Question

Find the period of a child's leg as it swings about the hip joint. Assume the leg is 0.39 m long and can be treated as a uniform rod.

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Estimate the child's walking speed.
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help me?

Answer 1

Assume that the angle from the vertical ? is small enough so that:

(1) ? ~= sin (?)

Then we can use the simple harmonic motion model for the leg. Additionally, the leg is an example of a compound pendulum, where the rod is not massless and may have any shape.

From ref 1, the equation for the period T of a compound pendulum is given by:

(2) T = 2 * ? * ? (I / (m * g * Lcm) )

where:

Lcm = the distance from the pivot point to the center of mass = 0.39 / 2 = 0.195m
and
(3) I = the moment of inertia of a uniform-mass rod pivoting about one of its ends (see ref. 2)

= m * Lr^2 / 3,

where Lr is the length of the rod = 0.39m

Substituting the above into (2):

(4) T = 2 * ? * ? (I / (m * g * Lcm) )

= 2 * ? * sqrt ( (m * Lr^2 / 3) / (m * g * Lr/2) )

= 2 * ? * sqrt ( (Lr^2 / 3) / (g * Lr/2) )

= 2 * ? * sqrt ( (Lr / 3) / (g / 2) )

= 2 * ? * sqrt ( (2 * Lr ) / ( 3 * g ) )

= 2 * 3.14 * sqrt ( (2 * 0.39 ) / ( 3 * 9.81 ) )

T = 0.9998 so let's just call it 1.00s . . . . . .<<<=== period of one complete oscillation (forth and back)


Next, the child's walking speed V:

(5) V = Distance / Time

How much distance does the child cover during one period? Experiment with this: It turns out that, for one leg to go through one cycle it requires two strides (or steps).

Each step is ? radians, so, continuing (5)

= 2 steps * ? rad/step * 0.37m/rad / 1s

= ? * 0.74 m/s . . . the child's estimated walking speed