Past exam of the physics 1 paper of the University of Oxford / 2019 / 8 / c / Solution

The diagram places the force centre on the circumference, not at the centre of the circle. Choose the polar axis through the circle's geometric centre. Its equation is $r=2a\cos \theta$, so $u=\sec \theta /(2a)$ andSubstitute into Binet's equation:Consequently a circular orbit through the force centre is generated byThe result applies to the regular arc where $r>0$ and $J\ne 0$. The force is singular at the point where the geometrical circle meets the origin, so the curve is not a regular periodic orbit continued through that point. Indeed, $U=-C/(4r^4)$ and this trajectory has zero total energy, with kinetic and negative potential terms diverging as the centre is approached.