gravity07

7.3.8 Relationship between F and U; between g and ϕ

To understand how g is related to ϕ:

Similarly, compare $g = - \frac{G M}{r^{2}}$ and $ϕ = - \frac{GM}{r}$ in the above table.
If we differentiate $ϕ = - \frac{GM}{r}$ with respect to r, we will get $\frac{d ϕ}{d r} = - \frac{G M}{(- r^{2})}$ , which has the same expression as g.
Hence, mathematically $\frac{d ϕ}{d r} = \frac{G M}{r^{2}} = - g$
To understand the meaning of $g = - \frac{d ϕ}{d r}$ observe the two graphs carefully, on the right side where r is positive, the gradient of ϕ vs r graph is positive but the value of g will be negative. And on the left side where r is negative, the gradient of ϕ vs r graph is negative but the value of g is positive. Thus, $g = - \frac{d ϕ}{d r}$

Similarly, it can be concluded that by multiplying both sides by test mass m.

$m g = - \frac{d m ϕ}{d r}$
thus $F = - \frac{d m ϕ}{d r}$

7.3.8.1 Activity To do

ICT inquiry worksheet 1 (E), as well as the "G field and potential" EJS here. The HTML5 version is below, the Java worksheet customization to HTML5 is work in progress.

7.3.8.2 Summary

symbol	$g = - \frac{G M}{r^{2}}$	$ϕ = - \frac{GM}{r}$
name	Field strength	Potential
units	N kg^-1 or m s^-2	J kg^-1
meaning	Gravitational force per unit mass	Gravitational potential energy per unit mass
quantity	vector	scalar
equation	$\|g\| = \frac{G M}{r^{2}}$ towards the centre of the source mass	$ϕ = - \frac{GM}{r}$
relationship to mass	Force, $F = \frac{G M_{1} M_{2}}{r^{2}}$ = mg	Potential energy, $U = -m \frac{G M}{r}$ = mϕ
graph
computer model if M = 500.	-6.67500/(abs(r)r)	-6.67*500/abs(r)
relationship between g and ϕ	$g = - \frac{d ϕ}{d r}$
relationship between F and U	$F = - \frac{d U}{d r}$

7.3.8 Relationship between F and U; between g and ϕ

7.3.8.1 Activity To do

7.3.8.2 Summary

7.3.8.3 Model