Question

Question: Even when the head is held erect, as inFigure 9.41, its center of mass is not directly over the principal point of support (the atlantooccipital 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. (a) Calculate the force exerted by these muscles using the information in the figure. (b) What is the force exerted by the pivot on the head?

Short answer

(a) The force exerted by the muscles is 25N.

(b) The force exerted by the pivot on the head muscles is 75N upward.

Step-by-step solution

Force

A force is an external factor that can change the rest or motion of a body. It has a size and a general direction.

Diagram

The diagram for the biceps muscle is shown below:

The weight is w = 50N.

Calculation of the force

(a)

As the force applied by the biceps muscle is perpendicular to the forearm, applying the condition of equilibrium under torque, we can write,

$\begin{array}{rcl}{F}_M\left(0.050\right)-w\left(0.025\right)& =& 0\\ F_M& =& \frac{w\left(0.025\right)}{\left(0.050\right)}\\ & =& \frac{50\text{N}\times 0.025}{0.050}\\ & =& 25\text{N}\end{array}$

Hence, the force is 25N.

Calculation of the force by the pivot

(b)

Applying the condition of equilibrium under force, we can write,

$$\begin{gathered} F_J-F_M-w=0 \\ {F}_J=F_M+w \\ {F}_J=25+50 \\ {F}_J=75\text{N} \end{gathered}$$

Hence, the force by the pivot is 75N.