Asked by j
Even when the head is held erect, as in the figure below, its center of mass is not directly over the principal point of support (the atlanto-occipital joint).The muscles at the back of the neck should therefore exert a force to keep the head erect. That is why your head falls forward when you fall asleep in the class.
Calculate the force (in N) exerted by these muscles. (Assume
w = 48 N, r1 = 4.8 cm, and r2 = 2.0 cm.)
What is the force (in N) exerted by the pivot on the head?
All Answers 1
Answered by
GPT-5 mini
AI
Take moments about the atlanto‑occipital joint for rotational equilibrium:
F_muscle · r2 = w · r1
F_muscle = w r1 / r2 = 48 N · (4.8 cm / 2.0 cm) = 48 · 2.4 = 115.2 N ≈ 1.15×10^2 N.
Vertical force balance: F_muscle + F_pivot_y − w = 0 ⇒ F_pivot_y = w − F_muscle = 48 − 115.2 = −67.2 N.
So the neck muscles exert about 1.15×10^2 N (115 N). The pivot exerts a vertical reaction of 67.2 N directed downward (magnitude ≈ 67 N).
F_muscle · r2 = w · r1
F_muscle = w r1 / r2 = 48 N · (4.8 cm / 2.0 cm) = 48 · 2.4 = 115.2 N ≈ 1.15×10^2 N.
Vertical force balance: F_muscle + F_pivot_y − w = 0 ⇒ F_pivot_y = w − F_muscle = 48 − 115.2 = −67.2 N.
So the neck muscles exert about 1.15×10^2 N (115 N). The pivot exerts a vertical reaction of 67.2 N directed downward (magnitude ≈ 67 N).
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