How are a Bicycle’s Translational and Rotational Motion Related?

Since we started talking about bicycle mechanics last week, I thought we’d continue and look at how the translational motion of the bicycle/bicycler system is related to the rotational motion of the bicycle’s gears and pedals.

Suppose a male bicycle sprinter and his bicycle have a total mass of 74 kg. The bicyclist wins the 1000-meter sprint with a time of 56.3 seconds. What is his average acceleration? What force does the bicycler/bicycle system exert horizontally? How much work does the system do?

The bicycler’s position x(t) is 0 at time 0 and 1000 meters at time 56.3 seconds. At the start of the race, he has velocity 0. Using the equation x(t) = x(0) + v(0)t + ½ at^2 we see that

x(56.3) = 1000 m = 0 + 0 + ½ a* (56.3 sec)^2 –> 1000*2/(56.3)^2 = a = .63 m/sec^2

Now that we know the acceleration, we can calculate the system’s force as:

F = ma –> F = 74 kg * .63 m/sec^2 = 46.62 Newtons (N)

Total work is the product of the force and the distance over which it is applied:

W = F*d –> W = 46.62 N * 1000 m = 46,620 N-m or Joules (J)

OK, that was pretty straightforward. Now how do we relate this translational motion to the rotational motion of the gears and the pedals?

By pushing on the pedals with his feet, the bicycler exerts something called torque on the chainring. The rotation of the chainring pulls the bicycle chain forward, exerting a torque on the rear cog (axle) and propelling the bicycle forward. The torque TAU is defined the moment of the (perpendicular) force acting on an object at a radius r from an axis of rotation, or

TAU = F(perpendicular) * r, measured in N-m

Still confused? Let’s do a thought experiment. You need to tighten a bolt on your bicycle. You can tighten it a little with your fingers, but you can tighten it really tightly with your ratchet. Why? The total torque needed remained constant, but once you introduced the radial distance of the ratchet, the force you have to exert is much smaller!

Now that we understand torque TAU, we can use rotational motion to calculate the work W:

W = TAU * THETA

THETA is the total angular measure (in radians) rotated through during the period of interest. In order to find THETA, we’re going to need some more information about our bicycle. A typical racing bike has a tire radius of .35 meters. We’re going to assume a gear ratio of one, that is, for every complete rotation of the pedals, the rear tire makes a complete rotation. One rotation of the pedals corresponds to 2*pi radians. It also corresponds to one rotation of the rear tire, so the bicycle traveled one circumference of the tire in linear distance, or 2*pi*.35 meters = .7*pi meters. Therefore, we can calculate that the total number of rotations the pedals needed to complete in order to travel 1000 meters is:

1000 m /( .7*pi m/rotation) = 454.7 rotations

THETA is just the number of radians the pedals/wheels rotated through during the race, so

THETA = 2*pi* 454.7 rotations = 2857.0 radians.

The amount of work done by the bicycle/bicycler system is fixed at 46,620 Joules, so now we can determine TAU:

TAU = W/THETA = 46,620 Joules/2857.0 –> TAU = 16.32 N-m of torque

In order to determine the amount of force applied at a typical crankarm length (radius) of .17 m, this translates

F = TAU/r = 16.32 N-m/.17 m –> F = 96 N of force

Stop. Does this calculation make sense? This is over twice the force we found using translational motion. We need to go back and calculate the force exerted at the tire radius:

F = TAU/r = 16.32 N-m/.35 m –> F = 46.63 N of force!

So the force exerted at the tire radius is equal (within rounding error) to the translational force we calculated originally, so that makes sense. But the force needed at the pedal radius is over twice that realized at the tire radius! How about that?

What we’ve learned is that even the simple bicycle has some complicated physics governing its motion. Today we’ve been able to relate its translational and rotational physics. But there are many other interesting principles illustrated by bicycles in motion, like staying balanced and the gyroscopic effect. For those of you interested in learning more, check out the book Bicycling Science by David Gordon Wilson et al.