To get a plot like the one you show, I have to correct a sign error in your equation. After doing so, the equation is easily solved with NDSolveValue
. Like so.
rF =
NDSolveValue[
{r'[ϕ] == -(r[ϕ] - 1)^2/r[ϕ]/Sqrt[2] Sqrt[r[ϕ]^3 + r[ϕ]], r[0] == 100},
r, {ϕ, 0, 8 π}]
PolarPlot[rF[ϕ], {ϕ, 0, 8 π}, PlotRange -> {{-2, 8}, {-1.5, 1.5}}]

Update
The following is added to address concerns raised by the OP in a comment to this answer.
Mathematica is has so much stuff in it that it is indeed hard for beginners to find there way around the app. The documentation is quite extensive and there really is a lot of introductory material in it, but again, there is so much of it that it hard for beginners to use it.
I recommend that you begin your further exploration of Mathematica by following this link (or its equivalent in the built-in Documentation Center).
Now let's look into how Manipulate
can be used to make a demonstration of a particle moving along the spiral shown in the polar plot. The main point is add an Epilog
option to the plot which will draw the moving point.
Manipulate[
PolarPlot[rF[ϕ], {ϕ, 0, 1080 °},
Epilog -> {Red, AbsolutePointSize[8], Point[rF[Φ °] {Cos[Φ °], Sin[Φ °]}]},
PlotRange -> {{-2, 6}, {-1.5, 1.6}},
PerformanceGoal -> "Quality",
ImageSize -> Medium],
{{Φ, 5, "ϕ (deg)"}, 5, 1080, 5, Appearance -> {"Large", "Labeled"}}]

Notes
- I have an engineering background, so I like to see polar plots read out in degrees. If I didn't have this prejudice, the demo code could be made a little simpler. If you have a scientific mind, you can simplify the code by removing all references to degrees.
- What looks like a simple slider controling the position of red point is actually an animator control. If you click on the plus ( + ) button at its right end, it will reveal a full set of animation controls. You should also click on the plus button at the top-right of the demonstration panel and see what it reveals.
- The
Point
graphics primitive in the Epilog
specification must be in expressed in Cartesion coordinates. Hence, Point[rF[Φ °] {Cos[Φ °], Sin[Φ °]}]
- The
PerformanceGoal
option is given to keep the spiral from being distorted when the slider is moving.
r
as an explicit function ofphi
(e.g. replace everyr
withr[phi]
). Then find an expression for the derivativer'[phi]
. This would give a differential equation, that you might solve analytically withDSolve
or numerically withNDSolve
. PuttingM=1
to normalise your coordinates would probably help. $\endgroup$