Question 6.8: Finding 3-D potential energies (a) Show that the uniform gra...
Finding 3-D potential energies
(a) Show that the uniform gravity field F = −mgk is conservative with potential energy V = mgz.
(b) Show that any force field of the form
F =h(r) \widehat{ r }
(a central field) is conservative with potential energy V = −H(r), where H(r) is the indefinite integral of h(r). Use this result to find the potential energies of (i) the 3-D SHM field F =-\alpha r \hat{ r }, and (ii) the attractive inverse square field F =-\left(K / r^{2}\right) \widehat{ r }, where α and K are positive constants.
Since the potential energies are given, it is sufficient to evaluate − grad V in each case and show that this gives the appropriate F. Case (a) is immediate. In case (b),
\frac{\partial H(r)}{\partial x}=\frac{d H}{d r} \frac{\partial r}{\partial x}=H^{\prime}(r) \frac{x}{r}=h(r) \frac{x}{r},
since r=\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2} \text { and } H^{\prime}(r)=h(r). Thus
-\operatorname{grad}[-H(r)]=h(r)\left(\frac{x}{r} i +\frac{y}{r} j +\frac{z}{r} k \right)=h(r) \frac{ r }{r}=h(r) \widehat{ r },
as required.
In particular then, the potential energy of the SHM field F =-\alpha r \hat{ r } \text { is } V= \frac{1}{2} \alpha r^{2} , and the potential energy of the attractive inverse square field F =-\left(K / r^{2}\right) \widehat{ r } is V = −K/r.