Question

A comet moves in a counter-clockwise orbit around the Sun. A portion of the orbit is shown below (Ignore all gravitational forces acting on the comet other than that by the Sun.) 2. C The position vector 7 of the comet at a time t is shown in the diagram at right. In the diagram, draw a vector dr representing the infinitesimal displacement of the comet between timet and time (t + dt) a. Sun r Show that the magnitude of the vector cross-product Fxdf is directly proportional to the area dA swept out by the comet from time t to time (tdt). Explain your reasoning in words and with one or more b. Comet diagrams (Hint Show that the magnitude of Fxdr is equal to the area of the parallelogram formed by f and dr How does the area dA swept out by the comet compare to that of the parallelogram?) On the basis of your work in part b, show that the rate dA/dt at which the comet sweeps out area along its orbit is directly proportional the angular momentum of the comet, and determine the constant of proportionality between the two quantities C.

Answer 1

Date Page No. Ans Sun a) Position vertor at given tume tone t+dt) The imtimitesimal displacement comet between, tims and twne t (t+ dr) s- Jume This de perpendicular to tray ectery the Sun ++ dr dr bi Now cross product a t amd dr x d Magaitude tt dt dr ndr sine angte and di where betueem Now, peupendicular hight ef dsine is to Sunce makes drsime the parallelogram ABCD of paxallelagam hemce Area height x(dnsine) dh ls AxRa= base x Ara cathotion of plet stould be the strinate sn of human eatence.-aRAnbechar

Page No.: Date / Aea swept by comet Area =2A = 2ndnsine A = 2 Hxd Ahea of parallelogr amo comet Swept by Area A Oc ixdi 1 ndrsine A = 2 Suept qusen by C) since Now area calculated aboue ndrsime A = 2 t dr supe dA elodg Since, Ixvsime 2 dA dlr mursine ALso momemtum L Angular L vasune Erom 0&2 dA o L alt 2m Constant ot proporlionality 2m comet (m=mas Do net say Tt ts moetrg end disnbs ft with a ne ef yestendey See k for the flat tdne as a nowtom child that has no name,"-Rablndrunath Thgore