Question

A particle follows a spiral orbit given by r = c (theta)² under an unknown force law. Prove that such an orbit is possible in a central field. Also find the force law?

Answer 1

Answer

u=$\frac{1}{r}=\frac{1}{c\theta^2}$

For finding force law

$\frac{d^2u}{d\theta^2}+u=-\frac{Fm}{l^2u^2}$

Putting value of

$\frac{d^2u}{d\theta^2}=\frac{6}{c\theta^4}$

And u in above equation

$\frac{6}{c\theta^4}+\frac{1}{c\theta^2}=-\frac{Fm}{l^2u^2}$

Changing in r form

Then force law become

$F=-\frac{m}{l^2}(6c+r)$