The potential energy function (in joules) of a particle in a region of space is given as\[U = (2x^2 + 3y^3 + 2z).\]Here \( x \), \( y \), and \( z \) are in meters. The magnitude of the \( x \)-component of force (in newtons) acting on the particle at point \( P (1, 2, 3) \) m is:
- 2
- 6
- 4
- 8
The Correct Option is C
Approach Solution - 1
To find the magnitude of the \( x \)-component of force acting on the particle at point \( P(1, 2, 3) \), we can use the concept of potential energy and force relationship. The force components are given by the negative gradient of the potential energy function.
The potential energy function is given as:
\(U = (2x^2 + 3y^3 + 2z)\)
The force acting on the particle is given by:
\(\vec{F} = -\nabla U\)
Here, \(\nabla U\) represents the gradient of the potential energy function \(U\). The gradient in three dimensions is:
\(\nabla U = \left( \frac{\partial U}{\partial x}, \frac{\partial U}{\partial y}, \frac{\partial U}{\partial z} \right)\)
Let's calculate each partial derivative:
Partial derivative with respect to \(x\):
\(\frac{\partial U}{\partial x} = \frac{\partial}{\partial x}(2x^2) = 4x\)
Partial derivative with respect to \(y\):
\(\frac{\partial U}{\partial y} = \frac{\partial}{\partial y}(3y^3) = 9y^2\)
Partial derivative with respect to \(z\):
\(\frac{\partial U}{\partial z} = \frac{\partial}{\partial z}(2z) = 2\)
Substituting these partial derivatives, the force is:
\(\vec{F} = - (4x, 9y^2, 2)\)
To find the \(x\)-component of the force at \(P(1, 2, 3)\), substitute \(x = 1\), \(y = 2\), and \(z = 3\):
\(F_x = -4x = -4(1) = -4 \, \text{N}\)
The magnitude of the \(x\)-component of force is:
\(|F_x| = 4 \, \text{N}\)
Thus, the magnitude of the \(x\)-component of the force acting on the particle at point \(P(1, 2, 3)\) is 4 N.
Approach Solution -2
Step 1: Given Potential Energy Function
\[ U = 2x^2 + 3y^3 + 2z \]
Step 2: Calculate the \(x\)-Component of Force
The force in the \(x\)-direction is given by \(F_x = -\frac{\partial U}{\partial x}\). Differentiating \(U\) with respect to \(x\):
\[ F_x = -\frac{\partial}{\partial x}(2x^2) = -4x \]
Step 3: Evaluate \(F_x\) at \(x = 1\)
Substitute \(x = 1\):
\[ F_x = -4 \times 1 = -4 \]
The magnitude of \(F_x\) is 4 N.
So, the correct answer is: 4 N
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