Question

The potential energy of a certain particle is given by $U=\frac{1}{2}(x^2-y^2)$. The force on it is:

• A. $-x\hat{i}+y\hat{j}$
• B. $-x\hat{i}+z\hat{k}$
• C. $-x\hat{i}-y\hat{j}$
• D. $x\hat{i}-y\hat{j}$

Answer 1

The correct option is A $-x\hat{i}+y\hat{j}$

$F_x=\frac{-\partial U}{\partial x}=\frac{-\partial }{\partial x}\left(\frac{x^2-y^2}{2}\right)=-x$
$F_y=\frac{-\partial U}{\partial y}=\frac{-\partial }{\partial y}\left(\frac{x^2-y^2}{2}\right)=y$
$\therefore \overrightarrow{F}=-x\hat{i}+y\hat{j}$