Walking speed can be expressed by the equation, (walking speed) = (leg swing fre
ID: 1843290 • Letter: W
Question
Walking speed can be expressed by the equation, (walking speed) = (leg swing frequency) times (gate length), where (gate length) is twice of one step size. If we assume the minimal use of the leg muscle, the (leg swing frequency) in the above equation would correspond to the natural pendulum frequency of the leg swing, and the (walking speed) would be the idle walking (the slowest natural walking) speed. The natural pendulum frequency of the leg swing is known to be proportional to Squareroot g/L, where g is the gravitational acceleration and L is the length of the leg. The above equation can also be used to estimate the maximum walking speed, if we assume the maximal use of the leg muscles to obtain the (leg swing frequency). By modeling the muscle force as a spring force where the magnitude of the force can be represented by a spring constant k, provide an argument to show that (maximum walking speed) alpha Squareroot k/m, where m is the mass of the leg.Explanation / Answer
As it is given, the walking speed is the product of frequency times the distance of two consecutive steps. Considering the gate length of the walking person to be normal or average among different leg lengths, the variable parameters is the swing frequency of the leg.
The modelling of the leg as a simple pendulum indicates the swing is inverse in proportion to the leg length which inturn affects the gate length of foot steps. So, a tall person will walk at greater leg swings than a shorter person to achieve equal walking speed of a shorter person.
This concludes the fact that walking speed is directly proportional to the frequency of the leg swing,
walking speed = frequency * gate length
or walking speed = sqrt (g / L) * C where c is the gate length constant for a particular person.
Since the above frequency is a linear frequency converting it into a circular is done by,
omega = 2*pi*f
So, f = omwga / 2* pi
Replacing the above relation with the circular frequency we get,
walking speed = 2 * pi * sqrt (g / L) ---- (1)
Also, the circular frequency of a spring-mass system is given by,
omega = sqrt ( k / m ) ---- (2)
Now check the dimensional correctness of the eqn 1 and 2,
eqn 1 : 2 * pi * sqrt (g /L) = sqrt (LT^-2 / L) = sqrt (1/T)
eqn 2: sqrt (k/m) = sqrt (MLT^-2 / ML) = sqrt (1/T)
The dimensional correctness of the above relation shows that walking speed is proportional to circular frequency of the leg omega.
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