- #1
mbrmbrg
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[SOLVED] given potential: find F
Find the force for the following potential energy function:
[tex]V=ce^{-(\alpha x+\beta y+\gamma z)}[/tex]
[tex]\mathbf{F}=-\nabla V[/tex]
[tex]F_x=- \frac{\partial V}{\partial x}[/tex]
[tex]F_y=- \frac{\partial V}{\partial y}[/tex]
[tex]F_z=- \frac{\partial V}{\partial z}[/tex]
By the chain rule:
[tex]F_x=-\frac{\partial V}{\partial x}=-[-(\alpha x+ \beta y + \gamma z)ce^{-(\alpha x+\beta y+\gamma z)}(-\alpha)][/tex]
[tex]F_y=-\frac{\partial V}{\partial y}=-[-(\alpha x+ \beta y + \gamma z)ce^{-(\alpha x+\beta y+\gamma z)}(-\beta)][/tex]
[tex]F_z=-\frac{\partial V}{\partial z}=-[-(\alpha x+ \beta y + \gamma z)ce^{-(\alpha x+\beta y+\gamma z)}(-\gamma)][/tex]
Additionally,
[tex]\mathbf{F}=F_x+F_y+F_z[/tex]
So add it up and factor out the common factors and get
[tex]\mathbf{F}=-(\alpha x+\beta y+\gamma z)ce^{-(\alpha x+\beta y +\gamma z)}(\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k})[/tex]
And that would be my final answer, except that the back of the book says that
[tex]\mathbf{F}=ce^{-(\alpha x+\beta y +\gamma z)}(\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k})[/tex]
Can you help me find my error?
Thanks!
Homework Statement
Find the force for the following potential energy function:
[tex]V=ce^{-(\alpha x+\beta y+\gamma z)}[/tex]
Homework Equations
[tex]\mathbf{F}=-\nabla V[/tex]
[tex]F_x=- \frac{\partial V}{\partial x}[/tex]
[tex]F_y=- \frac{\partial V}{\partial y}[/tex]
[tex]F_z=- \frac{\partial V}{\partial z}[/tex]
The Attempt at a Solution
By the chain rule:
[tex]F_x=-\frac{\partial V}{\partial x}=-[-(\alpha x+ \beta y + \gamma z)ce^{-(\alpha x+\beta y+\gamma z)}(-\alpha)][/tex]
[tex]F_y=-\frac{\partial V}{\partial y}=-[-(\alpha x+ \beta y + \gamma z)ce^{-(\alpha x+\beta y+\gamma z)}(-\beta)][/tex]
[tex]F_z=-\frac{\partial V}{\partial z}=-[-(\alpha x+ \beta y + \gamma z)ce^{-(\alpha x+\beta y+\gamma z)}(-\gamma)][/tex]
Additionally,
[tex]\mathbf{F}=F_x+F_y+F_z[/tex]
So add it up and factor out the common factors and get
[tex]\mathbf{F}=-(\alpha x+\beta y+\gamma z)ce^{-(\alpha x+\beta y +\gamma z)}(\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k})[/tex]
And that would be my final answer, except that the back of the book says that
[tex]\mathbf{F}=ce^{-(\alpha x+\beta y +\gamma z)}(\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k})[/tex]
Can you help me find my error?
Thanks!