Part A
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: 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.
Model: Point Particle Dynamics.
Approach:
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:
\begin
[\sum F_
= N \sin\theta = \frac{mv^{2}}
][\sum F_
= N \cos\theta - mg = 0 ] \end
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 ay must be zero. The x-component of the acceleration will of course not be zero because the object is following the curve.
Because ay = 0 here, it is not appropriate to assume N = mg cosθ. That equation follows from assuming the object accelerates along the incline.