Question

A particle in a certain conservative force field has a potential energy given by $V = \frac{20xy}{z}$. The force exerted on it is

• A. $\left(\frac{20y}{z}\right)\hat{i}+\left(\frac{20x}{z}\right)\hat{j}+\left(\frac{20xy}{z^{2}}\right)\hat{k}$
• B. $-\left(\frac{20y}{z}\right)\hat{i}-\left(\frac{20x}{z}\right)\hat{j}+\left(\frac{20xy}{z^{2}}\right)\hat{k}$
• C. $-\left(\frac{20y}{z}\right)\hat{i}-\left(\frac{20x}{z}\right)\hat{j}-\left(\frac{20xy}{z^{2}}\right)\hat{k}$
• D. $\left(\frac{20y}{z}\right)\hat{i}+\left(\frac{20x}{z}\right)\hat{j}-\left(\frac{20xy}{z^{2}}\right)\hat{k}$

Answer 1

Answer: (A)

The potential energy function $U ( x )$ for a system in which a one - dimensional force $F$ acts on a particle, we can find the force as
$F ( x )=\frac{ dU }{ dx }=\frac{ dU _{ x }}{ dx }+\frac{ dU _{ y }}{ dx }+\frac{ dU _z}{ dx } $
$\text { Apply } U =20 \frac{ xy }{ z } $
$=\frac{20 y }{ z } i +\frac{20 x }{ z } j -\frac{20 xy }{ z ^2} k$