Question

At what speed is a bicyclist travelling when his 27 inch diameter tires are rotating at an angular speed of 5pi radians?

Answer 1

circumfrence of tires (27''*pi)
angular velocity 5pi rad/min
circumfrence*angual velocity = speed
27*pi*pi*5=1332.39in/min=111.03ft/min

Answer 2

Primaballerina, where you given a rate with your radians? Was it 5pi radians per second?

The translational speed is related to the angular speed by\[v=r \omega\]where v is the translational speed, r the radius of the rotating object and omega the angular speed.

If we assume 5pi radians/second as your angular velocity, you would have,\[v=\frac{27}{2}5\pi \frac{inches}{second}=\frac{135 \pi}{2}\frac{inches }{second}\]You can convert this to something more standard (i.e. miles/hour) by \[v=\frac{135\pi}{2}\frac{inches}{second} \times \frac{3600 seconds}{hour} \times \frac{mile}{63360inches}\]\[=\frac{675 \pi}{176}miles/hour \approx 12 miles / hour\]