Walking Speed of a Dinosaur: How Does Leg Length Affect Natural Walking Pace?

[AlonsoMcLaren]

Homework Statement

All walking animals, including humans, have a natural walking pace, a number of steps per minute that is more comfortable than a faster or slower pace. Suppose this natural pace is equal to the period of the leg, viewed as a uniform rod pivoted at the hip joint. A) How does the natural walking pace depend on the length L of the leg, measured from hip to foot? B) Fossil evidence shows that Tyrannosaurus rex, a two-legged dinosaur that lived about 65 million years ago at the end of the Cretaceous period, had a leg length L = 3.1 m and a stride length (the distance from one foot-print to the next print of the same foot ) S = 4.0 m. Estimate the walking speed of Tyrannosaurus rex.

Homework Equations

I=(m*L^2)/3
T=2*pi/w

The Attempt at a Solution

T=2*pi/w=2*pi/sqrt(Lmg/I)

I=(m*L^2)/3

T=pi*sqrt(4L/3g)=2.04s

v=s/T=4m/2.04s=1.96m/s

But the answer says that T=2pi*sqrt(2L/3g)=2.88s
v=s/T=4m/2.88s=1.4m/s

 

[AlonsoMcLaren]

Nobody?

 

[BruceW]

you've said, you've seen the answer is T=2pi*sqrt(2L/3g), But you have used: T=2pi*sqrt(L/3g), so this is where the problem is. I guess the problem stems from the first line, where you seemed to use w=sqrt(Lmg/I), but this is not true (which is why you end up with an incorrect answer for the period of the motion).

 

[AlonsoMcLaren]

you've said, you've seen the answer is T=2pi*sqrt(2L/3g), But you have used: T=2pi*sqrt(L/3g), so this is where the problem is. I guess the problem stems from the first line, where you seemed to use w=sqrt(Lmg/I), but this is not true (which is why you end up with an incorrect answer for the period of the motion).

Why is w=sqrt(Lmg/I) incorrect?

 

[BruceW]

That equation for the angular frequency is correct for a pendulum where there is a small bob on the end of a light, inextensible string. But in this question, the pendulum is a uniform rod. So that equation for the angular frequency is not correct.

Are you meant to derive the equation for the angular frequency of a pendulum made of a uniform rod? If not, you can guess what it is, since you've seen the answer. The rest of your working is correct, its just the angular frequency which was wrong.