A motorcycle has to move with a consatnt speed on an overbridge which is in the form of as circular arc of radius R and has a total length L. Suppose the motorcycle starts from the highest point. a. what can its maximum velocity be for which the contact with the road is not broken at the highest point? b. If the motorcycle goes at speed `1/sqrt2` times the maximum found in part a. where will it lose the contact with the road? c. What maximum uniform speed can it maintain on the bridge if it does not lose contact anywhere on the bridge?

Answer

A, B, C, D

Explanation

R=radius of the bridge
L=total lengthof the over bridge ltbr.a At the highest point
`mg=(mv^2)/r`
`rarr v^2=Rg`
`rarr v=sqrt(Rg)`

b. Given `v=(1/sqrt2)sqrt(Rg)`
Suppose it loses contact at B.
`So at B, mg costheta= (mv^2)/R`
`rarr v^2=Rgcostheta`
`rarr sqrt((Rg)/2)^2)=Rgcostheta`
`rarr (Rg)/2=Rg.costheta`
`-. costheta=1/2`
`rarr theta=60^0=pi/3`
`theta=1/R`
`rarr Rtheta=(pir)/3`
so it will lose contact at a distance `(pir)/3` from highest point.
c. Let the uniform speed on the bridge by v. The changes of losing contact is maximum at the end of the bridge for which,
`alpha=L/(2R)`
`So, (mv^2)/R=mgcosalpha`
`rarr v=sqrt(gRcos(L/(2R))`