) = x^2 - 5x \]
We differentiate \( U \) with respect to \( x \):
\[ \frac{dU}{dx} = \frac{d}{dx}(x^2) - \frac{d}{dx}(5x) = 2x - 5 \]

### Step 4: Find the expression for force
Using the relationship from Step 2:
\[ F = -\frac{dU}{dx} = -(2x - 5) = -2x + 5 \]

### Step 5: Set the force to zero for equilibrium
For equilibrium, we set the force \( F \) to zero:
\[ -2x + 5 = 0 \]

### Step 6: Solve for \( x \)
Rearranging the equation gives:
\[ 2x = 5 \implies x = \frac{5}{2} = 2.5 \]

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