Question

In a simplified version of musculature action in leg raises, the abdominal muscles pull on the femur (thigh bone) to raise the leg by pivoting it about one end. When you are lying horizontally, these muscles make an angle of approximately with the femur, and if you raise your legs, the muscles remain approximately horizontal, so the angle increases. Assume for simplicity that these muscles attach to the femur in just one place, 10 cm from the hip joint (although in reality the situation is more complicated). For a certain 80-kg person having a leg 90 cm long, the mass of the leg is 15 kg and its center of mass is 44 cm from the his hip joint as measured along the leg. If the person raises his leg above the horizontal, the angle between the abdominal muscles and his femur would also be about .

1. With his leg raised to , find the tension in the abdominal muscle on each leg. Draw a free body diagram.
2. When is the tension in this muscle greater: when the leg is raised to or when the person just starts to raise it off the ground? Why? (Try this yourself.)
3. If the abdominal muscles attached to the femur were perfectly horizontal when a person was lying down, could the person raise his leg? Why or why not?

Short answer

1. The tension is 370 N.
2. The tension is greater when he just starts to raise his leg off the ground.
3. The person could not raise his leg.

Step-by-step solution

The given data

Given that when you are lying horizontally, these muscles make an angle of approximately with the femur, and if you raise your legs, the muscles remain approximately horizontal, so the angle increases. Assume for simplicity that these muscles attach to the femur in just one place, 10 cm from the hip joint. For a certain 80-kg person having a leg 90 cm long, the mass of the leg is 15 kg and its center of mass is 44 cm from the his hip joint as measured along the leg. If the person raises his leg above the horizontal, the angle between the abdominal muscles and his femur would also be about $60°$.

Mass of person, m = 80 kg

Length of leg L = 40 m

Formula used

Torque $\tau =FI$

Where Fis force exerted and l is distance.

(a)Step 3: Find tension in the abdominal muscle

Applying condition of equilibrium,

Net torque is zero.

$\sum _{}\tau =0$

This gives,

$$\begin{gathered} T=\frac{4.4\left(1.5\mathbf{kg}\right)\left(9.8\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}\right)}{\tan \theta } \\ Put\theta =60° \\ T=\frac{4.4\left(1.5\mathbf{kg}\right)\left(9.8\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}\right)}{60°}=370\mathbf{N} \end{gathered}$$

(b)Step 4: Find when the tension is greater

For $\theta =60°$

Tension in the muscle is 370 N

When theperson just starts to raise it off the ground .

Then tension is$T=\frac{4.4\left(1.5\mathbf{kg}\right)\left(9.8\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}\right)}{\tan 5°}=7400\mathbf{N}$

Hence, the tension is much greater when the person just starts to raise his leg off the ground.

(c)Step 5: Find if person could raise a leg if femur was perfectly horizontal

Tension in the leg is inversely proportional to 0, so if$\theta$ approaches zero, then tension will approach to infinity.

If the leg is horizontal then$\theta =0$ , the moment arm for tension is zero, so there is no torque to rotate the leg.

Hence, the person could not raise his leg.