()2sin0dmrdtφα=±Say constant 2sinmr==±A23cosmrmgmrα=−A±±212ddrdrrrdtdrdr===±±±±r223cos2mdrdrmgdrmrA±222cosmrmgrCmr−+±AThe constant of integration C is the total energy of the particle: kinetic energy due to the component of motion in the radial direction, kinetic energy due to the component of motion in the angular direction, and the potential energy. 22cos2Urmgrmr=+AFor , A, and turning points occur at 0φ≠±0≠0r=±Then 220co2mgrCmr−+As232cos02mgrCrm−+=AThe above equation is quadratic in r (2r4∝A) and has two roots. 10.24Note that the relation obtained in Problem 10.23, 2cos02lmgrCrm, defines the turning points. For the particle to remain on a single horizontal circle, there
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