chapter 2 unifying approaches - doane college

Physics Including Human Applications

Chapter 2 Unifying Approaches

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Chapter 2 UNIFYING APPROACHES

GOALS When you have mastered the content of this chapter, you will be able to achieve the following goals: Definitions Define each of the following terms, and use each term in an operational definition: equilibrium, restoring force, conservation laws, inertia, oscillatory motion, feedback, gradient, linear system, natural frequency, current, superposition and resonance. Inertia Give an example of a physical system that has mechanical, thermal, and electrical inertia. Energy Transfer Explain how you would maximize the transfer of energy at the interface between two systems. Superposition Solve problems making use of the superposition principle-given the proper physical variables of the systems. PREREQUISITES Before beginning this chapter you should have achieved the goals of Chapter 1, Human Senses. If you have not recently been working with Cartesian coordinate graphs and dimensional relations, you may wish to review the material on graphs and dimensional analysis in the mathematical background supplement in the appendix. To help you assess your readiness for this chapter you may use the following self-check. Graphing and Dimensional Analysis Self-Check 1. The following table is taken from a drivers manual and shows data for stopping an automobile on dry pavement.



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-------------------------------------------------------------------------------------------------- Velocity Thinking Distance Total Stopping Distance (m/sec) (m) (m) ________________________________________________________ 8.8 6.7 14 13 10 27 18 13 43 23 17 61 27 20 86

-------------------------------------------------------------------------------------------------- a. Draw a graph of thinking distance (y-axis) versus velocity (x-axis), and find the slope

of the curve at the point on the curve where x = 15 m/sec. b. Draw a graph of the total stopping distance (y axis) versus the velocity (x axis), and

find the slope of the curve at the point where x = 20 m/sec. 2. We can define length, mass, and time as fundamental dimensions in a system of

measurement. What are the SI (System International) units for a. Length b. Mass c. Time

The SI units are related to each other by multiples of ten, and the units are represented by the fundamental unit with the proper prefix. What are the relationships between the fundamental unit and the following common prefixes? d. The prefix centi- means_____, so one tesla = _____centiteslas. e. The prefix milli means _____, so one liter =_____milliliters. f. The prefix kilo- means _____, so one watt =_____kilowatts.

Graphing and Dimensional Analysis Self-Check Answers If you had difficulty in correctly solving these problems, please study Appdendix

Section A.4, Cartesian Graphs, and A.5, Dimensional Analysis. 1. a. 0.60 sec; b. 4.8 sec 2. a. meter; b. kilogram; c. second; d. 10-2, 102; e. 10-3, 103; f. 103, 10-3



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Chapter 2 UNIFYING APPROACHES

2.1 Introduction As you reach for the mug of coffee on the table in front of you, has it ever suddenly evaded your grasp? It is not likely that you have had that happen. In fact, your experiences with coffee mugs at rest upon the table have taught you quite a bit about what to expect from natural events. The mug seems perfectly content to remain at rest as long as you do not try to push it around. In fact, a gentle push on the mug is met with the cup's determination to remain at its present location. If you reach across the top of the mug to get a doughnut and accidently tip the mug slightly, you know that it will bang back down on the table, the surface of the coffee will slosh around for a short time, probably spilling out some, and then it will settle back down to a restful state. While you would be quite surprised if some invisible, mysterious power quickly removed the mug from your grasp, you think nothing at all of leaving your hot coffee mug on the table only to find that some time later the mug and coffee have cooled noticeably. Has a mysterious power been quickly removing something from your mug to make it feel cool? Suppose you reach out to pour some more coffee from a coffee pot into your mug. How does your brain manage to get your hand to go to the proper location to perform that task? After you have poured more coffee into you mug, what has happened to the coffee in the pot? What is the point of all this rhetoric? Since birth you have been continuously interacting with the physical environment in which you have lived. You already have considerable knowledge about how nature works. It is the purpose of this chapter to introduce you to some mental constructs that you can use to unify your approach to studying nature. These mental constructs, or models, are introduced first in the use of words that can be explained by operations performed on a specific system. These words are to be defined in terms of specific experiences, not in terms of other abstract words. We call such definitions operational definitions. In giving an operational definition for a word you should choose a specific system, a hot coffee mug sitting on a table, for example, and explain the characteristics of the system that are described by the word you are defining. There are several unifying approaches that can be made in the study of physical and living systems. These approaches make use of the qualitative relationships that exist between the various properties of a system. Your own physical experiences and observations must be used to incorporate these approaches into your understanding of your environment. 2.2 Equilibrium Your physics textbook at rest upon your desk is said to be in equilibrium. The book is in equilibrium because the sum of all forces acting on the book is zero, and there are no forces tending to make the book rotate (see Figure 2.1a). Another example of



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equilibrium is an object hanging on a spring (see Figure 2.1b).

In daily life you have seen many systems in equilibrium. In general a system is in a state of equilibrium when all the influences acting on the system are cancelled by others, resulting in a balanced, unchanging system. Consider a solid right circular cone like the one in Figure 2.2. The cone may rest on a table on its base, on its side, or on its apex in a vertical position. In each case the cone is in equilibrium. However, we find these positions are different types of equilibrium. We distinguish between these different types of equilibrium by the answer to the following question: If the system is changed slightly, what is the tendency of the system to return to its original state? With slight displacement from base position as in Figure 2.2a the cone tends to return to its original position. From side position (Figure 2.2b) there is no tendency to return to original position. From apex position (Figure 2.2c)there is a tendency to go further from its original position. When resting on its base, the cone is said to be in stable equilibrium; on its side, to be in neutral equilibrium; and on its apex, to be in an unstable equilibrium condition. Physical systems in stable equilibrium tend to return to their original state if they are slightly changed from their equilibrium state. This indicates that there is a force tending to restore the changed system to equilibrium.

2.3 Inertia Lift one rock in each hand, rocks that are alike except in size. If rock A is larger than rock B, a larger force is required to lift rock A than to lift rock B. Now shake the two rocks back and forth rapidly. What difference do you notice in the feel of the rocks? Do you notice that rock A is more difficult to start moving in one direction and that after it is moving it is more difficult to stop rock A and cause it to move in the opposite direction? You probably find it is much easier to shake rock B back and forth than it was rock A. The property of these rocks that measures their resistance to changes in their



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state of motion is called mechanical inertia. The rock with the larger mechanical inertia has the larger mass. We can think of the mass of an object as a measure of its mechanical inertia. The inertia of a system tends to maintain it in an equilibrium condition. A given system may have several different inertial variables, each appropriate to a specific physical property of the system. For example, a rock has mechanical inertia (called mass) which resists a change in its state of motion. A rock has thermal inertia; the larger its thermal inertia (called heat capacity) , the more difficult it is to change its temperature. A rock has electrical inertia. Again, the larger its electrical inertia (reactance), the more difficult it is to get electrical charge to move through it. In general, inertia is the property of a system that is a measure of the system's resistance to change. The inertial property of a system determines how fast the system responds to external forces. That is, the inertial property for a given physical variable determines the response time of the system for changes in that variable. Early in their mastery of the physics of pool, players discover another characteristic of inertial properties. Consider the example of the head-on collision between two pool balls. You will observe that ball A approaches ball B, which is at rest, with a certain speed and that after a collision, ball A comes to rest and ball B moves away with essentially the same speed as ball A originally had. This is an example of the behavior of a contact interaction at the interface between the two balls (systems). In this case, the mechanical inertia of both systems, that is, the mass of both pool balls, is the same and nearly all the original energy of motion of ball A is transferred to ball B. On the other hand, consider the collision between a moving bowling ball and a stationary tennis ball. In this case the mechanical inertia (mass) of the moving bowling ball is much, much greater than the mass of the resting tennis ball. The motion of the bowling ball is hardly changed at all by a collision with a tennis ball. The amount of energy of motion transferred from the bowling ball to the tennis ball is small. The general rule that applies to such interactions between systems is that the maximum energy transfer occurs between the two systems when their inertial properties for the transferred variable are equal. This is a useful rule with many applications. The transfer of electrical energy is greatest when the source has the same reactance (similar to inertia in a mechanical system) as the external circuit. The maximum energy is transferred to a stereo speaker when its reactance is equal to that of the input amplifier. You can now see that it is necessary to study the inertial properties of a system if you wish to understand its physical behavior. Questions 1. Name some systems in equilibrium. 2. Name some systems which tend to return to equilibrium if slightly displaced. What is

the "restoring force" in each system? 3. Consider a social system such as your physics class. List possible inertial properties

for the class and some possible consequences of these forms of inertia. 2.4 Oscillatory Motion and Restoring Force Consider a child at rest in a swing. If she is displaced (Figure 2.3) from the equilibrium position (the position where she is nearest to the ground) and released, she will tend to return to equilibrium. Why? There is a force acting on her to return her to the equilibrium position. This force is called a restoring force. In this case the restoring force is a gravitational force. The pull of gravity tends to bring her back to her equilibrium



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position.

In many systems the restoring force is directly proportional to the displacement x of the system from equilibrium position. Then there is a restoring force F acting to return the system to equilibrium. (See Figure 2.4) F = -kx (2.1) where k is the force constant of the system. If the system is moved to a distance twice as far from equilibrium, 2 x, then the restoring force will be twice the original restoring force, or 2 F.

This system is then said to be linear. An example of a linear system with which you are acquainted is a spring balance. This means that the force required to stretch the spring is directly proportional to the stretch. This is shown in the graph, Figure 2.5. Think of a weight hanging on a coil spring. The weight is moved slightly from equilibrium position and released. The system responds to the linear restoring force by oscillating about the equilibrium position and, in all real systems, gradually comes back to rest at equilibrium.

When an automobile is traveling at a high speed on a smooth highway and hits a sharp bump in the road the auto begins to oscillate up and down. However, the shock absorbers subtract out a portion of the impact of the bump, and on each oscillation the car vibrates less widely. The auto finally returns to smooth motion until it strikes another bump. Many physical systems exhibit this same kind of behavior. Nonlinear forces cause this damped oscillatory motion. Friction is one example of nonlinear force. Each of the examples, the girl on the swing, the weight on a spring, and the undamped car, illustrates the oscillatory motion that is characteristic of systems that are linear, those in which the restoring force is proportional to the displacement of the system from equilibrium. The frequency of the to and fro motion of such linear systems is completely determined by the restoring force constant (force per unit displacement) and the inertial property of the system for the displacement involved. This frequency, the number of complete oscillations per second, is called the natural frequency of the



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system. Almost every physical system behaves as a linear system when it is subjected to small displacements from equilibrium. Hence, most systems show oscillatory motion for small displacements from equilibrium. 2.5 Current and Gradient What happens to rainwater when it hits the ground? What happens to the rainwater that collects in your backyard, or that collects at the upper end of your street, or that falls on the mountains? How do you explain what you see happening to rainwater as it flows along the ground? What happens to the handle of a silver spoon when it is placed in a bowl of hot gravy? How do you explain the flow of heat energy through a solid? What happens to a storage battery when you connect it to a light bulb? How do you explain the flow of charge in an electrical circuit? These are only a few of the examples of the motion of something from one part of the universe to another. This type of motion occurs in many aspects of nature and exhibits similar properties in its many different appearances in nature. If all parts of a system are the same, nothing happens, and the status quo is maintained. The flow of something always happens in response to a difference in some property between two different parts of the system. In the above examples, the water flows as a result of a difference in elevation between two locations; the heat flows as result of a difference in temperature between two places; the electrical charge flows as a result of a difference in electrical potential between two different points in a circuit. The rate of flow, or current is related to the changes of some property from one part of the system to another. In general,

current = {Δ(quantity)}/{Δ(time) }where Δ means change in (2.2) For example, the greater the temperature difference across a given length of silver spoon handle the faster the heat energy flows through the handle. Because of the importance of the change of the properties of systems with location, let us define a new term called a gradient. Gradients have both size and direction. Such quantities are called vectors, and they will be discussed in Chapter 3. The size of the gradient of a physical quantity is determined by how rapidly the quantity changes with position. magnitude of gradient = Δ(physical quantity)/Δ(position) (2.3) The direction of the gradient is in the direction of greatest positive change. For example, consider a metal bar with one end in ice water (0oC) and the other end in boiling water (100oC) (Figure 2.6a). The graph of the temperature along the bar as a function of distance from the ice is shown in Figure 2.6b.

The slope of the temperature vs. distance curve is defined as the magnitude of the



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gradient. The direction of the gradient is positive from the cold end to the hot end. This direction is opposite to the direction of the current produced by the gradient. For this example, the current is flow of heat energy from the 100oC end to the ice bath end. (Note that cold is the absence of heat, and consequently there is no cold current.) In many cases the magnitude of a current, that is, the rate of flow, is directly proportional to a gradient. (We will discuss specific examples of such transport phenomena in later chapters.) The proportionality constant K between the current and gradient is a basic physical property of the material in which the current occurs. current = -K • gradient (2.4a) The minus sign indicates that the direction of the current is opposite the direction of the gradient. For our example this becomes ΔQ/Δt) = -K • (ΔT/Δx ) (2.4b) where ΔQ is the change in heat energy, ΔT, the change in temperature, K, the thermal conductivity, Δt, the change in time, and Δx, the change in distance. 2.6 Conservation Laws An important property of physical systems is that certain properties are neither created or destroyed within the system. This property is the basis of the conservation of physics. These conservation laws are expressed by the simple mathematical statement: (change in physical property)/(change in time) = 0 (2.5) The concept of conservation is one of the early mental developments of human beings. Several tasks are used with young children to test their conservation reasoning. In one of them two horizontal rows of checkers, one row of six black and one row of six red checkers, are made. After the child is convinced that there are the same number of each, the red checkers are stacked in a vertical pile. The child is asked if there are the same number of red and black checkers. In other words, is the number of checkers of one color independent of their arrangement in space? This is conservation of number, change in number/change in arrangement = ΔN/Δa = 0 (2.6) In another test six matches are laid end to end as in Figure 2.7a, and six other matches are laid in some other configuration as shown in Figure 2.7b. A child is asked if the distance along the matches from A to B is the same. This is conservation of length, change in length/change in arrangement = ΔL/Δa = 0 (2.7)



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Suppose you give a child two green sheets of paper of equal size, representing two pastures of grass. You also give the child a number of blocks of equal size, representing barns. The child places one square on each sheet (Figure 2.8). When the child is convinced that each sheet has equal areas of pasture for the two configurations shown in Figure 2.8, a number of blocks are used, and the child is given two configurations such as shown in Figure 2.9. The child is then asked, "In which pasture will a horse have more grass to eat? Which one has more green grass or are they the same?" This is an example of conservation of area, change in area/change in arrangement = ΔA/Δa = 0 (2.8)

Consider another example. A child is shown two balls of modeling clay of equal mass (Figure 2.10a). The questioner then shapes one of the balls into a pancake form (Figure 2.10b) and asks, "Which one contains more clay?" This is an example of conservation of mass, change in mass/change in shape = ΔM/ΔS = 0 (2.9) The laws of conservation in physics refer to the constancy of a system variable in time. A variable that does not change in time is said to be conserved. For example, in classical physics we find that the mass, the energy, the momentum, and the electrical charge of a system are conserved. In relativity physics, the conservation laws of mass and energy become one combined law. 2.7 Feedback You can buy a camera that will automatically adjust the exposure for the intensity of light and the type of film you are using. The film requires a given amount of light energy for the proper exposure. The amount of energy reaching the film depends upon the size of the aperture, time of exposure, and the intensity of light. In using the camera, you set the aperture. Then the intensity of light produces an input to the electrical circuit that controls the exposure time. In another type of camera the exposure is set, and the aperture automatically adjusts for the speed of the film. This second type is parallel to the system of the human eye. In the eye the light incident upon the eye is the stimulus to the muscles that control the size of the pupil of the eye (Figure 2.11). You can observe the change in the size of the pupil of your eyes by looking in a mirror in a darkened room and then turning on the lights.



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Feedbackis said to exist in a given system if part of the output from the system is returned as an input into the same system. This feedback may be either positive or negative, that is, additive or subtractive. In general, if the feedback is negative, the system will return to a stable situation. If the feedback is positive, the system will oscillate or go out of control. Figure 2.12 illustrates positive and negative feedback loops for two cases in which one automobile follows another and the lead vehicle slows down or stops. InFigure 2.12a the lead automobile is sighted, and the message is delivered to brain of the following car. The second driver's brain commands that brakes be applied. Brakes are applied, and the second automobile is slowed or stopped so that a safe distance between the cars is maintained. Thus a steady system is approached. In Figure 2.12b the proper execution is not made. The visual stimulus results in more gasoline going into the engine so that the car speeds up, and the distance between the cars decreases. A collision will occur if this feedback continues. That is, the system goes out of control, and a collision occurs.

The thermal regulation systems for a house and for the human body are examples of feedback used to maintain equilibrium. In a house, a thermal sensing element in the thermostat provides a feedback signal. When the temperature falls below a set point, the feedback signal turns on the furnace to supply more heat. The heat raises the temperature of the house above the set point, and the feedback signal is shut off. In a similar way the body's thermal sensors provide data to a portion of the brain (the hypothalamus) that regulates the body metabolism to maintain the skin temperature of the body in the equilibrium range of about 33 to 38oC in a room of 27oC. Questions 4. Consider yourself as a system in the physics classroom environment. What are the

"input" and "feedback" for you? For your instructor?



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2.8 Superposition Principle Picture a rope with two people holding opposite ends of the rope. Each person independently snaps his end of the rope and sends a pulse down the rope (Figure 2.13). What is the resultant displacement of the rope as the result of the activities of the pair? Let y1 be the displacement of a given point in the rope at a given time as given by Alan and y2be the displacement of the same point at the same time as given by Barbie. The resultant displacement then is the sum of y1 and y2, that is, y = y1+ y2 (2.10) This procedure assumes that the pulses do not interact with one another, that each propagates as though the other were not present. This summation process is called the superposition principle.

We can use the superposition principle in an analysis of a situation only when the systems involved are independent and interact linearly. For cases in which these criteria are satisfied, the analysis of complex systems is possible by means of the superposition principle. The complex system can then be broken into simpler parts, and the sum of the properties of the parts is taken to be the behavior of the complex system. For example, the interaction of the earth and the solar system is equal to the sum of the interactions of the earth with each individual constituent of the solar system. The pressure in a gas filled container is equal to the sum of the partial pressures produced by each of the gases in the container.



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Perhaps you have seen the stakes for a circus tent being driven. Let us assume that a stake is driven by three men, each hitting the stake in succession. The force contributed by each man is a function of time. See Figure 2.14 . The total impulse exerted is the sum of the separate contributions (see Figure 2.14d).

There are many nonlinear physical systems. Electronic amplifiers, for example, can be operated in a nonlinear way. Such an amplifier distorts the input signal by amplifying different frequencies by different amounts. Most of the human senses seem to respond nonlinearly. Questions 5. Consider a married couple as a system. Does this system obey the superposition

principle? Is this system equivalent to the sum of individual interactions? Explain your answer.

6. List some physical, biological and social systems, and determine whether they are linear systems with respect to the significant variables for the given system.

2.9 Applications of Unifying Approaches Consider a cork floating in a motionless pool of water. It is in equilibrium because it remains at rest. We drop another cork onto the first, producing a small displacement of the first cork. This cork will oscillate about its equilibrium position. The bouyant force of the water is a restoring force. The amplitude of the oscillation, the amount of maximum displacement from the equilibrium position, will depend upon the inertia of the cork (its mass) and on the inertia of the dropped cork. If the corks have the same mass, we will obtain a maximum oscillation amplitude. The oscillation of the cork sets up a water wave, which transports energy through the water. If a repeating external force is applied to the floating cork, we will observe another phenomena. As we change the repetition rate of the applied force, we will find a particular repetition rate at which the amplitude of the cork reaches a maximum creating turbulent water waves. This condition of maximum oscillation under the influence of a repeating force is called resonance. Resonance occurs when the energy transferred from the applied force to the oscillating system is maximum. Resonance is observed in many physical situations, and



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in each case the externally applied resonance frequency matches the natural frequency of the oscillating system. The natural frequency of any system is determined by its inertial properties and by its restoring force constant. The restoring force constant is the ratio of the restoring force to displacement. restoring force constant = restoring force/displacement (2.11) For a linear system the restoring force constant is a constant given by the slope of the force versus the displacement shown in Figure 2.5. You have undoubtedly experienced a resonance phenomena in your automobile. At a particular speed on a bumpy road your automobile can experience large vibrations. This occurs when the repeating forces applied to the automobile by the road have the same frequency as the natural vibration frequency of the suspension system of your automobile. 2.10 Homeostasis The broad applicability of the various principles discussed in this chapter becomes clear when we consider the biological concept of homeostasis. Higher animals are able to survive in a wide variety of external environments because they carry their own environment with them in the form of fluids that bathe their cells. The constancy of this internal environment in spite of variation in the external environment is characteristic of all higher forms of life. The term homeostasis refers to the stability of the internal environments of organisms. The concept of homeostasis, or steady-state control, is now recognized as one of the fundamental principles of biology. Multicellular animals maintain homeostasis by means of feedback. Consider the homeostasis and feedback involved in human hunger. What determines your hunger equilibrium point? How do you determine your hunger inertia? It is now known that the feeling of hunger is stimulated by the concentration of glucose in the blood, which varies from high levels just after eating to lower levels several hours later. What is an important gradient in the feedback cycle for hunger control? The stimulation of the hunger nerve center in the brain causes the feeling of hunger. Eating raises the blood sugar level and reduces the stimulation to the hunger center. At the same time it provides stimulation to the satiety nerve center and actively counteracts the feeling of hunger. Hence, we alternate between hunger and its absence, and we eat enough to maintain the proper glucose concentrations in the blood for the healthy functioning of the body cells. SUMMARY Use these questions to evaluate how well you have achieved the goals of this chapter. The answers to these questions are given at the end of this summary with a reference to the section where you can find the related content material. Definitions 1. The property of a system that measures resistance to change is called

a. gradient b. current c. equilibrium d. inertia e. superposition



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2. When a system is displaced from___ and the___ is proportional to the displacement, the system is called a___ . Such systems will display oscillatory motion with a___ determined by its restoring force constant and inertial property.

3. Fluid flow through a tube is proportional to the pressure difference between one end of the tube to the other. The current in this example is_____. The gradient magnitude would be _____, and the direction of the gradient is ______.

4. If a measurable quantity is thought to obey a conservation law, how could you verify that the quantity is conserved?

5. Give an example of feedback as it applies to human vision. 6. On a TV commercial, Ella Fitzgerald is shown shattering a glass with her voice. This

is an example of resonance. What are the conditions necessary for this occurrence? Inertia 7. Resistance to _____ change would be the thermal inertia of a metal rod. 8. Resistance to _____ change would be the mechanical inertia of a metal rod. 9. Resistance to _____ would be the electrical inertia of a metal rod. Energy Transfer 10. If you plotted the energy transfer of two interacting systems against the inertial property ratio of the two systems, what kind of qualitative curve would you expect to get? Superposition 11. If two interactions produce the following responses when acting alone on a linear

system, sketch the response they produce when both interactions act on the same system at the same time.

Answers 1. d (Section 2.3) 2. equilibrium, restoring force, linear system, natural

frequency (Sections 2.2 and 2.4) 3. fluid flow rate, pressure difference/tube length, from low

pressure to high pressure (Section 2.5) 4. Measure the quantity at different times, and find the same

value within experimental error (Section 2.6) 5. The iris closes rapidly in bright light to protect the retina

and opens (much more slowly) to improve dark vision. Feedback loops control this response. (Section 2.9)

6. Voice frequency must equal the natural frequency of vibration of glass (Section 2.7)

7. temperature (Section 2.3) 8. state of motion (Section 2.3) 9. electrical charge flow (Section 2.3)



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ALGORITHMIC PROBLEMS Listed below are the important equations from this chapter. The problems following the equations will help you learn to translate words into equations and to solve single concept problems. Equations

restoring force = -kx (2.1) current = Δ(quantity)/Δ(time) (2.2) gradient = Δ(physical quantity)/Δ(position) (2.3) current = - K • gradient (2.4)

y = y1+ y2 (superposition for linear system interactions) (2.10) Problems 1. A spring has a force constant of 10.0 newton/meter (N/m). What is the force

necessary to stretch the spring 0.250 m? 2. A force of 200 N is required to draw a bow 0.50 m. What is the value of k for the bow?

What are the units of k? 3. A teakettle holds water at 100oC, the surface of the stove is at 200oC, and the thickness

of the bottom of the teakettle is 0.150 cm. What is the temperature gradient in the bottom of the teakettle?

4. If the temperature of the air in a room is 20.0oC and the air outside the window is -4.00oC, what is the temperature gradient in the glass of 3.00 mm thickness?

5. The difference in potential between two metal plates 2.00 mm apart is 300 volts. What is the potential gradient in a glass dielectic that fills the space between the plates.

6. The pressure is measured at two points in a horizontal water system. It is found that the pressure reading at one position is 80.0 N/cm2and at the second position it is 60.0 N/cm2. These positions are 10.0 m apart. What is the average pressure gradient between the two points?

7. Two people push on opposite sides of a box with forces of 90.0 N to the right and 60.0 N to the left. Find the resultant force due to these two people. That is, give both the magnitude and the direction of the force.

8. In Figure 2.13, the displacement of a point P in the rope by Alan is 1.00 cm and the displacement by Barbie is 2.00 cm. What is the resultant displacement?

Answers 1. 2.50 N 2. 400 N/m 3. 667 oC /cm 4. 8.00 oC /mm

5. 150 V/mm 6. a decrease of 2.00 N/ cm2 for each meter of distance 7. 30.0 N to the right 8. 3.00 cm



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EXERCISES These exercises are designed to help you apply the ideas of this chapter to physical situations. Where appropriate the quantitative answer is given at the end of the exercise. Section 2.1 1. Reread the introduction to this chapter, and discuss the questions asked in the

introduction in terms of the words and concepts that have been defined in this chapter.

Section 2.2 2. Describe a system, other than the one given in this text, that exhibits stable, unstable,

and neutral equilibrium. Section 2.3 3. Use the mechanical inertial properties of persons to explain the characteristics of large

and small football players, include starting, stopping, and collision actions. Section 2.4 4. In carrying out an experiment in the laboratory a student obtains the following data

when she loads a pan suspended on a spring balance:

Plot the curve with load as ordinate (vertical) and index reading as abscissa. What is the value of the slope of the curve? What is the physical significance of the slope? Describe the motion of the pan if it is displaced slightly. [slope = 8.33 g/cm; slope = spring constant.]

Section 2.55. Consider three triangular supports with equal altitudes as shown in Figure 2.15, and with bases in ratio of 3:2:1. Compare the slopes of the three. In which case is the gradient the largest? If equal drops of water, that is, equal inertia are placed on each slope at the highest point, in which case will the drop reach the bottom first? Why?

[The a - a triangle has steepest slope; a - 3 a triangle has the smallest slope. Gradient is

largest for a - a. Drop will reach bottom first on triangle a - a.] Section 2.6 6. Use the conservation of mass to calculate the amount of water produced when 4 g of

hydrogen combine with 32 g of oxygen. [36 g H2O] Section 2.7 7. Construct a feeback loop for light reaching the film in a camera. Include shutter speed

and aperture size. [Refer to any set of encyclopedias for further information.]



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8. Construct a feedback control loop for regulating the size of the iris in the human eye. [Refer to any introductory biophysics or physiology book.]

9. Construct a feedback loop for both positive and negative feedback for change in the socioeconomic population spectrum of your home community. Indicate the nature of the feedback in each case.

Section 2.8 10. A man buys a new house at a cost of Q0dollars. He decides that in order to maintain

the quality of his house, he must spend money at the yearly rate of three percent of the original value. The total money spent on up keep is then Q = 0.03 Q0T where Tis the time in years. The aging process will decrease the value of his house. He estimates that his house decreases in value with age according to

Qa= Q0x √[(40 - T)/40] for the first 40 years where Tin years. Use the superposition principle to draw a graph of the value of his house as a function of Tfor 40 years. Estimate the age of his house when it has its maximum value. Assume there is neither inflation or recession during the 40 years! [value = Qa + Q ]

11. Two independent disturbances are simultaneously impressed upon body A. These disturbances were give by: y1= 8 and y2 = 4(1 - t). What is the form of the y axis displacement of body A as a function of t? Plot the curve.

PROBLEMS The following problems involve more than one physical concept. 12. Describe a simple experiment to illustrate:

a. the property of inertia of a body b. a system in equilibrium c. the action of a restoring force d. the principal of superposition

13. a. Heat flows from a stove burner through the bottom of a pan. The gradient producing the heat current will be ______. [(burner temp minus inside temp)/thickness of pan] b. When the switch of a flashlight is turned out, the gradient across the lamp is ______ and the current is given in ______. [electric potential/m; electric charge/sec] c. Diffusion is a process involving mass flow (current = mass transferred/sec). What kind of gradient might produce such a current? [mass density/cm] d. What kind of gradient produces the water flow from the root system of a tree up into its limbs and leaves? [pressure/distance]

14. There are a wide range of homeostasis systems within the human body. Use the operational definitions you have learned in this chapter to describe some of the physical parameters of these systems. Consider the following examples: a. gases in the blood b. water content of the blood c. excretion d. thirst

e. temperature control of the body f. circulation g. respiration

15. Construct a model using feedback to explain threshold phenomena for a biological

system.

chapter 2 unifying approaches - doane college

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