2.5 Froude number - a deeper dive

The Froude number (sounds like “food” with an “r”) is a useful parameter to study how locomotion changes with body size. It is a dimensionless variable that relates a scalar dimension of the body (typically leg length) to the forward velocity of the body. Theoretically, animals of different sizes that rely on pendulum-like mechanics for gait will use the same form of locomotion at a given Froude number (e.g., a T-Rex and a chicken traveling at the same Froude number would look the same). The Froude number is defined as:

where  = the speed of movement (in m/s)

= the acceleration due to gravity (in m/s)

= the effective length of the limb (in m)

(a) Using the simple inverted pendulum model from the previous question, explain why the walk-run transition theoretically occurs at . (Note: the model is redrawn below with the linear speed  and gravitational acceleration  shown.)

(b) Dr. Evil (a somewhat normal adult male with a leg length of 0.9m) and Mini-Me (an exact 1/8 replica of Dr. Evil) need to go to the other side of their secret lair in Nevada to feed their sharks and repair some laser beams. Dr. Evil demands that Mini-Me hold his hand so they can remain side-by-side as they travel to the shark tank. Dr. Evil walks to the shark tank at optimum walking speed (Fr = 0.25). Once arriving at the destination, Mini-Me says that he is too tired from running to keep up with Dr. Evil to feed the sharks. How can you explain this?

(c) Dr. Evil is tired of having a miniature version of himself and orders the biomechanical engineers on his lunar base to build a robot that looks, walks, and behaves exactly how he does on Earth. Given that gravity = (1/6)gravity, why is the lunar engineering team doomed to failure? 

(d) Walking speed is a good indicator of mobility and is often used in research of gait disorders. It has been suggested that walking speed should be reported in terms of the Froude number to remove the effects of individual subject height in experiments. Suppose a patient with cerebral palsy walks with excessive knee flexion (this is called crouch gait). Knowing that she needs leg length to compute Fr, the clinician measures the patient's leg length while he lies on the exam table. She also measures his walking speed. She is pleased when she computes Fr = 0.25, which suggests a reasonable walking speed for the patient’s stature and thus good energy efficiency. However, after measuring the patient's oxygen consumption, she finds that his gait is not energetically efficient at all. Explain this discrepancy in her results using the Froude number. Based on this information, do you think that Fr is a good clinical indicator of walking speed?