The Bowling Ball and the Feather

Dropping a feather and a ball in a vacuum to show that they fall at the same rate is a classic physics demonstration. We have a version of this demo at Oakland University, but it is not very effective. A small ball and a feather are in a tube about 1 meter long and a few centimeters in diameter. We have vacuum pump to remove the air, but it is difficult to see the objects from the back of the room, and often they bump into the wall of the tube, slowing them down. I have never found it useful. Yet, the physical principle being demonstrated is fundamental. The gravitational mass in Newton’s universal law of gravity and the inertial mass in Newton’s second law of motion cancel out, so that all objects fall downward with acceleration g = 9.8 m/s².

This result is unexpected because in everyday life we experience air friction. When you include air friction, objects do not all fall at the same rate. Russ Hobbie and I illustrate this point in Problem 28 of Chapter 2 in the 4th edition of Intermediate Physics for Medicine and Biology.

Problem 28 When an animal of mass m falls in air, two forces act on it: gravity, mg, and a force due to air friction. Assume that the frictional force is proportional to the speed v.

(a) Write a differential equation for v based on Newton’s second law, F = m(dv/dt).

(b) Solve this differential equation. (Hint: Compare your equation with Eq. 2.24.)

(c) Assume that the animal is spherical, with radius a and density ρ. Also, assume that the frictional force is proportional to the surface area of the animal. Determine the terminal speed (speed of descent in steady state) as a function of a.

(d) Use your result in part (c) to interpret the following quote by J. B. S. Haldane [1985]: “You can drop a mouse down a thousand-yard mine shaft; and arriving at the bottom, it gets a slight shock and walks away. A rat is killed, a man is broken, a horse splashes.”

If we ignore air fraction, v = gt; the acceleration is g and does not depend on mass. With air friction, objects reach a terminal velocity that depends on their mass. We are all so used to seeing a feather float downward with its motion dominated by air friction that it is difficult to believe it could ever fall as fast as a ball. To persuade students that this behavior does indeed happen, to convince them that in a vacuum a feather drops like a rock, we need a powerful demonstration. The result is so significant, and so nonintuitive, that the demo must be dramatic and memorable.

Now we have it. Watch this amazing video with British Physics Professor Brian Cox. He found the biggest vacuum chamber in the world—a large room used by NASA to test space vehicles—and inside it he dropped a bowling ball and a feather simultaneously from the same height. When the room was filled with air, the feather slowly fluttered to the ground. When the room was evacuated, the feather stayed right beside the bowling ball all the way down. The visual effect is stunning. Cox has a fine sense of drama, building the tension until the final sensational experiment. The video is less than five minutes long. You’ve gotta see it.

The MCAT and IPMB

The Medical College Admission Test, famously known as the MCAT, is an exam taken by students applying to medical school. The Association of American Medical Colleges will introduce a new version of the MCAT next year, focusing on competencies rather than on prerequisite classes. How well does the 4th edition of Intermediate Physics for Medicine and Biology prepare premed students for the MCAT?

The new MCAT will be divided into four sections, and the one most closely related to IPMB deals with the chemical and physical foundations of biological systems. Within that section are two foundational concepts, of which one is about how “complex living organisms transport materials, sense their environment, process signals, and respond to changes that can be understood in terms of physical principles.” This concept is further subdivided into five categories. Below, I review the topics included in these categories and indicate what chapter in IPMB addresses each.

MCAT: Translational motion, forces, work, energy, and equilibrium in living systems

IPMB: Chapter 1 discusses mechanics, including forces and torques, with applications to biomechanics. Work and energy are introduced in Chapter 1, and analyzed in more detail in Chapter 3 on statistical mechanics and thermodynamics (parts of thermodynamics are included under another foundational concept dealing mostly with chemistry). Periodic motion is covered in Chapter 11, which discusses the amplitude, frequency and phase of an oscillator. Waves are analyzed in Chapter 13 about sound and ultrasound.

MCAT: Importance of fluids for the circulation of blood, gas movement, and gas exchange

IPMB: Chapter 1 analyzes fluids, including buoyancy, hydrostatic pressure, viscosity, Poiseuille flow, turbulence, and the circulatory system. Much of this material is not covered in a typical introductory physics class. Chapter 3 introduces absolute temperature, the ideal gas law, heat capacity, and Boltzmann’s constant.

MCAT: Electrochemistry and electrical circuits and their elements

IPMB: Chapters 6 and 7 cover electrostatics, including charge, the electric field, current, voltage, Ohm’s law, resistors, capacitors, and nerve conduction. Chapter 8 discusses the magnetic field and magnetic forces.

MCAT: How light and sound interact with matter

IPMB: Sound is analyzed in Chapter 13, including the speed of sound, the decibel, attenuation, reflection, the Doppler effect, ultrasound, and the ear. Chapter 14 covers light, photon energy, color, interference, and the eye. This chapter also describes absorption of light in the infrared, visible, and ultraviolet. Chapter 18 analyzes nuclear magnetic resonance.

MCAT: Atoms, nuclear decay, electronic structure, and atomic chemical behavior

IPMB: Chapter 17 is about nuclear physics and nuclear medicine, covering isotopes, radioactive decay, and half life. Atoms and atomic energy levels are explained in Chapter 14.

MCAT: General mathematical concepts and techniques

IPMB: Chapter 1 and many other chapters require students to estimate numerically. Chapter 2 covers linear, semilog, and log-log plots, and exponential growth. Metric units and dimensional analysis are used everywhere. Probability concepts are discussed in Chapter 3 and other chapters. Basic math skills such as exponentials, logarithms, scientific notation, trigonometry, and vectors are reinforced throughout the book and in the homework problems, and are reviewed in the Appendices.

The MCAT section about biological and biochemical foundations of living systems includes diffusion and osmosis (discussed in Chapters 4 and 5 of IPMB), membrane ion channels (covered in Chapter 9), and feedback regulation (analyzed in Chapter 10).

Overall, Intermediate Physics for Medicine and Biology covers many of the topics tested on the MCAT. A biological or medical physics class based on IPMB would prepare a student for the exam, and would reinforce problem solving skills and teach the physical principles underlying medicine, resulting in better physicians.

I’m a realist, however. I know premed students take lots of classes, and they don’t want to take more physics beyond a two-semester introduction, especially if the class might lower their grade point average. I have tried to recruit premed students into my Biological Physics (PHY 325) and Medical Physics (PHY 326) classes here at Oakland University, with little success. Perhaps if they realized how closely the topics and skills required for the MCAT correspond to those covered by Intermediate Physics for Medicine and Biology they would reconsider.

To learn more about how to prepare for the physics competencies on the MCAT, see Robert Hilborn’s article “Physics and the Revised Medical College Admission Test,” published in the American Journal of Physics last summer (Volume 82, Pages 428–433, 2014).

Faraday, Maxwell, and the Electromagnetic Field

Faraday, Maxwell, and the
Electromagnetic Field,
by Nancy Forbes and Basil Mahon.

Michael Faraday and James Clerk Maxwell are two of my scientific heroes. So, when I saw the book Faraday, Maxwell, and the Electromagnetic Field displayed in the new book section of the Rochester Hills Public Library, I had to check it out. In their introduction, Nancy Forbes and Basil Mahon write

It is almost impossible to overstate the scale of Faraday and Maxwell’s achievement in bringing the concept of the electromagnetic field into human thought. It united electricity, magnetism, and light into a single, compact theory; changed our way of life by bringing us radio, television, radar, satellite navigation, and mobile phones; inspired Einstein’s special theory of relativity; and introduced the idea of field equations, which became the standard form used by today’s physicists to model what goes on in the vastness of space and inside atoms.

I have read previous biographies of both Faraday and Maxwell, so their story was familiar to me. But one anecdote about Faraday I had never heard before.

The Royal Institution’s Friday Evening Discourses had by now become an institution in their own right. The lecture on April 3, 1846, turned out to be a historic occasion, although none of the audience recognized it as such and the whole thing happened by chance in a rather bizarre fashion. Charles Wheatstone was to have been the latest in a long line of distinguished speakers, but he panicked and ran away just as he was due to make his entrance[!]. Although amply confident in his professional dealings as a scientist, inventor, and businessman, Wheatstone was notoriously shy of speaking in public, and Faraday had taken a gamble when engaging him to talk about his latest invention, the electromagnetic chronoscope—a device for measuring small time intervals, like the duration of a spark. The gamble had failed, and Faraday was left with the choice of sending disappointed customers home or giving the talk himself. He chose to talk, but he ran out of things to say on the advertised topic well before the allotted hour was up.

Caught off-guard, he did what he had never done before and gave the audience a glimpse into his private meditations on matter, lines of force, and light. In doing so, he draw an extraordinary prescient outline of the electromagnetic theory of light, as it would be developed over the next sixty years….

Writing about Maxwell’s electromagnetic theory, Forbes and Mahon say

The theory’s construction had been an immense creative effort, sustained over a decade and inspired, from first to last, by the work of Michael Faraday. Thanks to Faraday’s meticulous recording of his findings and thoughts in his Experimental Researches in Electricity, Maxwell had been able to see the world as Faraday did, and, by bringing together Faraday’s vision with the power of Newtonian mathematics, to give us a new concept of physical reality, using the power of mathematics. But mathematics would not have been enough without Maxwell’s own near-miraculous intuition; witness the displacement current, which gave the theory its wonderful completeness. The theory belongs to both Maxwell and Faraday.

Russ Hobbie and I discuss electricity and magnetism in the 4th edition of Intermediate Physics for Medicine and Biology. Chapters 6 and 7 show how electrostatics can be used to describe how nerves and muscles behave. Chapter 8 discusses magnetism and electromagnetism. Chapter 9 examines in more detail how electromagnetic fields interact with the body, and Chapter 18 describes how magnetism leads to magnetic resonance imaging. So, it’s safe to say that IPMB has Maxwell and Faraday’s influence throughout.

If you want to learn more about Maxwell’s work, I suggest Maxwell on the Electromagnetic Field: A Guided Study by Thomas K. Simpson. He reproduces Maxwell’s three landmark papers, and provides the necessary context and background to understand them. Forbes and Mahon talk briefly at the end of their book about the scientists who came after Maxwell and firmly established his theory. For more on this topic, read The Maxwellians, one of the best histories of science I know. I enjoyed Faraday, Maxwell, and the Electromagnetic Field. It provides a great introduction to a fascinating story in the history of science.

Low Reynolds Number Flows

In Chapter 1 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the Reynolds number.

The importance of turbulence (nonlaminar) flow is determined by a dimensionless number characteristic of the system called the Reynolds number N_R. It is defined by N_R = LVρ/η where L is a length characteristic of the problem, V a velocity characteristic of the problem, ρ the density, and η the viscosity of the fluid. When N_R is greater than a few thousand, turbulence usually occurs….

When N_R is large, inertial effects are important. External forces accelerate the fluid. This happens when the mass is large and the viscosity is small. As the viscosity increases (for fixed L, V , and ρ) the Reynolds number decreases. When the Reynolds number is small, viscous effects are important. The fluid is not accelerated, and external forces that cause the flow are balanced by viscous forces. Since viscosity is a form of internal friction in the fluid, work done on the system by the external forces is transformed into thermal energy. The low-Reynolds-number regime is so different from our everyday experience that the effects often seem counterintuitive. They are nicely described by Purcell (1977).

“Life at Low Reynolds Number,”
by Edward Purcell.

Edward Purcell’s 1977 paper in the American Journal of Physics provides much insight into low Reynolds number flow, and is a classic. But to learn from this paper you have to read it. Nowadays, students often want to learn from videos rather than reading text (don’t get me started...). Fortunately, a good video exists to explain low-Reynolds-number flow, and it has been around for many years. Click here to watch G. I. Taylor illustrate low Reynolds flow. Sir Geoffrey Ingram Taylor (1886–1975) was an English physicist and an expert in fluid dynamics. He contributed to the Manhattan Project by analyzing the hydrodynamics of implosion needed to develop a plutonium bomb. Among his many contributions is the description of Taylor-Couette flow between two rotating cylinders.

The video shows a beautiful example of reversibility of low Reynolds number flow. A blob of dye is placed into the fluid between the cylinders, one of the cylinders is rotated, and the dye spreads throughout the fluid. They rotation is then reversed, and the dye eventually returns to its original localized blob. This demonstration always reminds me of the formation of a spin echo during magnetic resonance imaging (see Chapter 18 of IPMB), where all spins begin in phase after a 90 degree radio-frequency magnetic field pulse. Then, because of slight heterogeneities in the static magnetic field, the spins dephase as they all rotate at slightly different Larmor frequencies. If you reverse their positions using a 180 degree RF pulse, the spins eventually return to their original configuration, all in phase (the echo). When you think about it, the formation of spin echoes during MRI is nearly as “magical” as the reformation of the dye blob in Taylor’s cylinder.

The video also analyzes how small machines can “swim” at low Reynolds numbers. He even has built small devices, one a machine that zooms through water but just sits there in a viscous fluid, and another that has a helix for a tail and that swims—slowly but steadily—through the viscous fluid. This example reminds me of both Purcell’s article and the research of Howard Berg, who studies how E coli bacteria swim.

To learn more about Taylor’s life and work, watch Katepalli Sreenivasan’s lecture, or read The Life and Legacy of G. I. Taylor by George Batchelor.