A student bends her head at 40.0 from the vertical while intently reading her ph

Question

A student bends her head at 40.0 from the vertical while intently reading her physics book, pivoting the head around the upper vertebra (point P in the figure below(Figure 1) ). Her head has a mass of m = 4.70 kg (which is typical), and its center of mass is 11.0 cm from the pivot point P. Her neck muscles are x = 1.55 cm from point P, as measured perpendicular to these muscles. The neck itself and the vertebrae are held vertical.

Find the tension in her neck muscles.

T=

How do I set up the formula in regards to a free body diagram?

Explanation / Answer

Force of Gravity Fg = m*g
Force of Gravity Fg = (4.7 kg) * (9.8 m/s²) = 46.06 N

Torque () on pivot point P, due to force of gravity =

Force * Perpendicular distance from the center of mass to the pivot

= F*r*sin() = (46.06 N) * (0.11 m) * sin(40°) = 3.256 N*m

This is the torque that the girl's neck muscles balance. Assuming that the neck muscles apply this force directly perpendicular to the 40° angle of the neck.

Force that Girl's Neck Muscles Apply =

= F*r*sin()
3.256 n*m = F*(0.0155 m)*sin(90) = 210.11 N

The neck muscles maintain a constant tension of 210.11 N , which is spread across the area of the muscle.

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