biomechanics - angular kinetics

Question

penny farthing bicycle has front and rear tires with diameters of 140 and 44cm, respectively. If each tire is rotated 180°counterclockwise, what is the magnitude of the linear distance travelled for a point on each tire?

Answer 1

When a wheel rotates, the linear distance traveled is equal to the arc length (S = r𝛉). Since the wheels have different diameters, they will rotate at different speeds to cover the same linear distance. So if the bike were rolling down a road, the smaller wheel will rotate faster than the larger wheel to travel the same linear distance, so the tires would never turn 180° at the same time. However, the distance linear distance from each tire rotating 180° is calculated below.

Convert 180° into radians:

𝛉 = 180° * (𝝿 / 180°) = 𝝿

front tire:

x = r𝛉 = 140cm * 𝝿 = 140𝝿 cm = 439.82 cm

rear tire:

x = r𝛉 = 44cm * 𝝿 = 44𝝿 cm = 138.23 cm