An 17 cm long bicycle crank arm. with a pedal at one end. is attached to a 24-cm

Question

An 17 cm long bicycle crank arm. with a pedal at one end. is attached to a 24-cm-diameter sprocket, the toothed disk around which the chain moves. A cyclist riding this bike increases her pedaling rale from 63 rpm to 90 rprn in 12 s. What is the tangential acceleration of the pedal? Express your answer to two significant figures and include the appropriate units. What length of chain passes over the top of the sprocket during this interval? Express your answer to two significant figures and include the appropriate units.

Explanation / Answer

Given
   length of the crank arm is l = 0.17 m

sprocket diameter d = 24 cm, radius r = 0.12 m

initial angular velcoity w1 = 63 rpm = 63*2pi/60 rad/s = 6.6 rad/s

final angular velocity is w2 = 90 rpm = 90*2pi/60 rad/s = 9.42 rad/s

time taken to reach from 63 rpm to 90 rpm is t = 12 s

a)

now the tangential acceleration is a = r*alpha

   alpha is angular acceleration = (w2-w1)/(t2-t1) rad/s2

                   = (9.42-6.6)/(12-0) rad/s2

                   = 0.235 rad/s2
now the tangential accelerationis a = 0.17*0.235 m/s2 = 0.03995 m/s2

b) length of chain passes over the top of the sprocket during this interval is

   from S = R*theta

   L = r*d(theta)

for d theta using equations of motion

   d(theta) = w1t+0.5*alpha*t^2

       = 6.6*12+0.5*0.235*12^2 rad

       = 96.12 rad

now length L = 0.12*96.12 m = 11.5344 m

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