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Physics For The Life Sciences , 2nd UK Ed. Edition Class Notes

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    1. PHYSICS AND THE LIFE SCIENCES IF NOTHING ELSE, MY STUDENTS SHOULD LEARN… 1. A physical model of a system is a description focusing on its most significant properties. The description is built using observable phenomena, has a minimum number of assumptions, and should have predictive power. 2. The accuracy of a measurement refers to how close it is to the accepted or correct value of the parameter being measured. The precision of a measurement refers to how repeatable the measurement is, in other words how close different measurements of the same parameter conducted under identical conditions, are to each other. In general, the accuracy and precision of a set of measurements are independent of each other. 3. An order of magnitude refers to a rough numerical estimate of a physical quantity which is accurate to within a factor of 10, or to within an order of magnitude. 4. Dimensional analysis is a mathematical procedure in which information about relevant physical variables in an equation can be obtained through manipulation, by equating the physical dimensions of both sides of an equation. 5. Scientific notation is a method of expressing very large or small numbers in a convenient and compact fashion, also displaying all the relevant significant figures. It is very useful when making order of magnitude estimates. 6. Special mathematical functions such as the trigonometric functions, the exponential and logarithmic functions, the square root function, and others, must always have dimensionless arguments. LEARNING OBJECTIVES Students should be able to:  Describe what a physical model is. (understand)  Describe the characteristics that a good physical model must possess. (understand)  Describe how significant figures are used in basic mathematical operations. (understand)

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    Chapter 1 Physics for the Life Sciences 2  Explain the concept of accuracy. (understand)  Explain the concept of precision. (understand)  Explain the difference between the accuracy and precision of a set of physical measurements. (understand)  Be able to do order of magnitude estimates. (analyze)  Be able to do basic mathematical operations using scientific notation. (analyze)  Be able to explain how scientific notation is used in order of magnitude estimates. (understand)  Have an understanding of - and be able to do mathematical operations in - the metric system. (understand)  Be able to convert units in the metric system. (understand)  Be able to do dimensional analysis to extract useful information from an equation relating relevant physical parameters. (understand)  Be able to recognize when to use special mathematical functions such as trigonometric, exponential, and logarithmic functions. (evaluate) WHY IS THIS CHAPTER IMPORTANT TO SCIENTISTS?  In this chapter, the fundamental ideas of the scientific method are presented. In particular, the concept of a physical model is introduced. The advantages of physical models are explained, as well as the conditions that all good physical models must satisfy.  The concepts of accuracy and precision in physical measurement are introduced, as well as their importance, and their very different natures.  The power and efficiency of order of magnitude estimates are introduced and illustrated with relevant examples. A close connection between this tool and the scientific notation is established.  The metric, or SI system is introduced, and its power and elegance in the physical sciences are explained with relevant examples.

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    Physics for the Life Sciences Chapter 1 3  The deep implications and simplicity of the method of dimensional analysis are introduced. Emphasis is placed on the idea that the method can produce crude mathematical approximations of relevant physical parameters with little if any understanding of the basic physics of a given problem.  The concepts of conversions between units in the metric system - and between systems - are introduced. The sometimes catastrophic real consequences of improper unit conversions are analyzed with relevant examples.  A thorough introduction to the mechanics of significant figures is introduced, along with the motivation behind their use. Many practical examples are presented. WHY SHOULD STUDENTS CARE?  In this chapter, we introduce fundamental concepts needed for all of the sciences, such as the metric system, dimensional analysis, significant figures, accuracy and precision, etc.  These tools allow the student to get a feel for what the solution to a problem should look like before, or without, solving the problem. For example, an order of magnitude estimate can give one a numerical answer which is accurate to within a factor of 10, all with minimal effort.  Another important example of a labor saving tool is dimensional analysis. This technique can give the student a rough idea of how the relevant physical parameters fit into an important equation with very little if any knowledge of the underlying physics. The method does have limitations, for example it cannot give information on constants of proportionality.  Unit conversions within the metric system, as well as within different systems (i.e. metric vs. imperial) are introduced in this chapter. Examples of the sometimes catastrophic consequences of improper unit conversions are presented so students can begin to appreciate the potential implications of doing this incorrectly.

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    Chapter 1 Physics for the Life Sciences 4 WHAT ARE COMMON STUDENT MISCONCEPTIONS/STUMBLING BLOCKS? 1. Students typically consider – incorrectly – that a physical model of a system under study is a better model to the extent that it is more complicated and makes more, rather than fewer assumptions. In fact, the more complicated the model, the more parameters or factors to keep track of. The difficulty here is that in an attempt to take into account these additional parameters, the few really relevant ones can be relegated. Therefore it is important to stress that good physical models are simple, they make few assumptions, and they focus only on a few relevant parameters while ignoring the rest. 2. Students are notorious for confusing mathematical operations involving significant figures. The typical example goes something like this: the student is asked to find the area of a square having a measured side of 7.23 m. They proceed to correctly square the side and merrily report the area of the square to be 52.279 m 2 , when of course the correct answer is 52.3 m 2 ! In this example, the students made two errors, first their answer had more significant figures than the measured data, and second, they never rounded the last significant digit. It is important that students be made to understand how to manipulate significant figures. 3. Perhaps the two most confusing concepts for students in this chapter are accuracy and precision. Not only should students be made to understand that they are different concepts, they should also understand that they are independent of each other. The accuracy of a measurement refers to how much it differs from the correct, or agreed upon value for a given physical quantity. In essence, one only needs one measurement in order to talk about accuracy. Precision on the other hand, refers to the repeatability of a series of measurements. It is an indication of how close measurements performed under identical conditions, are to each other. It is a concept associated with an experimental uncertainty. In order to define a precision, a set of measurements is necessary. 4. Students initially tend to have enormous difficulties with the concept of orders of magnitudes. It takes them great effort to understand that an increase or decrease of one order of magnitude actually means an increase or decrease of a factor of 10! Since the concepts of logarithms and orders of magnitude are similar, students are often similarly confused by both. The typical confusion goes something like this: if the electrostatic force between two masses is 10 -9 N and the gravitational force is 10 -39 N, how much larger is the electrostatic force than the force of gravity? Student’s answer: it is about 30 times larger. Of course the correct answer is that it is about 10 30 times larger, or 10 followed by 30 zeroes! To write this number out would take about two lines of text. It is a number so large, that we don’t even have a name for it!

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    Physics for the Life Sciences Chapter 1 5 WHAT CAN I DO IN CLASS? Activity #1: Take a physical phenomenon such as the flight of a rocket or a tornado, and have students construct a physical model of the phenomenon. Remind students that their model should satisfy four conditions: a) It should be based on observable phenomena, b) It should focus, or track, just a few of the most relevant physical properties of the phenomenon (in the case of the rocket, these could be position, velocity, and acceleration for example), c) It should contain a minimum number of assumptions, and most importantly, d) It must have predictive power, in other words it has to be able to predict the future evolution of the phenomenon. Activity #2: Have student volunteers (those who are of the less shy type) come to the front of the class and have them solve problems in addition, subtraction, multiplication, and/or division involving significant figures. When the volunteers make the inevitable mistakes, then let other students in the class point out the errors and take this as a starting point for a discussion on the finer points of the mathematical rules of significant figure operations. Activity #3: Referring to example #4 above, ask students how much stronger the electrostatic force is than the gravitational force for the example given. Once the usual incorrect answers are given, proceed to explain first, the magnitude of their errors, and then the correct way to solve this problem. Activity #4: As an example of the catastrophic consequences of not being able to properly convert units, discuss the case of the “Gimli Glider”. This was a brand new state of the art Air Canada Boeing 767 aircraft that was involved in a notable aviation incident. On July 23, 1983, the aircraft ran out of fuel at 41,000 feet (12,500 m) of altitude, about halfway through its flight from Montreal to Edmonton. The crew was able to glide the aircraft safely to an emergency landing at a Canadian Air Force base in Gimli, Manitoba. The subsequent investigation revealed fuel loading was miscalculated through misunderstanding of the recently adopted metric system which replaced the imperial system. Luckily, not a single life was lost that day, and eventually the aircraft was repaired and returned to service.

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    2. KINEMATICS IF NOTHING ELSE, MY STUDENTS SHOULD LEARN… 1. Motion in the physical sense, can be analyzed through the parameters of position, time, velocity, and acceleration. 2. These four parameters are related. For example, the velocity of an object is its change of position in a given time interval. 3. Position, velocity, and acceleration, are vector quantities because they are specified by a magnitude and a direction. Speed on the other hand, is not a vector since it is just the magnitude of the velocity. 4. Graphing motion as a function of time can give an intuitive feel for the motion of an object. The slopes of the position and velocity functions give the instantaneous velocity and acceleration of the object. These graphs are representations of the kinematic equations of motion. 5. Motion in the universe, whether biological or not, depends on an object’s acceleration. Therefore we must pay special attention to developing a mathematical description of acceleration. LEARNING OBJECTIVES Students should be able to:  Understand that one dimensional motion takes place along a straight line. (understand)  Understand that a vector is a mathematical representation of a physical quantity that has both a direction, and a magnitude. (understand)  Understand that the position of an object is represented by a vector which begins at some defined origin, and ends at the object’s location. It points away from the origin. (understand)  Understand that the vector that begins at point A and ends at point B, is called the displacement vector from A to B. (understand)

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    Physic for the Life Sciences Chapter 2 7  Understand that the distance between two points A and B is simply the magnitude of the displacement vector between them. (understand)  Be able to compute the distance between two points A and B. (analyze)  Understand that velocity is a vector which gives information on how fast an object moves, or its rate of motion, and the direction of the motion. (understand)  Understand that the speed is the magnitude of the velocity vector. It is not a vector and therefore gives no indication of the direction of the motion. (understand)  Be able to compute speed, given the velocity. (evaluate)  Understand that the average velocity is the quotient of the displacement vector and the time interval over which the displacement took place. It is a vector. (understand)  Understand that the instantaneous velocity is the velocity at a given instant. By definition, it is given by the limit of the average velocity as the time interval goes to zero. It is a vector. (analyze)  Understand that the average speed is simply the magnitude of the average velocity. (understand)  If the average speed is zero, but there was overall motion, then the average speed is taken to be the distance covered divided by time. (analyze)  Be able to calculate the average speed given the average velocity. (evaluate)  Understand that the instantaneous speed is simply the magnitude of the instantaneous velocity. (understand)  Be able to calculate the instantaneous speed given the instantaneous velocity. (evaluate)  Understand that the tangent to a curve at a given point is the best straight line approximation to the curve, at the given point. (apply)  Remember that acceleration is the change in velocity per unit time. It is a vector. (remember)  Understand that the average acceleration is the quotient of the change in the velocity and the time interval over which the change took place. It is a vector. (understand)  Understand that the instantaneous acceleration is the acceleration at a given instant. By definition, it is given by the limit of the average acceleration as the time interval goes to zero. It is a vector. (analyze)  Remember that the acceleration of gravity is the acceleration that all freely falling bodies near the Earth’s surface experience. It is approximately equal to 9.8 m/sec2. (remember)

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    Chapter 2 Physics for the Life Sciences 8  Remember that the resultant vector is obtained by adding all the vectors acting at a particular point. (remember)  Remember that projectile motion refers to the motion of an object under the influence of a single force, the gravitational force. The trajectory of such motion is generally a section of a parabola, however in the special case special case in which there is no horizontal component, it is just a straight line. (remember)  Remember that uniform circular motion refers to motion in a circle with a constant speed. The velocity is changing constantly however. (remember)  Remember that the centripetal acceleration is the radially inward acceleration which any object in circular motion is subjected to. (remember) WHY IS THIS CHAPTER IMPORTANT TO SCIENTISTS?  The central idea in this chapter is the mathematical analysis of motion, which is itself a central unifying concept in all of physics. The basic mathematical structure of the study of motion - or kinematics - is introduced.  Special cases of motion which will be of particular interest in the analysis of biological systems are introduced. Examples of such special types of motion are rectilinear motion, free fall, projectile motion, and uniform circular motion.  We begin in this chapter the task of applying physical analysis to biological systems, the major thrust of this text. Two examples of this type of analysis are presented, a study of the acceleration sensors located in the human inner ear, and motion sensors located along the anatomy of certain species of fish. WHY SHOULD STUDENTS CARE?  In this chapter, the concept of motion of bodies is introduced. This is perhaps the single most important concept in all of physics. A minimum set of mathematical tools which allows a rigorous definition and analysis of motion is presented.  Special types of motion which will become useful in later analyses are presented in this chapter. Among these are rectilinear motion, free fall, projectile motion, and uniform circular motion.

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    Physic for the Life Sciences Chapter 2 9  In this chapter, we also introduce the graphical method of motion analysis, which helps give the student a feel for the type of motion being studied, without the added mathematical complications.  Most importantly, in this chapter we begin to make the connection between the biological sciences and physics, by introducing biological motion sensors such as the accelerometers located in the inner ear in humans, and motion sensors in fish. This connection is a motif we will continue to focus on throughout the remainder of the text. WHAT ARE COMMON STUDENT MISCONCEPTIONS/STUMBLING BLOCKS? 1. Students typically confuse - at least initially - the concepts of average velocity with instantaneous velocity. They typically have a good understanding of the concept of average velocity, but take longer to assimilate and have more trouble with, instantaneous velocity. Three facts need to be stressed to students in an attempt to get them to understand the concept instantaneous velocity: 1) the instantaneous velocity is the velocity vector at any given time, 2) as such, it is not an average of any sort, and 3) as such, it is constantly changing. This same sort of confusion can exist for students between the average and instantaneous accelerations, and the same remedies should be attempted. 2. Students will typically have early difficulties differentiating the concepts of displacement and distance. We start by pointing out that displacement is a vector, but distance is a scalar. That difference alone sets them apart. But there’s more; a simple way to illustrate further differences is this: take the example of a newspaper delivery boy who does his rounds and eventually ends up where he started. In this case the distance covered might be a few kilometres and real energy was expended, the boy sweats, he gets tired, his legs hurt, he consumed glucose in the process etc. However, because he ended up exactly at the point of origin, the displacement vector is zero! In this case, the average velocity over the entire delivery route is zero, but the average speed is the total distance divided by time which clearly is non zero since there was motion. So in this sense, displacement is related to average velocity in the same way as distance is to average speed. 3. Students can have difficulties with the concepts of the centripetal acceleration, and centripetal force. They will assume that when moving in a circular path such as when a car is taking a turn, there is a force that “pushes” objects outward, “the passengers are thrown outwards by the centrifugal force” they are heard to say often! This of course is incorrect. The “centrifugal” force as such does not exist, it is fictitious. In reality, it is the absence of a centripetal (i.e. radially inward) force which causes the passengers to travel in a tangential path with respect to the circle, thus giving the illusion that they are “thrown” out. If there is an adequate centripetal force such as from a tightly fitting seat belt, or bucket type seats that firmly contour the bodies of passengers, then the passengers will follow a circular path and not be “thrown” out!

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    Chapter 2 Physics for the Life Sciences 10 WHAT CAN I DO IN CLASS? Activity #1: Since the concepts of instantaneous velocity and acceleration are defined as limits as the interval Δ t goes to zero, this would be a good opportunity to either introduce, or review, the concept of a limit. This can be done with relative ease by picking out limit exercises from either a calculus or algebra textbook, and going over them in class with students. When doing these exercises, it should be emphasized that these instantaneous quantities are vectors, and they ARE NOT quotients, but rather limits! Even though they might look like quotients, they are not! Emphasize also that when taking limits, first you simplify the expression algebraically as far as possible, and only then is the limit to be taken. This means that to the extent possible, the Δ t’ s should be cancelled out before taking the limit. Activity #2: Discuss and analyze methods by which the brain when presented with acceleration information from the accelerometers in the inner ear, can quantify both velocity and distance travelled. In the discussion, make the argument that the human brain (and that of other species as well) basically has the ability to integrate (in the calculus sense) acceleration once to produce velocity estimates, and twice to produce displacement estimates.

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    3. FORCES IF NOTHING ELSE, MY STUDENTS SHOULD LEARN… 1. Force is defined by the interaction between separate objects. The term force is used to describe and measure the interaction. Force has several properties but is mainly characterized by its two quantitative properties: magnitude and direction. 2. There are four fundamental forces in nature: (1) gravity, (2) the electromagnetic force, (3) the strong nuclear force, and (4) the weak nuclear force. These forces are contact-free forces that do not require contact between objects; they are called field forces. All other forces are considered to be non-fundamental. 3. Muscle tissue serves the specific purpose of exerting forces; locomotion in turn is achieved when several muscles and other tissues, such as bone, cooperate. LEARNING OBJECTIVES Students should be able to:  Remember that muscles are divided according to form and function into three types: skeletal, smooth, and cardiac. (remember)  Understand that a force is an interaction between objects, such as a push or pull. (understand)  Understand that forces can act through direct contact between objects, and these are called contact forces. Examples are the normal force, tension on a string, and friction. (understand)  Understand that forces can act between objects over great distances without the need for contact. These are called field forces. Examples are gravity and the electric force. (understand)  Understand that force is a vector, thus it always has a direction and a magnitude. (understand)

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    Chapter 3 Physics for the Life Sciences 12  Evaluate the additive nature of forces. The effect of two simultaneous forces on the same object is the same as that of a single force equal to the addition of the forces. The addition is vector addition and the resultant force is called the net force. (evaluate)  Analyze forces acting on an object, i.e. changes in the velocity of the object. (analyze)  Understand that a restoring force in one which tends to bring an object back to its initial position, or to a position of equilibrium. An example of such a force is the elastic force of a spring, which always tends to return to its uncompressed or unstretched length when released. (understand)  Analyze how forces always act in pairs. (analyze)  Remember that the unit of force is the newton (N). The newton is a derived (SI) unit. In terms of fundamental units, 1 N = 1 kg·s2/m. (remember)  Remember that the gravitational force of the Earth on any object, is commonly called the object’s weight. (remember)  Solve mathematically the force of gravity between two masses, given by the expression: where m 1 and m 2 are the masses of the objects, r is the distance between the two objects, and G is the universal gravitational constant 6.674 ×10 − 11 N · m 2 /kg 2 . (apply)  Understand that “g” is the acceleration of gravity. At the surface of the Earth, it is approximately equal to 9.8 m/s2. It is the quotient of an object’s weight and its mass. (understand)  Determine mathematically the electric or Coulombic force between two charges, given by the expression: where q 1 and q 2 are the magnitudes of the charges of each object (in coulombs, “C”), r is the distance between the two objects, and the constant k is the electric force (or Coulomb) constant and is equal to k = 8.99 × 10 9 Nm 2 /C 2 . (analyze)  Remember that the electric force is attractive if the charges are of opposite sign, and it is repulsive otherwise. (remember)

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    Physics for the Life Sciences Chapter 3 13  Remember that the strong force holds protons and neutrons together in the nucleus. Since it overcomes the repulsive electric force, it is the strongest of the fundamental forces, but it has no significant effect outside the atomic nucleus. So the nuclear force is very strong and short-ranged. (remember)  Remember that the weak force - the last of the four fundamental forces - plays a role in the decay of certain nuclei, and it extremely short-ranged. (remember)  Classify examples of everyday non-fundamental forces such as the normal force, tension on a cable, the force of a spring, friction, air drag, and others. These forces are all macroscopic manifestations of microscopic electrical forces. (analyze)  Analyze a free body diagram which is a sketch of all forces acting on an object. It includes the object and vectors representing the applied forces on it. It shows all types of forces, both fundamental and convenience, exerted on the object. (analyze)  Understand that if the net force that acts on an object is zero, the object is in equilibrium, and the object stays at rest or continues to move without changing its velocity. (understand)  The mathematical condition for equilibrium is expressed as: where Fi is the ith force acting on the object.  Remember that the vestibular apparatus in our inner ears are capable of detecting the direction of gravity - and thus - of establishing up and down, independent of visual cues. (remember)  Remember that our skins are equipped with mechano-receptors capable of indicating the magnitude of forces acting on them. They work by correlating the magnitude of a force with the compression of the skin. (remember) WHY IS THIS CHAPTER IMPORTANT TO SCIENTISTS?  The concept of a force as an interaction between two objects is introduced. A classification of forces into fundamental and non-fundamental or of convenience is introduced. A description of each type of force is presented.  The properties of forces are introduced, and their vector nature is established. The muscular system as the origin of mechanical forces in the body is presented.

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    Chapter 3 Physics for the Life Sciences 14  The free body diagram is introduced as a useful analytical tool for solving mechanical problems.  The concept of mechanical equilibrium is introduced. The conditions required to established equilibrium are defined. WHY SHOULD STUDENTS CARE?  In this chapter, we introduce qualitatively, the mechanical concept of a force as an interaction between objects.  We distinguish the four known fundamental forces from other everyday forces which are macroscopic manifestations of the electromagnetic force.  We introduce within the human body, the muscular system as the origin of most mechanical forces. A detailed physiological model for muscle action is presented.  The basic properties of a force are discussed, along with the vector nature of all forces.  An introductory description of everyday – or convenience – forces such as tension, friction, normal, and drag forces, is presented.  The concept of a free body diagram is introduced, and its usefulness as an analytical tool is discussed.  The concept of mechanical equilibrium , and its consequences, is analyzed.  Physiological mechanisms which allow us to keep our balance, and judge the magnitude of external forces acting on our bodies, are discussed. WHAT ARE COMMON STUDENT MISCONCEPTIONS/STUMBLING BLOCKS? 1. Students can have a difficult time understanding the concept that forces ALWAYS act in pairs. There is no such thing as an isolated, single force acting on an object. When object A exerts a force on object B - whether through contact or through the action of a field (gravitational, electric, or magnetic) - object B exerts an identical force on object A, except for one characteristic: the force is antiparallel. Many examples can be given to substantiate this claim.

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    Physics for the Life Sciences Chapter 3 15 2. The concept of equilibrium may cause students some difficulty. If a body is in equilibrium, this does NOT mean there are no forces acting on it! Equilibrium is defined as a mechanical condition in which the NET force acting on an object is vectorially equal to zero, in other words they cancel out. The mechanical effect of this condition is equivalent to having NO force acting on the object, but the two conditions are different. Of course, a condition of equilibrium implies that the velocity of the object remains constant. 3. Friction is a concept that often creates trouble for students. For example, they can have some difficulty comprehending, or accepting, that the force of friction between two bodies is independent of the contact area between them, depending only on the normal forces pressing them together. There may also be some difficulty understanding that static friction is typically larger than sliding friction. WHAT CAN I DO IN CLASS? Activity #1: In simple illustrations of the limitations of the human vestibular apparatus, have a student spin around on his axis a few times, until he/she is clearly dizzy, but not overly so. Then ask the student to attempt to walk in a straight line, and observe the deviations or zig-zag - like path thy will describe. In a similar vein, have a student lie down on the floor for about 10 or 15 minutes (without going to sleep!) and ask him/her to quickly stand up and walk straight. Again there will be some zig-zag – like motion. Explain that what is happening here is that the fluid in the vestibular apparatus is being shaken to the point of becoming turbulent. This has the effect of causing rather large and sudden oscillations in the hair-like structures of the apparatus, which sends confusing electrical signals to the brain such that it compromises the ability to maintain proper balance. The effect of alcohol in the brain is somewhat similar to this. Activity #2: In an exercise designed to improve the students’ understanding of how friction works (and how it doesn’t) the following simple exercise may be useful. Take a book with a known mass and place it on a table. Tie a string around it such that the string does not make contact with the table, and tie the other end of the string to a spring balance. Begin slowly dragging the book across the table, measuring the force necessary to do so by reading it off the scale. As one knows from theory, if the normal force on the book is doubled, the sliding friction should more or less double as well. So next put an identical book on top of the original one (to double the normal force), and repeat the experiment. The force required to pull the two books across the table should roughly have doubled. This illustrates two key points discussed in the chapter: the first one being that friction is directly proportional to normal force; the second one being that it is independent of contact area (in other words, no change in contact area results in a doubling of friction).

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