Statics of the Body

Abstract

The statics of the body are investigated by first reviewing forces, torques, and equilibrium, and then applying these conditions to statics in a plane and a lever. Specific examples of statics in the body are analyzed, including that of the lower arm, hips, shoulder, knee, ankle, jaws, and lower back. Some of these are related to injuries in the body, such as those in the hips and lower back. Several of the discussions in this chapter form the basis for analyzing motion in the next chapter.

Problems

Statics of the Arm and Levers

2.1

A person holds a weight with her arm extended horizontally, using her deltoid muscles to balance the weight about the shoulder joint. What type of lever is this? Sketch and label this lever.

2.2

A person holds himself up in a pushup position. Consider one of the person’s arms, with the pivot point being the hand on the ground. Are the biceps brachii or triceps brachii involved? What type of lever is this? Sketch and label this lever.

2.3

For each of the following, identify the type of lever, and show in a sketch the locations of the applied force, the fulcrum, and the load being applied:

(a) cutting with a pair of scissors

(b) lifting a wheelbarrow

(c) picking up something with a pair of tweezers.

2.4

Analyze the force balance in the x and y directions and the torque balance in the z-direction for the lower arm, with a vertical upper arm and a horizontal lower arm (length 35 cm) (Fig. 2.9). As with Case 1 in the text, ignore the forearm weight and other muscles. Now the effect of the reaction force $\mathbf {N}$ on the joint (with components $N_{x}$ and $N_{y}$) is explicitly included (because it contributes no torque it did not have to be included in the earlier Case 1 analysis) and assume the biceps are attached 4 cm from the pivot at an angle $\theta$ to the lower arm. With $\theta =75^{\circ }$, find the muscle force M and the magnitude of the reaction force in terms of the weight W and the angle of this reaction force relative to the x (horizontal) axis.

Statics of the Hip and Leg

2.5

Using Fig. 1.15, what can you say about the height of the person whose leg is depicted in Fig. 2.17?

2.6

Show how the dimensions given in Fig. 2.16 change to those in Fig. 2.20 when the person holds a cane.

2.7

In examining the hip forces during the equilibrium of a man standing on one foot, we assumed that the effective angle of the hip abductor muscles to the x-axis was the same, $70^{\circ }$, without or with a cane. Would using more realistic (and different) values of this angle without or with a cane, significantly affect the conclusions concerning the effect of using a cane?

Fig. 2.51

Forces on hip and femoral head while standing on one leg and lifting a weight with the opposite hand. (Reprinted from [15]. Used with permission of Elsevier.) For Problem 2.8

2.8

A 200 lb man stands on his right foot while carrying a 100 lb bag in his left hand. The center of mass of the bag is 12 in from his center of mass (see Fig. 2.51).

(a) Show that the placement of the foot (as shown) leads to no net torque in the body.

(b) Find the force (its magnitude and direction) on the head of the support femur and the force in the hip abductor muscle by examining the right leg.

(c) Compare your answers in parts (a) and (b) with what was found for the man holding no mass—without and with a cane (for $W_{\mathrm {b}}=880$ N $=200$ lb). Are the forces here greater than for a (200 lb $+$ 100 lb $=$) 300 lb man (with no cane). Why? (The muscle angle and leg mass are only trivially different for the problem given here and those analyzed above.)

2.9

Redraw Fig. 2.51, changing all distances and forces into metric units and then do Problem 2.8.

2.10

Calculate the force on the hip abductor muscles for a person standing symmetrically on two feet, as a function of foot separation. For what position is this force zero?

2.11

In the arabesque position an initially upright gymnast kicks one leg backward and upward while keeping it straight (mass $m_{\mathrm {leg}}$), pushes her torso and head forward (mass $m_{{\mathrm {torso}}+{\mathrm {head}}}$), and propels her arms backward and upward while keeping them straight (each $m_{\mathrm {arm}}$). They, respectively, make acute angles $\theta _{\mathrm {leg}}$, $\theta _{{\mathrm {torso}}+{\mathrm {head}}}$, and $\theta _{\mathrm {arms}}$, to the horizontal. The centers of mass of the extended leg and vertical balancing leg are $x_{\mathrm {extended\;leg}}$ and $x_{\mathrm {balancing\;leg}}$ behind the vertical from the center of mass (in the midsagittal plane), and that of her upper body (torso/head/arms combination) is $x_{\mathrm {upper\;body}}$ in front of this vertical (so all of these distances are defined as positive). In achieving this arabesque position her center of mass drops vertically from height $y_{\mathrm {before}}$ to $y_{\mathrm {after}}$.

(a) Draw a diagram showing the gymnast before and during this maneuver.

(b) Find the equilibrium condition in terms of these masses and distances. (Hint: Analyze the torques about her center of mass in the arabesque position. You can ignore the contribution from her arms. Why? You may not need all of the information that is presented.)

(c) Assume the gymnast is 1.49 m (4 ft 11 in) tall and has a mass of 38 kg (weight 84 lb) and her free leg and arms make a $30^{\circ }$ angle with the horizontal. What angle does her upper body make with the horizontal? (Assume the anthropometric relations for a standard human.)

2.12

Redo Problem 2.11c if the free leg of the gymnast is horizontal and her arms are vertical. (This is the cheerleading position.)

2.13

(a) Calculate the torque of the diver of mass $m_{\mathrm {b}}$ about the axis through her toes (normal to a sagittal plane) when she is on a diving board and leaning over and about to dive, so the vertical axis through her center of mass is a distance x in front of her toes.

(b) Now say that the diver has a height H and that her body is proportioned as per the data given in Chap. 1. Calculate the torque in terms of H, with her body straight with arms stretched parallel to her torso. Assume that her body can be straight (and so everything in her body can be approximated as being in one plane) and ignore the change in position because she is on her toes.

(c) Redo this if her arms are instead along her sides. How does this torque differ from that in part (b)?

2.14

The split Russel traction device is used to stabilize the leg, as depicted in Fig. 2.52, along with the relevant force diagram [28]. The leg is stabilized by two weights, $W_{1}$ and $W_{2}$, attached to the leg by two cables. The leg and cast have a combined weight of $W_{1}=300$ N and a center of mass 2/3 of the way from the left, as shown. The cable for $W_{2}$ makes an angle $\beta =45^{\circ }$ with the horizontal. For equilibrium, find the tension in the cables $T_{1}$ and $T_{2}$ and angle the cable for $W_{1}$ makes with the horizontal, $\alpha$.

Fig. 2.52

The split Russel traction device. (From [28]) For Problem 2.14

Statics of the Shoulder, Knee, Ankle, and Jaw

2.15

The clavicular portion of the pectoralis major muscle exerts a 224 N force at an angle 0.55 rad above the horizontal and the sternal portion of it exerts a 251 N force at an angle 0.35 rad below the horizontal, both in the plane containing the vertical humerus. Find the resulting magnitude of the total force and its direction, and sketch the three force vectors in relation to the humerus. (From [9].)

2.16

You are able to hold your arm in an outstretched position because of the deltoid muscle (Fig. 2.53). The force diagram for this is shown in Fig. 2.54. Use the three equilibrium conditions to determine the tension T in the deltoid muscle needed to achieve this equilibrium, and the vertical and horizontal components of the force exerted by the scapula (shoulder blade) on the humerus. Assume the weight of the humerus is $mg=8$ lb and the deltoid muscle makes an angle of $\alpha =17^{\circ }$ to the humerus. (From [3].)

Fig. 2.53

Deltoid muscle during lifting with an outstretched arm. (From [3].) For Problem 2.16

Fig. 2.54

Force diagram for the deltoid muscle and reaction forces during lifting with an outstretched arm. (This is a simpler version of Fig. 2.21.) (From [3].) For Problem 2.16

2.17

Solve the more general shoulder problem with a weight in the hand, depicted in Fig. 2.21, by finding M, R, and $\beta$. Now evaluate the x and y components of the muscle force, the magnitude of the joint reaction force , and its angle for the following parameters: $a=15$ cm, $b=30$ cm, $c=60$ cm, $\theta =15^{\circ }$, $W=40$ N, and $W_{0}=60$ N.

2.18

(a) A gymnast of mass $m_{\mathrm {b}}$ suspends himself on the rings with his body upright and straight arms that are horizontal with which he clutches the rings. Each ring is suspended by a rope with tension T that makes an acute angle $\theta$ with his arms, and the rings are separated by a distance d. Solve for the T and $\theta$. Assume symmetry.

(b) If the gymnast weighs 600 N, $\theta =75^{\circ }$, and $d=1.8$ m, find T. (From [14].)

2.19

Derive (2.34)–(2.36) for the equilibrium of the lower leg with an ankle weight.

2.20

Determine the angle between the leg and the reaction force at the knee for the conditions given in the text for Fig. 2.24.

2.21

Analyze how the patellar tendon and reaction forces depend on the ankle weight (for a fixed leg weight) and leg angle for Fig. 2.24. During exercise , what are the advantages of varying this weight versus this angle?

2.22

Consider the equilibrium of the foot during crouching for a 200-lb person, with the force through the Achilles tendon, the reaction force of the tibia, and the normal force from the floor in balance, as in Fig. 2.55—neglecting the weight of the foot for simplicity. Take the angle $\alpha =38^{\circ }$.

(a) Why is the normal force from the floor 100 lb?

(b) Find the magnitude of the Achilles tendon tension T and the magnitude and direction of the reaction force $\mathbf {F}$.

Fig. 2.55

Forces on the foot during crouching. (From [3].) For Problem 2.22

2.23

The topic of Problem 2.22 is similar to the discussion of forces on the foot in Chap. 3, where we will assume that all forces are parallel or antiparallel to each other and normal to a bar, as in a lever. Is this totally valid? Why or why not?

2.24

Redo Problem 2.22, now including the mass of the foot. The distance from the bottom of the tibia (from where the normal force emanates) is 4 in and the center of mass of the foot is halfway between it and the ground. Use the data for the mass of the foot in Chap. 1.

2.25

In the crouching position, the lower leg is held in equilibrium through the action of the patellar ligament, which is attached to the upper tibia and runs over the kneecap . As depicted in Fig. 2.56, the forces acting on the lower leg are $\mathbf {N}$, $\mathbf {R}$, and $\mathbf {T}$. If the lower leg is in equilibrium, determine the magnitude of the tension $\mathbf {T}$ in the patellar ligament, and the direction and magnitude of $\mathbf {R}$. Assume that the tension acts at a point directly below the point of action of $\mathbf {R}$. Take the normal force equal to 100 lb (half the body weight), the weight of the leg $W_{\mathrm {leg}}$ as 20 lb, and the angle $\alpha =40^{\circ }$ (for the leg at a $45^{\circ }$ angle). (From [3].)

Fig. 2.56

Forces on the lower leg during crouching. (From [3].) For Problem 2.25

2.26

Derive (2.37) and (2.38) for the equilibrium of the kneecap.

2.27

Derive (2.39) and (2.40) for the equilibrium of the foot.

2.28

The lateral distances from the temporomandibular joint to the insertion of the masseter muscles, the first bicuspids, and the central incisors are $0.4\,L$, L, and $1.2\,L$, respectively.

(a) What type of lever is involved in biting and chewing with the masseter muscles?

(b) For biting in equilibrium with a masseter muscle force of 1,625 N, show that the force on the first bicuspid of 650 N, assuming there is no force on the central incisors [5]. Draw a force diagram for this.

(c) Under these conditions, show that the needed counter force on the central incisors is 540 N, now assuming no force on the first bicuspid. Draw a force diagram for this.

2.29

When you bite an apple with your incisors only, you exert a force of 650 N on it. When you bite an apple with your bicuspids only, you exert a force of 540 N on it. Find the force per unit area (which is called the stress) on the apple for both cases if the effective contact areas of the incisors and bicuspids are 5 mm$^{2}$ and 1 mm$^{2}$, respectively.

Statics of the Back

2.30

In analyzing bending, we assumed that the weight of the trunk (above the hips, excluding arms and head) is $W_{1}=0.4W_{\mathrm {b}}$ and the weight of the arms and head is $W_{2}=0.2W_{\mathrm {b}}$. Is this reasonable. Why?

2.31

Consider a woman of height 1.6 m and mass 50 kg.

(a) Calculate the reaction force on her lower vertebrae and the force in her erector spinae muscle when she is either upright or bent at $60^{\circ }$ (and consequently $30^{\circ }$ to the horizontal).

(b) Recalculate these forces when she is pregnant . Assume that during pregnancy the mass of her torso increases by 15 kg, but the center of mass of the torso is the same.

(c) The forces in part (b) are equivalent to those for the same nonpregnant woman who lifts a weight of what mass?

2.32

Describe the designs of the back of a chair that could lead to pain in the lumbar vertebrae and those that would give good lumbar support.

2.33

We showed that when the force on the lumbosacral (intervertebral) disc increases from 0 to 2,400 N, the disc height H decreases by 20% and the disc radius r increases by about 10%. When it is recognized that the load on the disc for a vertical person is really 530 N (and not 0), how do the disc dimensions really change when the person bends to an angle of 30$^{\circ }$ (and then to a load of 2,400 N)?

2.34

Why is it more difficult to lift bulky objects? A person lifts a package of mass 20 kg in front of her so the back of the package touches her abdomen. The horizontal distance from the person’s lumbar-sacral disc to the front of her abdomen in 20 cm. Calculate the bending moments (in N-m) about the center of mass of her disc caused by the lifted loads, assuming the package is alternatively 20 or 40 cm deep [25]. Draw force diagrams for these two cases. The other dimensions of the packages are the same and they both have uniform density. How does this show that the size of the lifted object affects the load on the lumbar spine?

2.35

Why is it better to stand erect when you hold an object? A person holds a 20 kg object while either standing erect or bending over. The mass of the person above his lumbar-sacral disc (his torso) is 45 kg. When upright, the center of mass of the torso is (horizontally) 2 cm in front of his disc and that of the object is 30 cm in front of his disc. When bent, the center of mass of the torso is 25 cm in front of his disc and that of the object is 40 cm in front of his disc. Draw force diagrams for these two cases. Calculate the bending moments (in N-m) about the center of mass of his disc caused by holding this load while either being upright or bent over [25]. How does this show that bending when lifting an object affects the load on the lumbar spine?

2.36

Why is it best to lift an object with bent legs and the object very close to you? A person lifts a 20 kg object while either bending over with legs straight, with bent knees and the object near to her body, or with bent knees and the object far from her body. The mass of the person above her lumbar-sacral disc (her torso) is 45 kg. When bent over with straight legs, the center of mass of her torso is (horizontally) 25 cm in front of her disc and that of the object is 40 cm in front of her disc. When bent over with bent knees and the object near her body, the center of mass of her torso is (horizontally) 18 cm in front of her disc and that of the object is 35 cm in front of her disc. When bent over with bent knees and the object far from her body, the center of mass of her torso is (horizontally) 25 cm in front of her disc and that of the object is 50 cm in front of her disc. Draw force diagrams for these three cases. Calculate the bending moments (in N-m) about the center of mass of her disc for these three lifting methods [25]. Which position is the worst? How does this show that the position when lifting an object affects the load on the lumbar spine?

2.37

One position during shoveling snow or soil is shown in Fig. 2.57. Assume this is an equilibrium position.

(a) If the shovel and contents have a mass of 10 kg, with a center of mass 1 m from the lumbar vertebra, find the moment about that vertebra.

(b) If the back muscles are 5 cm behind the center of the disc, find the magnitude and direction of the muscle force needed for equilibrium.

(c) Find the force on the intervertebral disc.

(d) If the abdominal muscles of the person are strong and can provide some upward force, would that help relieve stress to the back muscles and the disc? Why?

Fig. 2.57

Shoveling . (From [11].) For Problem 2.37

Fig. 2.58

Idealized resultant forces for six muscle groups in the leg. (Based on [19].) For Problem 2.38

Multisegment Modeling

2.38

(a) Sketch an (in-plane) multisegment model of the leg showing the forces on the upper leg, lower leg, and ankle—using the resultant forces in Fig. 2.58. Show the center of mass gravity forces on each segment, along with the normal forces (at each body joint and with the floor).

(b) Label all distances and angles needed to analyze the in-plane forces and torques. For each segment, label the distances starting from the proximal end. Label each angle between muscle and bone; use the angles as shown, with the acute angle when possible.

(c) Write the equilibrium force balance and torque equation for each of the segments.

(d) For each of these equilibrium conditions, sum the equations—such as the torque equations—for the upper leg, lower leg, and foot. Show that the three resulting balance equations are the correct equations for the entire leg.

Sense of Touch

2.39

(a) A 50 kg person stands on her fingertips. Assuming each finger makes a 1 cm$^{2}$ contact area with the ground, find the pressure on each finger tip.

(b) Which tactile sensors are sensing this pressure?