Question

The potential energy of a body of mass $m$ is given by $U=px+qy+rz$. The magnitude of the acceleration of the body will be

• A. $\frac{p+q+r}{m}$
• B. $\frac{\sqrt{p^2+q^2+r^2}}{m}$
• C. $\frac{\sqrt{p^3+q^3+r^3}}{m}$
• D. $\frac{\sqrt{p^4+q^4+r^4}}{m}$

Answer 1

The correct option is B $\frac{\sqrt{p^2+q^2+r^2}}{m}$

$F=-\frac{dU}{dr}=F_x=-\frac{\partial U}{{\partial }_X}=-p$,

$F_y=-\frac{\partial U}{\partial y}=-q,F_Z=-\frac{\partial U}{{\partial }_Z}=-r$

$=\overrightarrow{F}=F_x\hat{i}+F_y\hat{j}+F_Z\hat{k}=(-p)\hat{i}+(-q)\hat{i}+(-r)\hat{i}$

$=F=\sqrt{F_x^2+F_y^2+F_Z^2}=\sqrt{p_2^2+q^2+r^2}$

$\Rightarrow a=\frac{F}{m}=\frac{\sqrt{p^2+q^2+r^2}}{m}$