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 musttherefore 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 thepivot point is rw = 2.4 cm and the perpendicular distance between the line of action for theforce the muscles exert on the head and the pivot point is rMi=5.1 cm. determine each or thefollowina. (Assume the weight of the head is 50 N.)

We are asked to determine the force required by the neck muscle in order to keep the head in equilibrium. To do that we will add the torques produced by the muscle force and the weight of the head. We will use torque in the clockwise direction to be negative, therefore, we have:

[tex]\Sigma T=r_{M\perp}(F_M)-r_{W\perp}(W)[/tex]

Since we want to determine the forces when the system is at equilibrium this means that the total sum of torque is zero:

[tex]r_{M\perp}(F_M)-r_{W\perp}(W)=0[/tex]

Now, we solve for the force of the muscle. First, we add the torque of the weight to both sides:

[tex]r_{M\perp}(F_M)=r_{W\perp}(W)[/tex]

Now, we divide by the distance of the muscle:

[tex](F_M)=\frac{r_{W\perp}(W)}{r_{M\perp}}[/tex]

Now, we substitute the values:

[tex]F_M=\frac{(2.4cm)(50N)}{5.1cm}[/tex]

Now, we solve the operations:

[tex]F_M=23.53N[/tex]

Therefore, the force exerted by the muscles is 23.53 Newtons.

Part B. To determine the force on the pivot we will add the forces we add the vertical forces:

[tex]\Sigma F_v=F_j-F_M-W[/tex]

Since there is no vertical movement the sum of vertical forces is zero:

[tex]F_j-F_M-W=0[/tex]

Now, we add the force of the muscle and the weight to both sides to solve for the force on the pivot:

[tex]F_j=F_M+W[/tex]

Now, we plug in the values:

[tex]F_j=23.53N+50N[/tex]

Solving the operations:

[tex]F_j=73.53N[/tex]

Therefore, the force is 73.53 Newtons.