Normal Reaction in rotating frame.

Feb 2019
Hi there,

I'm new to this forum so forgive me if this is in the wrong section. The module I'm studying is a mechanics course as part of a mathematics degree (currently in year 3 of 4), however, I am not confident with physics at all.

I have a problem with the following question . . .

"A car of mass m is driven at constant speed v around a circular test track of radius a, which is banked at an angle α to the horizontal. With reference to the figure, consider the rotating frame, with origin at the centre of the track, in which the car is at rest. Determine the Coriolis and centrifugal forces on the car (these are −m× the corresponding acceleration terms). Assuming that the frictional force between the car and the track is negligible, use N2 to obtain two expressions involving the normal reaction on the car. Hence show that the speed of the car at which there is no radial force acting on it in the rotating frame is given by v=sqrt(ag tan(theta))."

My attempt,

I have the Force for the Coriolis and the Centrifugal, I got this using mass and the acceleration for Coriolis and Centrifugal.

It's messy but these are:

F(Coriolis)= ma = m(2w x dx/dt) = 2m theta(dot) r(dot) e(theta)
F(Centrifugal) = ma = m(w x (w x x)) = -m theta(dot)^2 r(dot) e(r)

My issue is I have no idea how to get two expressions for the normal reaction of the car ?.

Can I add the Forces for Coriolis, Centrifugal and the normal, then make this equal to the mass dotted with the acceleration for a particle in the rotating frame ?.

I'm not sure how I get to the last equation but this might become clearer once I have the two expressions required.

Any advice is appreciated as my knowledge in this area is very poor. Thanks.
Jun 2010
I forgot to say. With assumed "no friction" at the road surface, the
normal force would be perpendicular to the plane of the road.