Question

A certain hill must be climbed for an upcoming bicycle race. A cyclist know from experience that the maximum pedal cadence that he can maintain on his 17.6-cm diameter chainring is 84 rpm. if by race rules he must use the regulation 73.8 cmdiameter rear tires, what diameter sprocket must he install in order to achieve his desired  constant speed of 21 kilometers per hour up the hill?

Answer 1

angular speed of chainring, Wc = 84 rpm

radius of chainring, R = 17.6/2 cm = 0.176/2 m = 0.088 m

Let the diameter of sprocket be, d

So, for a bicycle, Wc*R = Ws*(d/2)

where Ws = angular speed of sprocket

So, Ws = Wc*R*2/d

So, speed of the rear tires, v = Ws*r = (Wc*R*2/d)*(0.738/2) = 21000 m/s

So, ((84*2*pi/60)*0.088*2/d)*(0.738/2) = 21000

So, d = 2.7 mm