35.109 Compute the curvature of r = 2 + sin 6.
r' = cos 6, r" = -sin 0. By the formula of Problem 35.108
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CHAPTER 35
35.110 Show that the spirals r = 6 and r = 1 /0 intersect orthogonally at (1,1)
For r = e, r'-I, and tan , = rlr' = 0 = 1. For r=l/0, r
1 =-1/0
2 =-1, and tant/> 2 = -l.
Hence, tan ( 35.111 Prove the converse of Problem 35.101.
If r/r' = l/c, then $ (drlr) = \ c dO, Inr = c0 + lno [a = const.], r = aece. Note that c = 0
gives a circle, a degenerate spiral that maintains the fixed angle -rr/2 with its radii.
35.112 Show that, if a point moves at a constant speed v along the equiangular spiral r = ae
c , then the radius r
changes at a constant rate.
But, along the curve, drldt = ace
ce (d0/dt) = cr(d6ldt); hence
or
const.