The potential energy of a particle under a conservative force is given by `U(x)=(x^(2)-3x)J`. The equilibrium position of the particle is at x m. The value of 10 x will be

Answer

To find the equilibrium position of the particle given the potential energy function \( U(x) = x^2 - 3x \), we will follow these steps:

### Step 1: Understand the Equilibrium Condition
In a conservative force field, the equilibrium position occurs where the net force acting on the particle is zero. The force \( F \) can be derived from the potential energy \( U \) using the relation:
\[ F = -\frac{dU}{dx} \]
At equilibrium, \( F = 0 \). Therefore, we set:
\[ -\frac{dU}{dx} = 0 \]

### Step 2: Differentiate the Potential Energy Function
Now we need to differentiate the potential energy function \( U(x) \):
\[ U(x) = x^2 - 3x \]
Calculating the derivative:
\[ \frac{dU}{dx} = \frac{d}{dx}(x^2) - \frac{d}{dx}(3x) = 2x - 3 \]

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

### Step 4: Solve for \( x \)
Solving the equation:
\[ 2x = 3 \\ x = \frac{3}{2} = 1.5 \text{ m} \]

### Step 5: Calculate \( 10x \)
Now, we need to find the value of \( 10x \):
\[ 10x = 10 \times 1.5 = 15 \]

### Final Answer
Thus, the value of \( 10x \) is:
\[ \boxed{15} \]

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