Question

An 19-cm-long bicycle crank arm, with a pedal at one end, is attached to a 21-cm-diameter sprocket, the toothed disk around which the chain moves. A cyclist riding this bike increases her pedaling rate from 63 rpm to 93 rpm in 11 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.

Answer 1

B). Initial angular velocity, $\omega_i= 63 rpm= 63 (\frac{2 \pi}{60}) rad/s= 6.594 rad/s$

Final angular velocity, $\omega_f= 93 rpm= 93 (\frac{2 \pi}{60}) rad/s= 9.734 rad/s$

time taken, t= 11s

Then let total radians covered in this much time be $\theta$ and angular acceleration be $\alpha$, using equations of motion

  $\alpha= \frac{\omega_f-\omega_i}{t}$

  $\alpha= \frac{9.734-6.594}{11}= 0.2854 rad/s^2$

Again using,

  $\omega_f^2-\omega_i^2= 2 \alpha \theta$

$(9.734)^2-(6.594)^2= 2 (0.2854)\theta$

  $\theta= \frac{94.75-43.48}{0.5708}= 89.821 rad$

Then radius of sprocket, r= diameter/2= (21/2)cm= 10.5cm

Then length of chain passes over the top of sprocket in this interval is

$l= r\theta= (0.105m)(89.821)= 9.4m$(ANS)