Ok, so we need to find the conditions that are required for the bicycle to stay on the track at
.
They are that the centripetal force supplied by the loop (
) needs to be greater than
.
The initial energy of the system is all gravitational potential which is
and at part
this energy is transformed into some kinetic (note that some potential still remains)
;
(where
is the minimum speed required to stay on the loop)
solving
=>
subbing in
in terms of
;
Total Energy required to complete loop in terms of
;
Since energy is conserved we can equate the initial gravitational potential (
) with this new expression.
cancelling
but since h needs to be greater than this height, otherwise it would give a 0 reaction force at point B.
so
I think it's right (had a similar problem at uni).