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 atlant-ooccipital 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 exerted by these muscles using the information in the figure.

What figure

To calculate the force exerted by the muscles at the back of the neck, we need to consider the torque acting on the head.

Torque is the product of force and the distance between the force and the axis of rotation. In this case, the axis of rotation is the atlanto-occipital joint, which is the point where the head pivots on the neck.

The figure does not provide specific distances, but we can estimate the distance between the atlanto-occipital joint and the center of mass of the head, since it is not directly over the joint.

Let's assume the distance between the atlanto-occipital joint and the center of mass of the head is d. We'll also need to know the weight of the head, which we'll represent as W.

The torque acting on the head is given by the formula: Torque = Force x Distance

In this case, the torque required to keep the head erect is the torque exerted by the muscles at the back of the neck.

Since the head is in equilibrium (not rotating), the torque exerted by the muscles must be equal in magnitude but opposite in direction to the torque exerted by the weight of the head.

So, the torque exerted by the weight of the head is given by: Torque_weight = W x d

The torque exerted by the muscles is equal in magnitude but opposite in direction, so: Torque_muscles = - W x d

We can rearrange the equation to find the force exerted by the muscles: Force_muscles = Torque_muscles / d

Substituting the negative torque and dividing by the distance, we have: Force_muscles = (-W x d) / d

Simplifying, we get: Force_muscles = -W

Therefore, the force exerted by the muscles at the back of the neck is equal in magnitude but opposite in direction to the weight of the head.