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
$F(x)=\frac{dU}{dx}=\frac{d{U}_x}{dx}+\frac{d{U}_y}{dx}+\frac{d{U}_z}{dx}$
$\text{ Apply }U=20\frac{xy}{z}$
$=\frac{20y}{z}i+\frac{20x}{z}j-\frac{20xy}{z^2}k$