Consider a system into which a small amount of heatQ flows. In many cases the temperature of the system rises [by an amount ΔT]… The heat capacityC of the system is defined as
C = Q/ΔT .
(3.39)
Heat capacity has units of J K-1. It depends on the size of the object and the substance it is made of. The specific heat capacity, c, is the heat capacity per unit mass (J K−1 kg−1).
The specific heat of air is 1006 J kg−1 K−1, a typical value for a gas… Water, however, … has a very high specific heat for a liquid… about 4200 J kg−1 K−1… It thus takes about four times as much heat to raise the temperature of a kilogram of water one degree as it does to raise the temperature of an equal mass of air.
A factor of four is significant, but frankly I would have expected an even bigger difference. The reason for the somewhat similar values of the specific heat capacity for air and water is that we are comparing heat capacity per unit mass. It may be more intuitive to compare heat capacity per unit volume. To convert from per kilogram to per cubic meter you must multiply the specific heat capacity per unit mass by the mass per unit volume, which is just the density, ρ. The density of air and water are very different. Air has a density of about 1.2 kg m−3, whereas water has a density of 1000 kg m−3.
We can express the specific heat capacity per unit volume as the product ctimes ρ. For air cρ is 1207 J K−1 m−3 but for water cρ is 4,200,000 J K−1 m−3. So water has a vastly higher specific heat capacity compared to air when expressed as per unit volume.
The relative similarity of specific heat between air and water can be misleading, however, because specific heat is measured on per-mass basis. One cubic meter of air weighs only about 1.2 kg, while a cubic meter of water weighs 1000 kg. It thus takes about 3500 times as much heat to raise the temperature of a given volume of water one degree as it does to raise the temperature of the same volume of air.
For similar volumes of air and water in thermal equilibrium, the heat stored in the air is negligible compared to that stored in the water. Biological tissue is mostly water, so that means air holds a lot less heat than tissue, per unit volume. This has implications for biological processes, such as heat exchange between air and tissue in the lungs.
The forces in the hip joint can be several times a person’s weight, and the use of a cane can be very effective in reducing them.
Indeed, a cane is very useful, as Russ and I show in Section 1.8 of IPMB. But what if you don’t have a cane handy, or if you prefer not to use one? You limp. In this post, we examine the biomechanics of limping.
When you limp, you lean toward the injured side to reduce the forces on the hip. The reader can analyze limping in this new homework problem.
Section 1.8
Problem 11 ½. The left side of the illustration below analyzes normal walking and reproduces Figures 1.11 and 1.12. The right side shows what happens when you walk with a limp. By leaning toward the injured side you reduce the distance between the hip joint and the body’s center of gravity, and your leg is more vertical than in the normal case.
Pertinent features of the anatomy of the leg: normal (left) and limping (right).
(a) Reproduce the analysis of Section 1.7 to calculate of the forces on the hip during normal walking using the illustration on the left. Begin by making a free-body diagram of the forces acting on the leg like in Figure 1.13. Then solve the three equations for equilibrium: one for the vertical forces, one for the horizontal forces, and one for the torques. Verify that the magnitude of the force on the hip joint is 2.4 times the weight of the body.
(b) Reanalyze the forces on the hip when limping. Use the geometry and data shown in the illustration on the right. Assume that any information missing from the diagram is the same as for the case of normal walking; For example, the abductor muscle makes an angle of 70° with the horizontal for both the normal and limping cases. Draw a free-body diagram and determine the magnitude of the force on the hip joint in terms of the weight of the body.
I won’t solve the entire problem for you, but I’ll tell you this: limping reduces the force on the hip from 2.4 times the body weight in the normal case to 1.2 times the body weight in the case of limping. No wonder we limp!
The main reason for the lower force when limping is the smaller moment arm. If we calculate torques about the head of the femur, then in the normal case the moment arm for the force that the ground exerts on the foot is 18 – 7 = 11 cm. When limping, this moment arm reduces to 9 – 7 = 2 cm. The moment arm for the abductor muscles (the gluteus minimus and gluteus medius) is the same in the two cases. Therefore, rotational equilibrium can be satisfied with a small muscle force when limping, although a large muscle force is required normally. The torque is a critical concept for understanding biomechanics.
What do you do if both hips are injured? When walking, you first lean to one side and then the other; you waddle. This reduces the forces on the hip, but results in a lot of swinging from side to side as you walk.
If you are having trouble solving this new homework problem, contact me and I’ll send you the solution.
Problem 5 ½. About 1 in every 2500 people is born with cystic fibrosis, an autosomal recessive disorder. What is the probability of the gene responsible for cystic fibrosis in the population? What fraction of the population are carriers of the disease?
To answer these questions, first we must know that an “autosomal recessive disorder” is one in which you only get the disease if you have two copies of a recessive gene. To a first approximation, there are often two variants (or alleles) of a gene governing a particular protein: dominant (A) and recessive (a). In order to have cystic fibrosis, you must have two copies of the recessive allele (aa). If you have only one copy (Aa), you are healthy but are a carrier for the disease: your children could potentially get the disease if your mate is also a carrier. If you have no copies of the recessive allele (AA) then you’re healthy and your children will also be healthy.
Let’s assume the probability of the dominant allele is p, and the probability of the recessive allele is q. Since we assume there are only two possibilities, we know that p + q = 1. Our goal is to find q, the probability of the gene responsible for cystic fibrosis in the population.
When two people mate, they each pass on to their offspring one of their two copies of the gene. The probability that both parents are dominant (AA), so the child is normal, is p2. The probability that both parents are recessive (aa), so the child has the disease, is q2. There are two ways for the child to be a carrier: A from dad and a from mom, or a from dad and A from mom. So, the probability of a child being a carrier (Aa) is 2pq. There are only three possibilities or genotypes: AA, Aa, and aa. The sum of their probabilities must equal one: p2 + 2pq + q2 = 1. But this expression is equivalent to (p + q)2 = 1, and we already knew that p + q = 1, so the result isn’t surprising.
The only people that suffer from cystic fibrosis have the genotype aa, so q2 is equal to the fraction of people with the disease. The problem states that this fraction is 1/2500 (0.04%). So, q is the square root of 1/2500, or 1/50 (2%; wasn’t that nice of me to make the fraction be the reciprocal of a perfect square?). One out of every fifty copies of the gene governing cystic fibrosis is defective (that is, it is the recessive version that can potentially lead to the disease). If q is 1/50, then p is 49/50 (98%). The fraction of carriers is 2pq, or 3.92%. The only reason this result is not exactly 4% is that we don’t count someone with the disease (aa) as a carrier, even though they couldpass the disease to their children (a carrier by definition has the genotype Aa).
If we are rounding off our result to the nearest percent, then 1 out of every 25 people (4% of the population) are carriers.
This calculation is based on several assumptions: no natural selection, no inbreeding, and no selection of embryos based on genetic testing. Cystic fibrosis is such a severe disease that often victims don’t survive long enough to have children (modern medicine is making this less true). The untreated disease is so lethal that one wonders why natural selection didn’t eliminate it from our gene pool long ago. One possible reason is that carriers of cystic fibrosis might be better able to resist other diseases—such as cholera, typhoid fever, or tuberculosis—than are normal people.
The most famous of medieval scientists was born in Somerset about 1214. We know that he lived till 1292, and that in 1267 he called himself an old man. He studied at Oxford under Grosseteste, and caught from the great polymath a fascination for science; already in that circle of Oxford Franciscans the English spirit of empiricism and utilitarianism was taking form. He went to Paris about 1240, but did not find there the stimulation that Oxford had given him…
Bacon is known for his support of the role of experiment in science. So much of medieval thought was based on religion and mysticism, and an emphasis on science and experiment is refreshing.
We must not think of him [Bacon] as a lone originator, a scientific voice crying out in the scholastic wilderness. In every field he was indebted to his predecessors, and his originality was the forceful summation of a long development. Alexander Neckham, Bartholomew the Englishman, Robert Grosseteste, and Adam Marsh had established a scientific tradition at Oxford; Bacon inherited it, and proclaimed it to the world. He acknowledged his indebtedness, and gave his predecessors unmeasured praise. He recognized also his debt—and the debt of Christendom—to Islamic science and philosophy, and through these to the Greeks…
Like Russ Hobbie and I, Bacon appreciated the role of math in science. Durant summarized Bacon’s view as “though science must use experiment as its method, it does not become fully scientific until it can reduce its conclusions to mathematical form.”
Bacon’s work on optics and vision overlaps with topics in IPMB. Durant notes that “one result of these studies in optics [performed by Bacon and others] was the invention of spectacles.” I can hardly think of a better example of physics interacting with physiology than eyeglasses. Durant concludes:
Experimenting with lenses and mirrors, Bacon sought to formulate the laws of refraction, reflection, magnification, and microscopy. Recalling the power of a convex lens to concentrate many rays of the sun at one burning point, and to spread the rays beyond that point to form a magnified image, he wrote:
We can so shape transparent bodies [lenses], and arrange them in such a way with respect to our sight and the objects of vision, that the rays will be refracted and bent in any direction we desire; and under any angle we wish we shall see the object near or at a distance. Thus from an incredible distance we might read the smallest letters…
These are brilliant passages. Almost every element in their theory can be found before Bacon, and above all in al-Haitham [an Arab scientist also known as Alhazen]; but the material was brought together in a practical and revolutionary vision that in time transformed the world. It was these passages that led Leonard Digges (d. c. 1571) to formulate the theory of which the telescope was invented.
I enjoy reading the Durants’ books. They contain not only the usual political and military history of the world, but also the history of science, art history, music history, comparative religion, linguistics, the history of medicine, philosophy, and literature. While The Story of Civilization may not be the definitive source on any of these topics, it is the best integration of all of them into one work that I am aware of. Had the Durants lived longer, future volumes (which they tentatively titled The Age of Darwin and The Age of Einstein) might have focused even more on the role of science in civilization.
I won’t finish The Story of Civilization anytime soon; I still have seven volumes to go. The series runs to over ten thousand pages, single-spaced, small font (I had to buy more powerful reading glasses for this project). I’ll continue to search for discussions of medical physics and biological physics throughout.
The Story of Civilization. 1. Our Oriental Heritage, 2. The Life of Greece, 3. Caesar and Christ, 4. The Age of Faith, 5. The Renaissance, 6. The Reformation, 7. The Age of Reason Begins, 8. The Age of Louis XIV, 9. The Age of Voltaire, 10. Rousseau and Revolution, and 11. The Age of Napoleon.
I am an emeritus professor of physics at Oakland University, and coauthor of the textbook Intermediate Physics for Medicine and Biology. The purpose of this blog is specifically to support and promote my textbook, and in general to illustrate applications of physics to medicine and biology.
