The potential energy (in joules ) function of a particle in a region of space is given as:
`U=(2x^(2)+3y^(2)+2x)`
Here x,y and z are in metres. Find the maginitude of x compenent of force ( in newton) acting on the particle at point P ( 1m, 2m, 3m).

Options

A. 2
B. 3
C. 0
D. 4

Answer

To find the magnitude of the x component of the force acting on the particle at point P (1m, 2m, 3m), we can follow these steps:

### Step 1: Understand the relationship between potential energy and force
The force acting on a particle in a potential energy field can be found using the formula:
\[ F_x = -\frac{dU}{dx} \]
where \( U \) is the potential energy function and \( F_x \) is the x-component of the force.

### Step 2: Write down the potential energy function
The potential energy function given is:
\[ U = 2x^2 + 3y^2 + 2z \]

### Step 3: Differentiate the potential energy function with respect to x
To find the x-component of the force, we need to differentiate \( U \) with respect to \( x \):
\[ \frac{dU}{dx} = \frac{d}{dx}(2x^2 + 3y^2 + 2z) \]
Since \( y \) and \( z \) are constants with respect to \( x \), their derivatives will be zero:
\[ \frac{dU}{dx} = 4x + 0 + 0 = 4x \]

### Step 4: Substitute the value of x at point P
Now, substitute \( x = 1 \) m into the derivative:
\[ \frac{dU}{dx} = 4(1) = 4 \]

### Step 5: Calculate the x component of the force
Now, we can find the x-component of the force:
\[ F_x = -\frac{dU}{dx} = -4 \]

### Step 6: Find the magnitude of the force
The magnitude of the force is:
\[ |F_x| = |-4| = 4 \text{ N} \]

### Final Answer
The magnitude of the x component of the force acting on the particle at point P (1m, 2m, 3m) is:
\[ \text{4 N} \]

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