First off, thanks a lot for your help.

Question

Secondly, will you do it again?

A bicyclist traveling with speed v = 3.2m/s on a flat toad is making a turn with radius r = 2.8 m. The forces acting on the cyclist and cycle are the normal force (N) and friction force (Ffr) exerted by the road on the tires and mg, the total weight of the cyclist and bicycle.

A) Explain why the angle theta the bicycle makes with the verticle must be given by tan(theta) = Ffr/N if the cyclist is to remain balanced.

B) Find theta for the values given.

C) If the coefficient of static friction between the tires and road is µs = 0.65, what is the minimum turning radius?

I got the centripetal acceleration X mass = Ffr.
I got mg equals to N(hypothenuse * sin(th)) - Ffr(hypotenuse * cos(th))
In order to find the theta, I need to use circular translational motion?
This is where I got stumped.

Answer 1

A) Draw yourself a head-on view of the tilted bike coming out of the page. The torque about the point where the wheel touches the road must be zero, otherwise the bike will tip one way or the other. There is an upward force N on the tire that equals the weight of bike and rider. The opposite force acts as if applied to the center of mass of the bike/rider combination. There is also a centripetal friction force Ff applied at the point where the wheel meets the road. The problem can be treated statically if one allows a centrifugal or "D'Alembert" force to be acting horizontally on the bike at the center of mass of the bike-rider combination.

B) In order for there to be no net torque about the point where the wheel touches the road, the two forces acting at the center of mass, N and Ff, must have a ratio
Fr/N = tan theta = V^2/(gr)= 0.373
theta = 20.5 degrees

C) For the minimum turning radius at that velocity, Ff/N = µs = 0.65.
The leaning angle theta will be larger, and you will now have
V^2/(gr) = 0.65
r = V^2/(0.65 g)= 1.61 m

The above analysis makes an approximation that the centripetal acceleration is the same for all parts of the bike and rider;however for turning radii his small, the acceleration varies from top to bottom of the bike. Bike and rider are not small compared to the turning radius.