# Homework Help: Walking legs as physical pendulum oscillations

1. Apr 21, 2010

### intermu

1. The problem statement, all variables and given/known data

The typical walking speed of a person walking at a relaxed pace can be estimated by modelling their legs as a physical pendulum. Assume that the length of a person's leg is L and it pivots about the hip and the leg is tapered (more mass towards the hip and less towards the foot). So that its rotational inertia of the leg is 1/6 MR2

a) Determine the frequency of the oscillations of the leg
b) Derive a formula for the walking speed based on the frequency and the maximum angle of the oscillation
c) What is the walking speed of a person whose legs are 1m long with a center of mass at 0.45m away from the hip who is walking so the maximum angle of their legs is 30deg away from the vertical?

2. Relevant equations

a) w = sqrt (mgl / I)

Where w = oscillation frequency, m = mass, g = acc due to g, l = length, I = rotational inertia

b) v = -Aw sin(wt+theta)

where A = amplitude.

3. The attempt at a solution

From the equation at a), w = sqrt (MgL / (1/6) ML2)
w = sqrt (6g / L)

for b), I would have thought to use the formula as I stated in b), but since the example said to use the frequency and the maximum angle, I'm not sure which formula to use.

I thought about using the equivalent of the formula v = u + at, but I don't have the angular acceleration with me. I also briefly thought to use angular momentum to find the angular velocity before converting it to linear velocity but it seems far-fetched.

I also had thought to use a differential equation going by Torque = Ia, where I = rotational inertia and a = angular acc, and equating it to torque = mgl. Then thus, having:

MgL = 1/6 ML^2 a,
6g = aL
a = 6g/L

But I have no idea how that ties in with the question.

I suppose I have the period by playing around with w = 2pi f, turning into w = 2pi / T.

If someone could help, it will be much appreciated. Thank you :)

2. Apr 22, 2010

### ehild

When calculating the frequency, take care that l is the distance between the rotation axis (at the hip) and the centre of mass of the leg. So it is not equal to the length of the leg.

Walking speed: Think how do you walk. You swing your leg, and at maximum angle you put it down to the ground and pull your body straight above it. Then you swing your other leg. One step is about a quarter period. The length of one step is the distance between your legs at maximum angle.

ehild

Last edited: Jun 29, 2010