As a person steps onto their “tiptoes” as shown in the figure, the force exerted

Question

As a person steps onto their “tiptoes” as shown in the figure, the force exerted by their Achilles tendon on their ankle, Ft , increases greatly in magnitude. (We will learn in chapter 8 that this increase in tendon force occurs to prevent the ankle’s rotation.) The Achilles tendon to good approximation obeys Hooke’s law.

(a.) A person starts from rest with their feet flat on the floor (where their Achilles tendons are unstretched). They tilt both feet upward so that they are on their tiptoes. When each ankle is 200 mm above the ground, they stop and stay there. Explain why this process cannot conserve mechanical energy and therefore requires the person to burn calories while doing it. For full points you must correctly identify every type of energy involved in this process and explain why it is increasing or decreasing.

(b.) Using a realistic value for your mass, estimate the number of times you would have to stand on your tiptoes to burn one calorie. Assume your entire body rises 200 mm. Hint: When standing on your tiptoes, the force Ft in each of your Achilles tendons is typically 75% of your weight. Also, a typical spring constant for a human’s Achilles tendon is 35000 N/m!

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Explanation / Answer

given force exerted by achilees tendon = Ft, which increases greatly in magnitude

a. displacement of ankles, d = 200 mm = 0.2 m

as the achilles tendon applies force Ft on the heel

this force produces a moment about point A which is cancelling the moemnt produced by the center of gravity of the person and normal reaction of the floor

now, for every applied force, the muscle fibres change shape and burn energy

and hence for constant application fo force the person needs to burn calories and hence the person will not be able to conserve energy

b. every time a person gets on tip toes, teh heels move by 200 mm = d

Ft = 0.75mg

le tteh weight of the person be 80 kg

hence

Ft = 588.6 N

spring constant k = 35000 N/m^2

now every time the center of mass of the body rises by d = 0.2 m, energy consumed = mgd

E = 80*9.81*0.2 = 156.96 J

now 1 cal = 4184 J

hecne to burn 1 cal, the person has to tiptoe 4184/156.96 = 26.65 times

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