Answer to Question #230794 in Classical Mechanics for alwande hadebe

Question #230794

*Conservative force and its potential energy function Consider the components of a position dependent force F(r): Fx = ayz + bx + c Fy = axz + bz Fz = axy + by. (a) Show that this force is conservative. (b) Find its corresponding potential energy function, V (r)


1
Expert's answer
2021-08-29T16:55:10-0400

Given:

Fx=ayz+bx+cFy=axz+bzFz=axy+byF_x=ayz + bx + c \\ F_y=axz + bz\\ F_z=axy + by


(a) For conservative field

curlF=0\bf curl\: F=0

In our case

curlF=i^j^k^xyzFxFyFz\bf curl\: F=\begin{vmatrix} \hat i & \hat j & \hat k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ F_x&F_y&F_z \end{vmatrix}

=i^j^k^xyzayz+bx+caxz+bzaxy+by=\begin{vmatrix} \hat i & \hat j & \hat k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ ayz + bx + c&axz + bz&axy + by \end{vmatrix}

=i^(ax+baxb)+j^(ayay)+k^(azaz)=0={\hat i}(ax+b-ax-b)+{\hat j}(ay-ay)+{\hat k}(az-az)\\=\bf 0

(b) The potential energy function satisfies equation

F=V{\bf F}=-\nabla V

So

V(x,y,z)=(axyz+b(x2+y2+z2)/2+cx)V(x,y,z)=-(axyz+b(x^2+y^2+z^2)/2+cx)


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