Question

An object moving in the xy-plane is acted on by a conservative force described by the potential-energy function U ( x,y ) = a(1/x^2 + 1/y^2) , where a is a positive constant.

Derive an expression for the force F(vector) expressed in terms of the unit vectors i and j .

Express your answer in terms of the given quantities

Answer 1

\[U(x,y)=a\left(x^{-2}+y^{-2}\right)\quad,\quad a>0\]\[\vec{F}=-\nabla U(x,y)=-\left(\frac{\partial U}{\partial x}\vec{i}+\frac{\partial U}{\partial y}\vec{j}\right)=-\left(-2ax^{-3}\vec{i}-2ay^{-3}\vec{j}\right)=\]\[=2a\left(x^{-3}\vec{i}+y^{-3}\vec{j}\right)=2a\left(\frac{1}{x^{3}}\vec{i}+\frac{1}{y^{3}}\vec{j}\right)\]