Question

The potential energy function of a particle in a region of space is given as :
$U=\left(2 x^{2} + 3 y^{3} + 2 z\right)J$
Here $x,y$ and $z$ are in meters. Find the force acting on the particle at point $P\left(\right.1m,2m,3m\left.\right):-$

• A. $-\left(\right.4\hat{i}+36\hat{j}+2\hat{k}\left.\right)N$
• B. $-4\hat{i}+36\hat{j}+9\hat{k}$
• C. $-8\hat{i}-18\hat{j}+2\hat{k}$
• D. $-4\hat{i}+18\hat{j}-9\hat{k}$

Answer 1

Answer: (A)

$U_{2}=\left(2 x^{2} + 3 y^{3} + 2 z\right)$
$\Rightarrow \overset{ \rightarrow }{F}=-\left(\frac{\partial U}{\partial x} \hat{i} + \frac{\partial U}{\partial y} \hat{j} + \frac{\partial U}{\partial z} \hat{k}\right)$
$=-\left(4 xi + 9 y^{2} \hat{j} + 2 \hat{k}\right)N$
so, $\left(\overset{ \rightarrow }{F}\right)_{\left(\right. 1 , 2 , 3 \left.\right)}=-\left(\right.4\hat{i}+36\hat{j}+2\hat{k}\left.\right)N$