Assuming the earth's orbit around the sun to becircular, show that the area swept byits radius vector in unit time (areal velocity of the earth) is a constant.

Assuming the earth's orbit around the sun to becircular, show that the

1.	Assuming the earth's orbit around the sun to becircular, show that the area swept byits radius vector in unit time (areal velocity of the earth) is a constant.
Answer» Solution :Suppose the earthmoves in a circular orbit of radius r with the sun at the centre (O) of the orbit[Fig. 1.5]. Let the radius vector OP , in an infinitesimal interval of TIME dt, describe an angle `d""THETA` at the centre. Hence are PQ =`rd""theta`. As value of PQ is very small, the arc PQ can be taken to be a straight line (chord PQ). `therefore` Area swept in time dt =area of TRIANGLE OPQ `=1/2 OP xxPQ =1/2rrd ""theta=1/2r^2d""theta` `therefore` Area swept per unit time `=1/2 r^2(d""theta)/(dt)=1/2 r^2omega` = constant [as orbit is circular, r and `omega` of the earth should be constants] `therefore` Area swept in unit time by the radius vector of the earth is a constant.