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9.7 Forces and Torques in Muscles and Joints

Muscles, bones, and joints are some of the most interesting applications of statics. There are some surprises. Muscles, for example, exert far greater forces than we might think. Figure shows a forearm holding a book and a schematic diagram of an analogous lever system. The schematic is a good approximation for the forearm, which looks more complicated than it is, and we can get some insight into the way typical muscle systems function by analyzing it.

Muscles can only contract, so they occur in pairs. In the arm, the biceps muscle is a flexor—that is, it closes the limb. The triceps muscle is an extensor that opens the limb. This configuration is typical of skeletal muscles, bones, and joints in humans and other vertebrates. Most skeletal muscles exert much larger forces within the body than the limbs apply to the outside world. The reason is clear once we realize that most muscles are attached to bones via tendons close to joints, causing these systems to have mechanical advantages much less than one. Viewing them as simple machines, the input force is much greater than the output force, as seen in Figure.

A forearm of a person holding a physics book is shown. The biceps and triceps muscles of the arm are visible. The elbow joint is the pivot point. The upper part of the arm is vertical and the lower part is horizontal. Biceps muscles are applying a force F B upward. The vertical bone of hand exerts a force F E on the pivot. At the midpoint of the lower part of the hand, the center of gravity of the hand is shown where the weight of the hand acts. The midpoint of the front face of the book is its center of gravity, where its weight acts downward. A free body diagram is also shown and the distances of the three forces F-B, C-G of arm, and C-G of book from the pivot are shown as r one, r two and r three.

Figure 9.7.1. (a) The figure shows the forearm of a person holding a book. The biceps exert a force \(F_B\) to support the weight of the forearm and the book. The triceps are assumed to be relaxed. (b) Here, you can view an approximately equivalent mechanical system with the pivot at the elbow joint as seen in Example.

Example: Muscles Exert Bigger Forces Than You Might Think

Calculate the force the biceps muscle must exert to hold the forearm and its load as shown in Figure, and compare this force with the weight of the forearm plus its load. You may take the data in the figure to be accurate to three significant figures.

Strategy

There are four forces acting on the forearm and its load (the system of interest). The magnitude of the force of the biceps is \(F_B\); that of the elbow joint is \(F_E\); that of the weights of the forearm is \(w_a\), and its loadis \(w_b\). Two of these are unknown \((F_B\) and \(F_E)\) so that the first condition for equilibrium cannot by itself yield \(F_B\). But if we use the second condition and choose the pivot to be at the elbow, then the torque due to \(F_B\) is zero, and the only unknown becomes \(F_B\).

Solution

The torques created by the weights are clockwise relative to the pivot, while the torque created by the biceps is counterclockwise; thus, the second condition for equilibrium \((net \space \tau = 0)\) becomes

\[r_2w_a + r_3w_b = r_1F_B.\]

Note that \(sin \space \theta = 1\) for all forces, since \(\theta = 90^o\) for all forces. This equation can easily be solved for \(F_B\) in terms of known quantities, yielding \[F_B = \dfrac{r_2w_a + r_3w_b}{r_1}.\]

Entering the known values gives

\[F_B = \dfrac{(0.160 \space m)(2.50 \space kg)(9.80 \space m/s^2) + (0.380 \space m)(4.00 \space kg)(9.80 \space m/s^2)}{0.0400 \space m}\]

which yields \[F_B = 470 \space N.\]

Now, the combined weight of the arm and its load is

\((6.50 \space kg)(9.80 \space m/s^2) = 63.7 \space N\), so that the ratio of the force exerted by the biceps to the total weight is

\[\dfrac{F_B}{w_a + w_b} = \dfrac{470}{63.7} = 7.38.\]

Discussion

This means that the biceps muscle is exerting a force 7.38 times the weight supported.

In the above example of the biceps muscle, the angle between the forearm and upper arm is 90°. If this angle changes, the force exerted by the biceps muscle also changes. In addition, the length of the biceps muscle changes. The force the biceps muscle can exert depends upon its length; it is smaller when it is shorter than when it is stretched.

Very large forces are also created in the joints. In the previous example, the downward force \(F_E\) exerted by the humerus at the elbow joint equals 407 N, or 6.38 times the total weight supported. (The calculation of \(F_E\) is straightforward and is left as an end-of-chapter problem.) Because muscles can contract, but not expand beyond their resting length, joints and muscles often exert forces that act in opposite directions and thus subtract. (In the above example, the upward force of the muscle minus the downward force of the joint equals the weight supported—that is, \(470 \space N - 470 \space N = 63 \space N\), approximately equal to the weight supported.) Forces in muscles and joints are largest when their load is a long distance from the joint, as the book is in the previous example.

In racquet sports such as tennis the constant extension of the arm during game play creates large forces in this way. The mass times the lever arm of a tennis racquet is an important factor, and many players use the heaviest racquet they can handle. It is no wonder that joint deterioration and damage to the tendons in the elbow, such as “tennis elbow,” can result from repetitive motion, undue torques, and possibly poor racquet selection in such sports. Various tried techniques for holding and using a racquet or bat or stick not only increases sporting prowess but can minimize fatigue and long-term damage to the body. For example, tennis balls correctly hit at the “sweet spot” on the racquet will result in little vibration or impact force being felt in the racquet and the body—less torque as explained in Collisions of Extended Bodies in Two Dimensions. Twisting the hand to provide top spin on the ball or using an extended rigid elbow in a backhand stroke can also aggravate the tendons in the elbow.

Training coaches and physical therapists use the knowledge of relationships between forces and torques in the treatment of muscles and joints. In physical therapy, an exercise routine can apply a particular force and torque which can, over a period of time, revive muscles and joints. Some exercises are designed to be carried out under water, because this requires greater forces to be exerted, further strengthening muscles. However, connecting tissues in the limbs, such as tendons and cartilage as well as joints are sometimes damaged by the large forces they carry. Often, this is due to accidents, but heavily muscled athletes, such as weightlifters, can tear muscles and connecting tissue through effort alone.

The back is considerably more complicated than the arm or leg, with various muscles and joints between vertebrae, all having mechanical advantages less than 1. Back muscles must, therefore, exert very large forces, which are borne by the spinal column. Discs crushed by mere exertion are very common. The jaw is somewhat exceptional—the masseter muscles that close the jaw have a mechanical advantage greater than 1 for the back teeth, allowing us to exert very large forces with them. A cause of stress headaches is persistent clenching of teeth where the sustained large force translates into fatigue in muscles around the skull.

Figure shows how bad posture causes back strain. In part (a), we see a person with good posture. Note that her upper body’s cg is directly above the pivot point in the hips, which in turn is directly above the base of support at her feet. Because of this, her upper body’s weight exerts no torque about the hips. The only force needed is a vertical force at the hips equal to the weight supported. No muscle action is required, since the bones are rigid and transmit this force from the floor. This is a position of unstable equilibrium, but only small forces are needed to bring the upper body back to vertical if it is slightly displaced. Bad posture is shown in part (b); we see that the upper body’s cg is in front of the pivot in the hips. This creates a clockwise torque around the hips that is counteracted by muscles in the lower back. These muscles must exert large forces, since they have typically small mechanical advantages. (In other words, the perpendicular lever arm for the muscles is much smaller than for the cg.) Poor posture can also cause muscle strain for people sitting at their desks using computers. Special chairs are available that allow the body’s CG to be more easily situated above the seat, to reduce back pain. Prolonged muscle action produces muscle strain. Note that the cg of the entire body is still directly above the base of support in part (b) of Figure. This is compulsory; otherwise the person would not be in equilibrium. We lean forward for the same reason when carrying a load on our backs, to the side when carrying a load in one arm, and backward when carrying a load in front of us, as seen in Figure.

In part a of the figure, a side view of a girl standing on a surface is shown. The weight of the girl is acting vertically downward and is in the line with her hips. A point above her legs is marked as the pivot point. The weight vector is in the direction of the pivot. In part b, a side view of a girl standing on a surface is shown. The girl is bending slightly toward her front. The weight of her upper body is acting downward and the line of action of weight is not passing through the upper body pivot point.

Figure 9.7.2. (a) Good posture places the upper body’s cg over the pivots in the hips, eliminating the need for muscle action to balance the body. (b) Poor posture requires exertion by the back muscles to counteract the clockwise torque produced around the pivot by the upper body’s weight. The back muscles have a small effective perpendicular lever arm, \(r_{b\perp}\), and must therefore exert a large force \(F_b\). Note that the legs lean backward to keep the cg of the entire body above the base of support in the feet.

You have probably been warned against lifting objects with your back. This action, even more than bad posture, can cause muscle strain and damage discs and vertebrae, since abnormally large forces are created in the back muscles and spine.

In image a, a man with a child on his shoulders is shown in which the child is holding the head of the man. The center of gravity is marked at the center of his body. In image b, a man with a long bag on his left shoulder and leaning toward the right is shown. The center of gravity is marked at the center of his body slightly left of the middle. In image c, a lady walking toward the right is shown. She is holding books in her hands. The center of gravity is marked at the center of her body above her legs.

Figure 9.7.3. People adjust their stance to maintain balance. (a) A father carrying his son piggyback leans forward to position their overall cg above the base of support at his feet. (b) A student carrying a shoulder bag leans to the side to keep the overall cg over his feet. (c) Another student carrying a load of books in her arms leans backward for the same reason.

Example: Do Not Lift with Your Back

Consider the person lifting a heavy box with his back, shown in Figure. (a) Calculate the magnitude of the force \(F_B\) - in the back muscles that is needed to support the upper body plus the box and compare this with his weight. The mass of the upper body is 55.0 kg and the mass of the box is 30.0 kg. (b) Calculate the magnitude and direction of the force \(F_V\) - exerted by the vertebrae on the spine at the indicated pivot point. Again, data in the figure may be taken to be accurate to three significant figures.

Strategy

By now, we sense that the second condition for equilibrium is a good place to start, and inspection of the known values confirms that it can be used to solve for \(F_B\) - if the pivot is chosen to be at the hips. The torques created by \(w_{ab}\) and \(w_{box}\) - are clockwise, while that created by \(F_B \) - is counterclockwise.

Solution for (a)

Using the perpendicular lever arms given in the figure, the second condition for equilibrium \((net \space \tau = 0)\) becomes

\[(0.350 \space m)(55.0 \space kg)(9.80 \space m/s^2) + (0.500 \space m)(30.0 \space kg)(9.80 \space m/s^2) = (0.0800 \space m)F_B \] Solving for \(F_B\) yields \[F_B = 4.20 \times 10^3 \space N.\]

The ratio of the force the back muscles exert to the weight of the upper body plus its load is

\[\dfrac{F_B}{w_{ub} + w_{box}} = \dfrac{4200 \space N}{833 \space N} = 5.04.\]

This force is considerably larger than it would be if the load were not present.

Solution for (b)

More important in terms of its damage potential is the force on the vertebrae \(F_V\) The first condition for equilibrium \((net \space F= 0)\) can be used to find its magnitude and direction. Using \(y\) fo4r vertical and \(x\) for horizontal, the condition for the net external forces along those axes to be zero

\[net \space F_y = 0 \space and \space net \space F_x = 0\]

Starting with the vertical \((y)\) components, this yields

\[F_{V_y} - w_{ub} - w_{box} - F_B \space sin \space 29^o = 0.\] Thus,

\[F_{V_y} = w_{ub} + w_{box} + F_B \space sin \space 29.0^o\]

\[= 833 \space N + (4200 \space N) \space sin \space 29.0^o\] yielding

\[F_{V_y} = 2.87 \times 10^3 \space N.\]

Similarly, for the horizontal \((x)\) components,

\[F_{V_x} - F_B \space cos \space 29.0^o = 0\] yielding

\[F_{V_x} = 3.67 \times 10^3 \space N.\]

The magnitude of \(F_V \) is given by the Pythagorean theorem:

\[F_V = \sqrt{F_{V_x}^2 + F_{V_Y}^2} = 4.66 \times 10^3 \space N.\]

The direction of \(F_V\) is

\[\theta = tan^{-1} \left(\dfrac{F_{V_y}}{F_{V_x}} \right) = 38.0^o.\]

Note that the ratio of \(F_V\) to the weight supported is \]\dfrac{F_V}{w_{ub} + w_{box}} = \dfrac{4660 \space N}{833 \space N} = 5.59.\]

Discussion

This force is about 5.6 times greater than it would be if the person were standing erect. The trouble with the back is not so much that the forces are large—because similar forces are created in our hips, knees, and ankles—but that our spines are relatively weak. Proper lifting, performed with the back erect and using the legs to raise the body and load, creates much smaller forces in the back—in this case, about 5.6 times smaller.

A man is bending forward to lift a box. The back muscles and vertebrae of the person are shown. The weight of the box is acting downward at its center of gravity. The vertebrae of the man are inclined vertical at sixty one degrees. A point on the joint of legs to the upper body is the pivot point. The distance between the center of gravity of the box and the pivot is fifty centimeters and perpendicular distance between the pivot and the weight of the man is thirty five centimeters.

 

 Figure 9.7.4. This figure shows that large forces are exerted by the back muscles and experienced in the vertebrae when a person lifts with their back, since these muscles have small effective perpendicular lever arms. The data shown here are analyzed in the preceding example, Example.

What are the benefits of having most skeletal muscles attached so close to joints? One advantage is speed because small muscle contractions can produce large movements of limbs in a short period of time. Other advantages are flexibility and agility, made possible by the large numbers of joints and the ranges over which they function. For example, it is difficult to imagine a system with biceps muscles attached at the wrist that would be capable of the broad range of movement we vertebrates possess.

There are some interesting complexities in real systems of muscles, bones, and joints. For instance, the pivot point in many joints changes location as the joint is flexed, so that the perpendicular lever arms and the mechanical advantage of the system change, too. Thus the force the biceps muscle must exert to hold up a book varies as the forearm is flexed. Similar mechanisms operate in the legs, which explain, for example, why there is less leg strain when a bicycle seat is set at the proper height. The methods employed in this section give a reasonable description of real systems provided enough is known about the dimensions of the system. There are many other interesting examples of force and torque in the body—a few of these are the subject of end-of-chapter problems.

Section Summary

  • Statics plays an important part in understanding everyday strains in our muscles and bones.
  • Many lever systems in the body have a mechanical advantage of significantly less than one, as many of our muscles are attached close to joints.
  • Someone with good posture stands or sits in such as way that their center of gravity lies directly above the pivot point in their hips, thereby avoiding back strain and damage to disks.

Conceptual Questions

Why are the forces exerted on the outside world by the limbs of our bodies usually much smaller than the forces exerted by muscles inside the body?

Explain why the forces in our joints are several times larger than the forces we exert on the outside world with our limbs. Can these forces be even greater than muscle forces?

Certain types of dinosaurs were bipedal (walked on two legs). What is a good reason that these creatures invariably had long tails if they had long necks?

Swimmers and athletes during competition need to go through certain postures at the beginning of the race. Consider the balance of the person and why start-offs are so important for races.

If the maximum force the biceps muscle can exert is 1000 N, can we pick up an object that weighs 1000 N? Explain your answer.

Suppose the biceps muscle was attached through tendons to the upper arm close to the elbow and the forearm near the wrist. What would be the advantages and disadvantages of this type of construction for the motion of the arm?

Explain one of the reasons why pregnant women often suffer from back strain late in their pregnancy.

Problems & Exercises

Verify that the force in the elbow joint in Example is 407 N, as stated in the text.