Question

2. 10 points Consider a body that is confined to move in a vertical plane, the x - z plane. The body has mass m and moves in the plane subject to the constant) gravitational force g in the – î direction) and an additional "central” force of the form f = - Ar-1/2, where y2 = 22 +22. This additional force is thus directed towards the origin. Choose the appropriate generalized coordinates and let the gravitational potential be zero along a horizontal line through the origin (z = 0). a) Find the equations of motion for the system. (7 pts) b) Show whether or not angular momentum about the origin is conserved. (3 pts)

Answer 1

Zlewe) (0,7) m Xitve taking the Plane polar Coonainate be our generalised (0,0) to Coordinate . 80, a=rcoso , z= z = rsino ginen force the Central f f = - Art CF → nadically outward unit vecton sos the Potentia energy U? -S f.dñ + c 0 Us A dan tc Soita Carr >> uz + c 3 U=2Ar te

Setting (angiuo to s.) یه at Voo we get FO + c =) c=0 € este in Potential d the gravitational energy 19 onishbu + = امه Now since , narcoso siz ; caso + rosina) e nė sino siz prod I & Quisa = o orod atonis & - n x²+ z² = (n cono-rosinaj? +(risinotro caso)?

x² + 2² siz 2²82-2rri sina casa tansi è sluo caso 2 si? giz of the body So, total kinetic energy having mass m is To tmst? / m (242²) T= 2 I'm (riz+m²82) total potential energy so V- ut og > v= 2Arr 2 Ar tingrsino so, the lagrangian La T - V L imbo +52 02) - 2016-mgrsine

(1) — – өчале + ощ 0 (0) 4 I o = 0(!) - + 0.44 + 944мъ - ( СЕ (! өe Te о « oS Q Өvp) 4 4 - - - ne т 94 - e vet (18) - 944 шт = р өe ebel = Не MON (D-lonis buat hal zoom . 4: G o = 0~зм + «ILS -04- 4. ( о с de ne - (EF (П'S le 65 % Ч Louw Не le (! Не (0)

= ) ö + coso 2 0 - - 89, eanation of motion are imgi = mn .8²_A F -ngsino 28 d q caso b) since the lagnangian is o is hence function of O, not Cyclic Coordinate 2L ao 70 so 06) (Px) + dt where Po is the angular momentum. so, angular monentuna is not conserved.