SKEDSOFT

Maths For Engineers - 1

Polar Curves-2:

1. Derive an expression for arc length in Cartesian form.

Solution: Let P (r,θ) and Q (r δr, θ δθ) be two neighbouring point on the graph of the functions r f (θ). So that they are at length S and S δs from a fixed point A on the curve.

∴ PQ = (S δs) - s = δs

Draw Pn Perpendicular to OQ,

From Δ*OPN,

PN/OP = Sin δθ

i.e PN/r = Sin δθ or PN = r Sin δθ

and ON/OP = Cos δθ

i.e ON/r = Cos δθ (or) ON = r Cos δθ

When Q is very close to P, the length of the arc PQ are equal to δs as the length of the chord PQ.

In Δ*PQN,

(PQ)2 = (PN)2 (QN)2

But PN = r Sin δθ

And QN = OQ - ON

∴           = r δr - r

∴          = δr

 

And (PQ)2 = (PN)2 (QN)2

(δS)2 ( rδθ )2 (δr)2............................(5)

Hence This is Beta-Gamma Functions.