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h3. Part A

!velodrome.jpg!
Velodromes are indoor facilities for bicycle racing (picture by Keith Finlay, courtesy of Wikimedia Commons). Olympic velodromes are usually ovals 250 m in circumference with turns of radius 25 m. The peak banking in the turns is about 42°. Assuming a racer goes through the turn in such a velodrome at the optimal speed so that no friction is required to complete the turn, how fast is the racer moving?

System:  {color:maroon}The rider will be treated as a point particle.  The rider is subject to external influence from the earth (gravity) and from the track (normal force).  We assume friction is not present, since we are told to determine the speed at which friction is unnecessary to complete the turn.{color}

Model:  [Point Particle Dynamics].

Approach:  
{color:maroon}Since we are assuming no friction is needed, we have the free body diagram shown above.  The corresponding equations of Newton's 2nd law are:

{latex}\begin{large}\[\sum F_{x} = N \sin\theta = \frac{mv^{2}}{r} \]\[\sum F_{y} = N \cos\theta - mg = 0 \] \end{large}{latex}

Where _r_ is the radius of the turn, _v_ is the speed of the racer and _m_ is the racer's mass (including bike and gear).
{note}The situation here is very different than that of an object sliding down an inclined plane. In the case of an object moving _along_ the plane, the acceleration will have both _x_ and _y_ components.  For an object moving along a banked curve, the object will not be moving up or down and so _a_~y~ must be zero.  The x-component of the acceleration will of course not be zero because the object is following the curve.{note}
{warning}Because _a_~y~ = 0 here, it is _not_ appropriate to assume _N_ = _mg_ cosθ. That equation follows from assuming the object accelerates along the incline.{warning} 

From the y-component equation, we find:

{latex}\begin{large} \[ N = \frac{mg}{\cos\theta} \]\end{large}{latex}

{note}Note both the similarity to the standard inclined plane formula and the important difference.{note}

Substituting into the x-component equation then gives:{color}

{latex}\begin{large} \[ v = \sqrt{gr\tan\theta} = 15 \: {\rm m/s} = 33 \:{\rm mph} \]\end{large}{latex}

{tip}Is this speed reasonable for a bike race?{tip}

{info}Note that theour result is independent of the mass of the rider.  This is important, since otherwise it would be impractical to construct banked curves;.  differentDifferent people would require different bankings!{info}

h3. Part B