- #1

Rococo

- 67

- 9

## Homework Statement

I'm having trouble understanding these notes. Can anyone help me understand how Equation 1 is arrived at, and then how Equation 2 is arrived at?

A central force field [itex] \underline{F}(\underline{r}) [/itex] is a field of forces, which are directed from a mass m inwards or outwards from a fixed point O, with magnitude depending only on [itex]r= \mid{\underline{r}}\mid[/itex] of m from O:

[itex] \underline{F}(\underline{r}) = f(r) \hat{\underline{r}} = f(r) \frac{\underline{r}}{r} [/itex]

As a consequence, [itex] \mid \underline{F}(\underline{r}) \mid = f(r)[/itex] = constant on a sphere of radius [itex] r = \mid \underline{r} \mid [/itex].

Potential for a central force field:

[itex] U(r) = -\int_{r_0}^{r} dr' f(r') + U(r_0)[/itex] (Equation 1)

with [itex]U(r_0)[/itex] = constant. (Often chosen such that [itex]U(r_0)=0[/itex])

Let us calculate the force field:

[itex] \underline{F}(\underline{r}) = -\underline{\nabla}_r U(r)[/itex]

[itex] \underline{F}(\underline{r}) = \left.(\underline{\nabla}r)f(r')\right|_{r'=r} = f(r)\underline{\nabla}r [/itex] (Equation 2)