Study Sheet for Final Exam

Transcription

1 Physics 121 Spring 2003 Dr. Dragt Study Sheet for Final Exam 17. Physics knowledge, like all subjects having some substance, is cumulative. You are still responsible for all material on the Study Sheets for Exams 1, 2, and 3. Make sure that you now know how to do Exams 1, 2, and 3 perfectly. Exams are excellent opportunities to discover and learn what you previously did not know. 18. Chapter 11, Fluids. (a) Section 11.1, Mass Density. The density ρ of a substance is the ratio ρ = M/V where M is the mass of some amount of the substance, and V is its volume. By definition, because of the way the metric system is set up, water has a density of 1 gram per cubic centimeter, or 1000 kilograms per cubic meter. The specific gravity of a substance is its density divided by the density of water, Specific gravity = ρ substance /ρ water. Consider, for example the case of mercury (chemical symbol Hg). The specific gravity of mercury is 13.5, which means that a given volume of mercury has a mass that is 13.5 times larger than that of the same volume of water. (b) Section 11.2, Pressure. A fluid (liquid or gas) can only exert forces that are perpendicular to a surface. If a fluid exerts a force F on an area A, we define its pressure P by the ratio P = F/A. 1

2 Evidently pressure has units of Newton/(meter) 2. However, it is common to honor Pascal by introducing the honorary unit of 1 pascal = 1 Pa = 1 Newton/(meter) 2. Americans, in their provincial way, also use the unit of pounds per square inch (psi). Atmospheric pressure has the value 1 atmosphere = 14.7 psi = Pa = kpa. (c) Section 11.3, Pressure and Depth in a Static Fluid. In class we did a gedanken experiment, and found that if a fluid is subject to gravity, the pressure varies with depth. If P s is the pressure at the surface of a fluid (often atmospheric pressure), then the pressure at depth d is given by the relation P (d) = P s + ρgd. For the case of (fresh) water in earth s gravity, we found that going to a depth of 10.2 meters (about 32 feet) increased the pressure by 1 atmosphere. Thus, at a depth of 32 feet in a lake, the pressure is 2 atmospheres. Also, it is not possible to suck water with a straw more that 32 feet high since it is really atmospheric pressure that pushes the water up the straw. In class we saw that if one fills a glass with water, places a card over the face of the glass, and then inverts the glass, the water does not run out because of air pressure pushing up on the card. This would only work if the height of the glass were less than 32 feet. (d) Section 11.4, Pressure Gauges. As described in class, pressures can be measured using a piston pushing on a spring. This is how a tire gauge works. See Figure 10.2 of your text. (This figure should really be in this section!) Pressure can also be measured using a manometer. See Figure If the manometer is filled with mercury (a common practice until we started to worry about mercury poisoning), it is customary to measure pressure in millimeters of mercury. There are the pressure relations 1 atmosphere = 14.7 psi = 101.3kPA = 760 mm Hg = 29.9 inches Hg, 2

3 which you should know. There is also the honorary unit named for Evangelista Torr, 1 mm Hg = 1 torr. Blood pressure is measured in mm of Hg (or, equivalently, in torr). Since most pressure gauges measure the pressure above atmospheric pressure (unless one end of the manometer is closed and evacuated), it is important to distinguish between gauge pressure and absolute pressure. They are connected by the relation Absolute pressure = gauge pressure + atmospheric pressure. If one end of a manometer is closed and evacuated, it can be used as a barometer. See Figure (e) Section 11.5, Pascal s Principle. Pascal made the observation that for a completely enclosed fluid the pressure is the same everywhere save for the variations with depth that we have already studied. Often the variation with depth is not important, and can be neglected. In this case we have the (approximate) statement that the pressure is the same everywhere. This is the principle behind hydraulic jacks. If for no other reasons than cultural, you should also know of Pascal s wager : Even under the assumption that God s existence is unlikely, the potential benefits of believing are so vast (and the cost relatively little) as to make betting on theism rational. (f) Section 11.6, Archimedes Principle. When an object of volume V is submerged in a fluid (and gravity is acting), there is an upward buoyant force whose magnitude is given by the relation F buoyant = weight of fluid displaced = Mg. Here M is the mass of a volume V of the fluid. (g) Sections 11.7 through You should read these sections. However, they will not be covered in the final exam. (h) Section 11.12, Concepts & Calculations. This section reviews previous results and gives further examples. 3

4 You should be able to work all class examples and all examples in the book and all assigned homework having to do with items a-h above. Understand all pictures and figures in chapter Chapter 12: Temperature and Heat (a) Section 12.1, Common Temperature Scales. Section 12.2, The Kelvin Temperature Scale, Section 12.3, Thermometers. As we will see later, temperature is a measure of the energy associated with the random motion of atoms or molecules. There is the fundamental fact that if two bodies A and B have temperatures T A and T B and they are brought into contact so that the atoms in A can hit the atoms in B (and vice versa), then heat energy will flow from A to B if T A > T B. That is, on average, the random motion of the atoms in A will slow down and the random motions of the atoms in B will speed up. Consequently, object A will become colder and object B will become warmer. This energy transfer continues until bodies A and B reach the same temperature. We say that on average heat flows from hot to cold. There are 3 temperature scales in common use: Fahrenheit, Celsius (once called centigrade), and Kelvin. They are connected by the relations T F = 32 + (9/5)T C, T C = T K Water boils at T C = 100 (T F = 212) and freezes at T C = 0 (T F = 32). All random motion stops at T K = 0. For this reason T K is also called absolute temperature. A thermometer is a device for measuring temperature. There are several ways to measure temperature. Some are related to phase transitions (boiling and freezing as described above), some are related to the behavior of gasses (to be discussed in chapter 14), some are related to thermal expansion (to be discussed in the next section), some are related to various electrical properties of various substances, some are related to the emission of light, etc. (b) Section 12.4, Linear Thermal Expansion. When the temperature T of an object of length L is changed by an amount T, the 4

5 length of the object changes by an amount L. These changes are connected by the relation L = αl T. The quantity α is called the coefficient of linear expansion. It is a property of the material out of which the object is made, and does not depend on the geometry of the object. Linear thermal expansion is a very important possibility to take into account when building. (c) Section 12.5, Volume Thermal Expansion. When the temperature T of an object of volume V is changed by an amount T, the volume of the object changes by an amount V. These changes are connected by the relation V = βv T. The quantity β is called the coefficient of volume expansion. It is a property of the material out of which the object is made, and does not depend on the geometry of the object. Volume thermal expansion is also a very important possibility to take into account when building. Typically α and β are connected by the relation β = 3α. This occurs because volume goes as the cube of any linear dimension. (d) Section 12.6, Heat and Internal Energy. As described in item a above, heat energy flows from hot to cold. Although this seems intuitive because it accords with common experience, it is not in fact obvious. There is a vast subject called Statistical Mechanics that is devoted to proving such facts and related facts starting from Newton s law of motion and various assumptions about atoms and molecules and the forces between them. Even after many decades of work, it is still a subject of important research: it is difficult, and there is still much to learn. (e) Section 12.7, Heat and Temperature Change: Specific Heat Capacity. If an amount Q of random energy is transferred to an 5

6 object of mass m (say by pounding or rubbing or touching with a flame, etc.), its temperature is changed by an amount T. These quantities are connected by the relation Q = cm T. ( ) The coefficient c is called the specific heat. It is a property of the substance out of which the object is made, and does not depend on its geometry or mass. That a relation like this should hold is also not obvious. One of the goals of Statistical Mechanics is to show that such relations should be expected and to predict what the specific heat for any substance should be. The relation above can be turned around to define a unit of energy (this was done before it was really understood what was going on, but now is a part of our cultural heritage). A (physics) calorie is defined to be the amount of energy required to raise the temperature of 1 gram of water by 1 degree Celsius. Experiment (and some day we hope theory, once the properties of water are understood) gives for water the result c water = 4186 J/(kgC ). It follows that there is the relation 1 cal = Joules. This relation connects heat energy (the calorie) with mechanical energy (the Joule) and is called the mechanical equivalent of heat. We also note that, by definition, 1 food calorie = 1000 physics calories. Energy conservation and the relation (*) above give rise to the subject of calorimetry. (f) Section 12.8, Heat and Phase Change: Latent Heat. During the course of a phase change (melting or boiling) the temperature does not change, but there is still heat transfer. (That there should be such things as phases and phase changes and that temperature should be constant during a phase change is also not 6

7 obvious, and demands the proof that Statistical Mechanics should provide.) Suppose a mass m of material melts or freezes. Then it is found that the energy transfer Q associated with this phase change is given by the relation Q = ml f. The quantity L f is called the heat of fusion. It is a property of the substance out of which the material is made. For water it has the measured value L f for water = J/kg = 80 cal/g. (A successful Statistical Mechanics should predict this value!) Heat energy must be added to cause melting, and heat energy is given off during freezing. All the while the temperature remains constant. Suppose a mass m of material boils or condenses. Then it is found that the energy transfer Q associated with this phase change is given by the relation Q = ml v. The quantity L v is called the heat of vaporization. It is a property of the substance out of which the material is made. For water it has the measured value L v for water = J/kg = 540 cal/g. (A successful Statistical Mechanics should also predict this value!) Heat energy must be added to cause boiling, and heat energy is given off during condensing. All the while the temperature remains constant. It is a remarkable property of water that it has a very high specific heat and also large heats of fusion and vaporization. The facts above can be combined with energy conservation to give many useful results. (g) Sections 12.9 and You should read these sections. However, they will not be covered in the final exam. 7

8 (h) Section 12.11, Concepts & Calculations. This section reviews previous results and gives further examples. You should be able to work all class examples and all examples in the book and all assigned homework having to do with items a-h above. Understand all pictures and figures in chapter Chapter 14: The Ideal Gas Law and Kinetic Theory (a) In class we considered the gedanken experiment in which N point particles of mass m (an ideal gas) moved about randomly in a box having volume V. We found (from Newton s law of motion) that the particles exerted a pressure P on the walls of the box, and that N, P, V, and the temperature T of the gas, were connected by the remarkable relation P V = NkT. ( ) This relation is called the Ideal Gas Law. Here k is Boltzmanns constant given by the relation k = Joules/degreeK. Also we defined the (absolute) temperature T (in degrees Kelvin) by the rule < K > ave =< (1/2)mv 2 > ave = (3/2)kT. Suppose T is held constant. Then, as a special case of (*) we have the relation P 1/V, which is called Boyle s law. Suppose P is held constant. Then, as a special case of (*) we have the relation V T, which is called Charles or the Charles-Gay-Lussac law. Finally, Suppose V is held constant. Then, as a special case of (*) we have the relation P T. 8

9 (b) Section 14.1, The Mole, Avagadro s Number, and Molecular Mass. When we use the word dozen, we mean the number 12. In the same spirit, a mole is a number, also called Avagadro s number. Avagadro s number, denoted by the symbol N A, is the (big) number Just as we say so too we say N A = a dozen doughnuts = 12 doughnuts, one mole of atoms = N A atoms = atoms. As you know, atoms consist of a heavy nucleus made of neutrons and protons (which are in turn are made of quarks and gluons) surrounded by light electrons. For example, a hydrogen atom is made of 1 proton and 1 electron. As another example, ordinary carbon atoms are made of a nucleus consisting of 6 protons and 6 neutrons, and the nucleus is surrounded by 6 electrons. This is called 12 C because there are 12 objects in the nucleus. (There is also 14 C, which has 6 protons and 8 neutrons in its nucleus. It is radioactive.) Avagadro s number is defined in such a way that 1 mole of 12 C atoms (= N A 12 C atoms) has a mass of exactly 12 grams. The number 12 is called the atomic mass of 12 C. Similarly, the atomic mass of hydrogen, 1 H, is This means that 1 mole of 1 H atoms (= N A 1 H atoms) has a mass of gram. The reason that the number of objects in the nucleus generally doesn t exactly match the atomic mass is that the atomic mass takes into account nuclear binding energies, which are negative, and (as Einstein discovered) mass and energy are related. What would be the mass of 1 mole of nitrogen gas? We know that ordinary nitrogen (the nitrogen in the atmosphere) is in the form 9

10 N 2 (diatomic). Also, atomic nitrogen has an atomic mass of 14. See the periodic table at the back of your book. Therefore 1 mole of N 2 molecules has a mass of 28 (= ) grams. Similarly, 1 mole of O 2 molecules has a mass of 32 (= ) grams. What about water vapor, gaseous H 2 O? We find 1 mole of H 2 O molecules has a mass of 18 (= ) grams. We see that water vapor is lighter than the rest of the atmosphere. That is why clouds float up high and rain comes from above! You should know these facts. (c) Section 14.2, The Ideal Gas Law. Suppose we rewrite the ideal gas law (*) in the form Now make two definitions: P V = (N/N A )(N A k)t. n = N/N A = number of moles of gas, R = N A k = ( )( Joules/degree K) = 8.31 Joules/degree K. With these definitions we can rewrite the perfect gas law in the form P V = nrt. ( ) As a special case, suppose n = 1. Also suppose T = 273 (0 degrees C), P = Pa = 1 atmosphere = 760 mm Hg. 10

11 These conditions are called STP. Then we can solve (**) for V to find the result V = (1)(8.31)(273)/( ) = (meter) 3 = (cm) 3 = ml = 22.4 liters. Thus, 1 mole of gas at STP occupies a volume of 22.4 liters. You should know this fact, Avagadro s number, and the values of k and R. (d) Section 14.3, Kinetic Theory of Gases. The Kinetic Theory of Gases is a part of Statistical Mechanics. It is an example of the grand enterprise of trying to describe and derive the macroscopic properties and behavior of matter from a few fundamental microscopic laws, and goes back to Newton. We have already seen in item a above that we can get the perfect gas law in this way. We also learn that the total energy (often referred to as the internal energy and called U) of a box of N particles at temperature T is given by the relation U = N < K > ave = (3/2)NkT = (3/2)nRT. (e) Sections 14.4, Diffusion. You should read this section, but it will not be covered on the final exam. (f) Section 14.5, Concepts & Calculations. This section reviews previous results and gives further examples. You should be able to work all class examples and all examples in the book and all assigned homework having to do with items a-f above. Understand all pictures and figures in chapter Chapter 16: Waves and Sound. (a) Many substances behave like collections of coupled harmonic oscillators. If one of the oscillators is excited, its excitation is transferred to a neighboring oscillator because of the coupling, and this excitation is in turn transferred to yet another neighbor, etc. Thus, disturbances can propagate through these substances. In 11

12 class you saw a pulse propagating on a string. Likewise, a sonic boom propagates through the atmosphere. It can be shown (from Newton s law of motion) that these disturbances propagate with a fixed velocity independent of the shape or amplitude of the pulse. Some examples are given below: i. For a string under tension τ and having mass µ per unit length, the velocity of propagation for transverse disturbances is given by v = τ/µ. (This is equation 16.2 of the text in disguise.) ii. For a liquid or a gas having bulk modulus B and density ρ, v = B/ρ. For a gas the speed of sound is related to the average thermal speed of its molecules. Thus, the speed of sound increases with temperature. iii. For a solid having Young s modulus Y and density ρ, v = Y/ρ. iv. For electromagnetic disturbances (radio waves, light, x-rays) traveling in vacuum, Maxwells equations (which are analogous to Newton s law in that, given initial electric and magnetic field values, they predict future electric and magnetic field values) give the result v = c = meters/sec. This velocity you should know. (b) A wave is a special kind of disturbance. A wave is a traveling sinusoidal disturbance among a collection of coupled harmonic oscillators. At any given moment (fixed time), the disturbance is periodic in space (sinusoidal), and the distance between successive maxima (or minima) is called the wavelength, and is commonly denoted by the symbol λ. At any given fixed position in space, the 12

13 disturbance is periodic in time, and the time T between successive maxima (or minima) is called the period. See figures 16.6 and in your text. As usual for periodic phenomena, we define the ordinary frequency f and angular frequence ω by writing f = 1/T, (i) ω = 2πf. (ii) Then, as shown in class and in your text (see figure 16.7), there is the relation v = fλ. (iii) For a wave propagating along the +x axis, let D(x, t) denote the disturbance at point x and time t. Then a precise definition of a such a wave is given by writing D(x, t) = A sin[ω(x/v t)]. (iv) Here A is the amplitude of the disturbance. Evidently we can keep D the same by simultaneously increasing t and x in such a way as to keep the quantity (x/v t) fixed. Thus we do indeed have a disturbance propagating in the +x direction, and a little further thought shows that the velocity of propagation is indeed v. Next, suppose we fix the time, say at t = 0. Then we find for D(x, 0) the result D(x, 0) = A sin(ωx/v) = A sin(2πx/λ). (v) Here we have used (ii) and (iii). Evidently D(x, 0) is periodic in x with wavelength λ as desired. Finally, suppose we fix the position, say at x = 0. Then we find for D(0, t) the result D(0, t) = A sin[ω( t)] = A sin(2πt/t ). (vi) Here we have used (i) and (ii). Evidently D(0, t) is periodic in t with period T as desired. 13

14 (c) Section 16.1, The Nature of Waves. This section covers some of the same ground as items a and b above. However, Cutnell is sloppy, and does not distinguish disturbances in general from the special kind of disturbances that are called waves. (d) Section 16.2, Periodic Waves. This section also covers some of the same ground as items a and b above. However, again Cutnell is not as fussy as he should be. By definition, all waves are periodic, there are no other kind of waves. (e) Section 16.3, The Speed of a Wave on a String. This section covers some of the ground of item a above. However, it does not make clear that all kinds of disturbances, including waves, propagate with the same speed. (f) Section 16.4, The Mathematical Description of a Wave. This section is Cutnell s version of section b above. It would be better if he had used the symbol D in his equations (16.3) and (16.4) since his symbol y suggests a spatial coordinate perpendicular to the x direction. This would be all right for a wave describing transverse physical displacements, but is not appropriate for longitudinal waves or electromagnetic waves. (g) Section 16.5, The Nature of Sound. This section discusses the nature of sound disturbances and sound waves in more detail. By this point you should know the difference between a transverse and a longitudinal disturbance. (h) Section 16.6, The Speed of Sound. This section covers some of the same ground as item a above. (i) Sections 16.7 through You should read these sections. However, they will not be covered on the final exam. (j) Section 16.12, Concepts and Calculations. This section reviews previous results and gives further examples. You are not responsible for Example 12. You should be able to work all class examples and all examples in the book and all assigned homework having to do with items a-j above. Understand all pictures and figures in chapter

15 22. Lab 9, Mechanical equivalent of Heat. You should understand the Theory for this experiment. You should also understand all concepts, equations, calculations, and figures associated with this lab. You are expected to know the specific heat of liquid water and the conversion between calories and joules. 15