Question

In a certain two-dimensional field of force, the potential energy of a particle have the form $U=\alpha x^2+\beta y^2$, where $\alpha$ and $\beta$ are positive constants whose magnitudes are different. Find out whether this field is central.

• A. No.
• B. Yes
• C. It is parallel
• D. Is perpendicular

Answer 1

The correct option is A No.

$F_x=-\frac{\delta u}{\delta x}=-2\alpha x$ and $F_y=-\frac{\delta u}{\delta v}=2\beta y$

so $\overline{F}=2\alpha \overline{x_i}-2\beta \overline{y_i}$ and $F=2\sqrt{a^2{x}^2+{\beta }^2{y}^2}$

for a central force $\overline{r}\times \overline{f}=0------------\left(1\right)$

here $\overline{r}\times \overline{f}=(\overline{xi}+\overline{yj})\times (-2\alpha \overline{xi}-2\beta \overline{yj})$

$=2\beta xy\overline{k}-2\alpha xy\overline{(}k)=0$

hence the force is not a central force