Question

Need help with a potential energy question!

Answer 1

find a scalar potential energy function \(U(x,y,z)\) so that \[\nabla U = \left\langle \frac{-kx}{\left( x^2+y^2+z^2 \right)^{3/2}}\, ,\, \frac{-ky}{\left( x^2+y^2+z^2 \right)^{3/2}}\, ,\, \frac{-kx}{\left( x^2+y^2+z^2 \right)^{3/2}} \right\rangle\]

Answer 2

so I'm not sure how I could go about this without a huge amount of algebra. First of all I was wondering if I am allowed to factor the denominator out to show that the vector field is conservative like this...\[\nabla U = \left\langle \frac{-kx}{\left( x^2+y^2+z^2 \right)^{3/2}}\, ,\, \frac{-ky}{\left( x^2+y^2+z^2 \right)^{3/2}}\, ,\, \frac{-kx}{\left( x^2+y^2+z^2 \right)^{3/2}} \right\rangle\]and then\[\frac{\partial U_1}{\partial y}=0=\frac{\partial U_2}{\partial x}\]\[\frac{\partial U_1}{\partial z}=0=\frac{\partial U_3}{\partial x}\]\[\frac{\partial U_2}{\partial z}=0=\frac{\partial U_3}{\partial y}\]so that \(\text{curl}\,\nabla U =0\)

Answer 3

oops, I mean to say "like this \[\nabla U = \frac{-k}{\left( x^2+y^2+z^2 \right)^{3/2}}\left\langle x\,,\,y\,,\,z\right\rangle\] and then..."

Answer 4

Are you sure the last numerator should not be -kz instead of -kx?

In that case you have an spherical inverse square law.

Potential should be

\( U= \LARGE \frac{k}{r}\large+cst= \Large\frac{k}{\left( x^2+y^2+z^2 \right)^{1/2}}\large+cst \)