Question

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 in the back of the neck must therefore exert a force to keep it erect. that is why your head falls forward when you fall asleep in class. if the perpendicular distance between the line of action for the weight of the head and the pivot point is ( r_{wperp}=2.7 cm ) and the perpendicular distance between the line of action for the force the muscles exert on the head and the pivot point is ( r_{mperp}=5.1 cm ), determine each of the following. (assume the weight of the head is 50 n.)
(a) the force exerted by these muscles

(b) the force on the point of support

Explanation:

Step1: Use torque equilibrium

For torque equilibrium, \(\tau_{weight}=\tau_{muscle}\). Torque formula is \(\tau = r_{\perp}F\). So \(r_{w\perp}W=r_{M\perp}F_{M}\).

Step2: Solve for \(F_{M}\)

Rearrange the formula \(F_{M}=\frac{r_{w\perp}W}{r_{M\perp}}\). Substitute \(r_{w\perp} = 2.7\ cm\), \(W = 50\ N\), \(r_{M\perp}=5.1\ cm\).
\(F_{M}=\frac{2.7\times50}{5.1}\)

Step3: Use force equilibrium

In vertical direction, \(F_{support}=F_{M}+W\).

Answer:

(a) \(F_{M}=\frac{2.7\times50}{5.1}\approx26.5\ N\)
(b) \(F_{support}=26.5 + 50=76.5\ N\)