Equilibrium and Forces in Physics: Contact, Field, and Thermal - Prof. Donald G. Luttermos, Study notes of Physics

Download Equilibrium and Forces in Physics: Contact, Field, and Thermal - Prof. Donald G. Luttermos and more Study notes Physics in PDF only on Docsity! Physics 2018: Great Ideas in Science: The Fall 2008 Physics Module Dr. Donald G. Luttermoser East Tennessee State University Edition 2.0 Abstract These class notes are designed for use of the instructor and students of the course Physics 2018: Great Ideas in Science. This edition was last modified for the Fall 2008 semester. 2. Velocity. a) The average velocity, v, is defined as the displacement divided by the time interval during which the displace- ment occurred: v = ∆x ∆t = xf − xi tf − ti or in vector notation: ~v = ∆~x ∆t = ~xf − ~xi tf − ti . (I-3) b) The average velocity of an object during the time interval ti to tf is equal to the slope of the straight line joining the initial and final points on a graph of the position of the object plotted versus time. c) When plotting the value of a variable as a function of some other variable, the path on that plot that the object takes is called a trajectory. d) The instantaneous velocity, ~v, is defined as the limit of the average velocity as the time interval ∆t becomes infinitesimally small. ~v ≡ lim ∆t→0 ∆~x ∆t ≡ d~x dt . (I-4) 3. Acceleration. a) The average acceleration, a, during a given time inter- val is defined as the change in velocity divided by that time interval during which the change occurs: ~a = ∆~v ∆t = ~vf − ~vi tf − ti . (I-5) b) The instantaneous acceleration of an object at a cer- tain time equals the slope of the velocity-time graph at I–3 that instant in time (i.e., tangent line of v(t) at that point). ~a ≡ lim ∆t→0 ∆~v ∆t ≡ d~v dt = d dt ( d~x dt ) = d 2~x dt2 . (I-6) C. Newton’s Laws of Motion. 1. Newton’s 1st Law: Law of Inertia: An object at rest re- mains at rest, and an object in motion continues in motion with a constant velocity, unless it is acted upon by an external force. a) Inertia is the resistance that matter has to changes in motion. b) The mass of an object measures that object’s inertia. =⇒ Mass is nothing more than a measure of matter’s resistance to changes in motion. 2. Newton’s 2nd Law: The acceleration (a) of an object is di- rectly proportional to the resultant force (F ) acting on it and inversely proportional to its mass (m). The direction of the ac- celeration is the same direction as the resulting force. ∑ ~F = m~a . (I-7) a) This is arguably the most important equation in physics and possibly all of science. b) Force is measured in newtons in the SI system: 1 N ≡ 1 kg·m/s2, (I-8) =⇒ or in the cgs system: 1 dyne ≡ 1 g·cm/s2 = 10−5 N, =⇒ or in the English system: I–4 1 lb ≡ 1 slug·ft/s2 = 4.448 N. 3. Newton’s 3rd Law: If 2 bodies interact, the magnitude of the force exerted on body 1 by body 2 is equal to the magnitude of the force exerted on body 2 by body 1, and these forces are in opposite direction to each other. a) Another way of saying this is “for every action, there is an opposite reaction.” b) Newton’s 3rd law is nothing more than the conservation of linear momentum. D. Equilibrium in Motion. 1. Keywords that tell you an object is in (motion) equilibrium =⇒ no acceleration. a) Body is at rest (not changing position). b) Body is static (not changing in time). c) Body is in steady state (no acceleration). 2. When objects are in (motion) equilibrium: ∑ ~F = 0 (I-9) or in component format ( ∑ means summation of all forces): ∑ Fx = 0 ∑ Fy = 0 ∑ Fz = 0 . I–5 heat and work. Thermodynamics is essentially the study of the motion of heat. 2. The First Law of Thermodynamics. a) In words: The change in internal energy of a system equals the difference between the heat taken in by the system and the work done on the system. b) When an amount of heat Q is added to a system, some of this added energy remains in the system increasing its internal energy by an amount ∆U . c) The rest of the added energy leaves the system as the system does work W . d) Mathematically, note that ∆U = Uf − Ui , and the First Law states ∆U = Q + W . (I-10) e) In thermo, there will always be two specific regions in which we will be interested in: i) The system — the region of interest where we wish to know the state parameters (e.g., P = pres- sure, T = temperature, and V = volume). Ther- modynamic variables relating to the system will re- main ‘unsubscripted’ in these notes (i.e., W = work on the system). ii) The environment (also called the universe) — the region that contains the system. Note that I–8 some scientists (including me) prefer to use the word environment over universe since in astronomy the word Universe means all of the cosmos and not just the immediate (i.e., nearby) surroundings which is what thermo means by “universe.” Ther- modynamic variables relating to the environment will be labeled with the ‘env’ subscript in these notes (i.e., Wenv = work on the environment). iii) Note that work done on the system W is the exact opposite of work done on the environment: W = −Wenv . f) Note that we will often be interested in systems that are completely isolated from the environment. Such a system is called a closed system. 3. The Second Law of Thermodynamics — The Classical Description. a) The second law of thermodynamics deals with how heat flows. It is essentially a description of change. i) Change: To make different the form, nature, and content of something. ii) Change has, over the course of time and through- out all space, brought forth, successively and suc- cessfully, galaxies, stars, planets, and life. iii) Evidence for change is literally everywhere. iv) Much of the change is subtle, such as when the Sun fuses hydrogen into helium sedately over bil- I–9 lions of years or when the Earth’s tectonic plates drift sluggishly across the face of our planet over those same billions of years. v) Indeed, our perception of time is nothing more than our noticing changes on Earth and in the Uni- verse as a whole. b) There are two classical formulations of this law (both es- sentially mean the same thing): i) Clausius statement of the second law: Heat cannot, by itself, pass from a colder to a warmer body. ii) Kelvin-Planck statement of the second law: It is impossible for any system to undergo a cyclic process whose sole result is the absorption of heat from a single reservoir at a single temperature and the performance of an equivalent amount of work. c) The 2nd law specifies the way in which available energy (also called “usable energy,” “free energy,” or “potential energy”) change occurs. i) This law’s essence stipulates that a price is paid each time energy changes from one form to another. ii) The price paid (to Nature) is a loss in the amount of available energy capable of performing work of some kind in the future. iii) We define here a new term to describe this de- crease of available energy: entropy, S. It is derived I–10 ii) Boltzmann’s law of entropy then signifies that or- dered states tend to degenerate into disordered ones in a closed system. c) As can be seen, the concept of “disorder” is very difficult since it requires a detailed knowledge of probability and statistics. i) The best way to learn about probabilities is through example. Let’s say we have one die from a set of dice. There are 6 sides with dots imprinted on the sides relating to the numbers 1, 2, 3, 4, 5, and 6. ii) The probability of one number (say ‘4’) coming up is p = n N = a given state total number of states , (I-13) and for this case, n = 1 and N = 6, so p = one side total number of sides = 1 6 = 0.167 , or a 16.7% chance that we would role the die with a ‘4’ landing on top. iii) The probability of an even number (2, 4, or 6) landing on top is (n = 3 and N = 6) p = 3 6 = 0.50 , or a 50% chance to role such a number. d) Through this concept of entropy, we can rewrite the sec- ond law of thermodynamics as any of the following state- ments: i) The entropy of the Universe as a whole increases in all natural processes. I–13 ii) Isolated systems tend towards greater disorder and entropy is a measure of that disorder. iii) In a closed system, entropy increases over time =⇒ less and less energy can be converted into work. iv) All of these statements are probabilistic in nature → on average this is true. e) Note that the second law written in this probabilistic way can be violated locally =⇒ entropy can decrease locally. Only over the whole isolated (or closed) system over a long enough period of time, will necessitate an increase in entropy. i) Note that on the Earth, entropy decreases all the time at the expense of an increase of entropy of the Sun. ii) As such, the second law of thermodynamics cannot be used as proof against the theory of biological evolution as some people have suggested. f) The bottom line is that we no longer regard things as “fixed” or “being,” or even that they “exist.” Instead, everything in the Universe is “flowing,” always in the act of “becoming.” All entities — living and non-living alike — are permanently changing. I–14 H. Transport of Heat Energy. 1. Before discussing equilibrium vs. non-equilibrium states, we need to see how heat moves through matter and/or space. 2. Thermal energy (i.e., heat) can only flow by one of three dif- ferent mechanisms: conduction, convection, and radiation transport. 3. Heat Transfer by Conduction. a) Conduction is the process by which heat is transferred via collisions of internal particles that make up the object =⇒ individual (mass) particle transport. i) Heat causes the molecules and atoms to move faster in an object. ii) The hotter molecules (those moving faster) col- lide with cooler molecules (those moving slower), which in turn, speeds up the cooler molecules mak- ing them warm. iii) This continues on down the line until the object reaches equilibrium. b) The amount of heat transferred ∆Q from one location to another over a time interval ∆t is ∆Q = P ∆t . (I-14) i) P ≡ heat transfer rate. ii) P is measured in watts when Q is measured in Joules and ∆t in seconds. I–15 b) The rate at which an object emits radiant energy is given by the Stefan-Boltzmann Law: Pem = σAeT 4 . (I-17) i) Pem ≡ power radiated (emitted) [watts]. ii) σ ≡ Stefan-Boltzmann’s constant = 5.6696×10−8 W/m2/K4. iii) A ≡ surface area of the object [m2]. iv) e ≡ emissivity [unitless] (e = 1 for a perfect ab- sorber or emitter). v) T ≡ temperature [K]. c) A body also can absorb radiation. If a body absorbs a power of radiation Pabs, it will change its temperature to T◦. i) The net power radiated by the system is then Prad = Pnet = Pem − Pabs (I-18) or Prad = σAeT 4 − σAeT 4◦ , Prad = σAe ( T 4 − T 4◦ ) . (I-19) ii) In astronomy, the total power radiated by an ob- ject over its entire surface is called the luminosity, L = Prad, of the object. Since the amount of energy falling on the surface of the Sun (or any isolated star) from interstellar space is negligible to that of the power radiated, L = Pem for isolated stars. I–18 iii) If an object is in equilibrium with its surround- ings, it radiates and absorbs energy at the same rate =⇒ its temperature remains constant =⇒ this ra- diative equilibrium results in the object being in thermal equilibrium: Prad = 0 =⇒ T = T◦ , where T◦ is the temperature of the surroundings. d) An ideal absorber is defined as an object that absorbs all of the energy incident upon it. i) In this case, emissivity (e) = 1. ii) Such an object is called a blackbody (see next section): Pbb = σAT 4 . (I-20) iii) Note that a blackbody radiator can be any color (depending on its temperature =⇒ red blackbod- ies are cooler than blue blackbodies), it does not appear “black” (unless it is very cold). iv) The energy flux of such a radiator is Fbb = Pbb A = σ T 4 . (I-21) v) This radiative flux results from the condition that in order to be in thermal equilibrium, the heat gained by absorbing radiation must be (virtually immediately) radiated away by the object. vi) This is not the same thing as reflecting the radi- ation off of the surface (which does not happen in I–19 a blackbody). The incident radiation does get “ab- sorbed” by the atoms of the object and deposited in the thermal “pool.” It is just that this radiation immediately gets re-emitted just after absorption. I. Blackbody Radiation. 1. Late in the 1800s and in the early part of the 20th century, Boltz- mann, Planck, and others investigated E/M radiation that was given off by hot objects. a) In general, matter can absorb some radiation (i.e., pho- tons converted to thermal energy), reflect some, and trans- mit some of the energy. b) The color of cool objects, objects that don’t emit their own visible light, is dictated by the wavelengths of light they either reflect, absorb, or transmit. i) A blue sweater is “blue” because the material reflects blue light (from either room lights or the Sun) more effectively than the other colors of the rainbow. ii) Coal is black because it absorbs visible light and reflects very little. iii) Glass is transparent because visible light is al- most completely transmitted through the glass with little absorption and reflection. 2. To make the physics a little less complicated, these scientists invented the concept of an ideal or perfect radiator. a) A hypothetical body that completely absorbs every kind of E/M radiation that falls on it. I–20 7. From the Stefan-Boltzmann’s and Wein’s Displacement Laws, we see that as an object get cooler, it gets redder and fainter. And, as an object gets hotter, it gets bluer and brighter. J. Equilibrium vs. Non-Equilibrium. 1. Definitions. a) The simplest definition of equilibrium is being in a state of balance. b) States in equilibrium have the lowest internal energy pos- sible for that given state. c) As a result, non-equilibrium simply means that an im- balance must be present for a given state, or two oppos- ing forces being unequal. 2. Types of Equilibria. a) Equilibrium Forces. These are forces that counteract other forces that cause motion either forcing an object to remain at rest or continuing its uniform motion (i.e., no acceleration). i) Say we have an object resting on a desk. The weight of the object, or the gravitational force down- ward, is balanced by the force of the desk pushing upward (as there must be to keep the object from moving in response to gravity). This upward force is called the normal force. “Normal” because this force is perpendicular to the surface of the desk. This normal force results from the electromagnetic forces between the atoms and molecules in the desk that make it rigid (i.e., a solid). I–23 ii) If an object hangs from wires or ropes, another equilibrium force (i.e., counteracting gravity) is the tension of the rope. iii) Bodies in motion often feel frictional forces (i.e., from surfaces, air, etc.) which retard their motion =⇒ frictional force is in the opposite direction of the direction of motion. If an ob- ject does not move on a surface (~a = 0), it may be experiencing a force of static friction, ~fs, with possible values of fs ≤ µs n , (I-25) =⇒ µs ≡ coefficient of static friction, n ≡ magnitude of the normal force. Since the object doesn’t move under applied force, ~F , ~F − ~fs = 0 . (I-26) If one continues to increase the applied force un- til the object is just on the verge of slipping (i.e., moving), we have reached the maximum of static friction fs, max = µs n . (I-27) Once the object is in motion, it experiences the force of kinetic friction, ~fk: fk = µk n , (I-28) =⇒ µk ≡ coefficient of kinetic friction. Note that µk does not have to equal µs for the same surface. Usually, µk < µs and ranges between 0.01 and 1.5. I–24 iv) Objects that stay suspended in water (neither rise or sink in water), have a buoyancy force counteract the weight of the object in the water. This buoyancy force arises from a water pressure differential between the top and bottom of the sur- face of the object. b) Thermal Equilibrium. i) Thermal equilibrium means that an object has the same temperature throughout its interior. ii) Objects in thermal equilibrium radiate as black- bodies. iii) For an object to remain in thermal equilibrium is it absorbs energy from the environment, it must emit that same amount of energy that it absorbs. c) Hydrostatic Equilibrium. i) Whenever you have a volume of gas, there are two competing forces: • The force per unit volume from the internal pres- sure: F/V = ∆P ·A/V . • The weight per unit volume of the gas trying to pull the gas to the lowest potential: Fg/V = −mg/V = −ρg. ii) Assume we have a column of gas, then A = πr2 and V = πr2 ∆z, where r is the radius of the col- umn cylinder and ∆z is the length of the column. I–25 Iλ = (c/λ 2) Iν , (I-35) since ν = c/λ. ii) The total intensity is I = ∫ ∞ 0 Iν dν = ∫ ∞ 0 Iλ dλ. (I-36) iii) It is related to the electric field of the EM wave by the equation Iν(z) = ◦c|Eν(z)|2 , (I-37) where ◦ is the permittivity of free space, c is the speed of light, and I is the intensity of the electric field at point z in space along the “path” of the EM wave. iv) Once a photon is emitted into a vacuum (say from a star’s surface), that is, it does not inter- act with matter during flight, its intensity remains the same at all points along its flight path −→ Iν independent of distance r (or d). θ dΩ n dσ φ I–28 f) The brightness or strength of light corresponds to the ra- diation flux: Fν = π Fν = ∫ 2π 0 ∫ 1 −1 Iν(µ, φ) µ dµ dφ. (I-38) i) µ = cos θ and dΩ = sin θ dθ dφ = dµ dφ. ii) Fν is called the astrophysical flux. iii) In an isotropic (i.e., same in all directions), Iν is independent of θ and φ =⇒ Fν = 0. iv) It is often useful to separate Fν [W/m 2/Hz in SI units and erg/s/cm2/Hz in cgs units] into an out- ward and an inward component in an atmosphere of gas: π F+ν = ∫ 2π 0 ∫ 1 0 Iν µ dµ dφ (I-39) (0 ≤ µ ≤ 1) outward flux π F−ν = ∫ 2π 0 ∫ 0 −1 Iν µ dµ dφ (I-40) (−1 ≤ µ ≤ 0) inward flux, from which we can write Fν = F + ν + F − ν . (I-41) v) If Iν is axisymmetric (i.e., independent of φ), then the flux equation becomes: Fν = 2π ∫ 1 −1 Iν µ dµ. (I-42) vi) The total radiation flux is then given by F = ∫ ∞ 0 Fν dν = ∫ ∞ 0 Fλ dλ . (I-43) I–29 vii) Note that 4π R2 Fν = Lν is the monochromatic luminosity, where R is the radius of a spherical ob- ject (e.g., a star) emitting the light. viii) Fν = π Iν if Iν is isotropic outward and zero inward. g) Unlike intensity, flux does scale with distance =⇒ an ob- ject gets fainter the farther away it is: fν = ( R? d )2 Fν = 1 4 α2? Fν . (I-44) i) fν is the observed flux of a star of radius R? at a distance d. ii) α? is the angular diameter (in radians) of the star as seen at distance d. h) If a photon is traveling within a medium (i.e., non-vacuum), its intensity does change as it propagates through the gas depending on the opacity of the gas. i) The opacity, χ [cm−1], of a gas measures how opaque the gas is. The opacity of a transition (or continuum) depends primarily on the probability that a transition can occur, which, of course, de- pends upon the wave functions of the transition. ii) It is the inverse of the mean-free-path, L [cm], of the photon =⇒ the distance a photon travels before it interacts with another particle. iii) The opacity dictates how deep we can see into a gas. As such, the optical depth along depth s (s I–30 n) For non-thermodynamic equilibrium gas, however, Sν is much more difficult to ascertain, since it will depend upon both the mean intensity (Jν = 1 2 ∫ Iν dµ) of the photons and the thermal nature of the gas in the volume of interest =⇒ the equation of transfer becomes a integral-differential equation in Iν! i) The differential portion of the radiative transfer is obvious from Eq. (I-50). The integral portion comes from the expression of the source function in these cases: Sν = rν φν + rν Scν + φν φν + rν S`ν , (I-53) where Scν is the continuum source function, S ` ν is the line source function, rν = χcν χ`ν◦ (I-54) is the fraction of the continuous opacity, χcν, over the line center opacity, χ`ν◦, φν = χ`ν χ`ν◦ (I-55) is the profile function (usually a Gaussian, Lorent- zian, or Voigt [combination of the first two] profile) of the line which is the fraction of the line opacity at some portion of the profile, χ`ν, over the line center opacity, χ`ν◦. ii) The continuum source function is determine by Scν = κνBν + σνJν κν + σν = cνBν + (1 − cν)Jν , (I-56) where Jν is the angle-averaged mean intensity (i.e., an integral over angle θ — see previous page), and I–33 cν = κν/(κν + σν) is the thermal fraction of the continuum emission. iii) The line source function is somewhat more diffi- cult to determine in non-equilibrium cases: S`ν = ( 1 − `ν ) ∫ φνJν dν +  ` νBν = ( 1 − `ν ) J + `νBν , (I-57) where J-bar (J) is the profile-integrated mean in- tensity, and `ν is the thermal fraction of the line emission. iv) The line source function also can be determined from S`ν = 2hν3 c2  ni gj nj gi − 1   −1 , (I-58) where ni and nj are the particle number densities of the lower (‘i’) and upper (‘j’) energy states, respec- tively, and gi and gj are their corresponding statis- tical weights which are calculated from quantum numbers. In this equation, the number density ra- tio is determined from the radiative and collisional rate equations and the solution to radiative rates equation require integrals to be solved (see below). Note that Eq. (I-58) reduces to the Planck function is the particle density ratio achieves its equilibrium value. o) Now we have to worry about the net rate equations. Let Ni be the number of atoms in a sample of gas who have electrons in a lower-energy state and let Nj be the number in the upper-energy state. If we ignore matter particle collisions, the rate in which the upper level either gains I–34 or loses an electron to the lower level is given by dNj dt = −NjAji − NjBjiρ(νij) + NiBijρ(νij) . (I-59) i) Aji represents the spontaneous emission rate. The number of particles leaving this state, per unit time, by this mechanism is NjAji. ii) The transition rate for stimulated emission is pro- portional to the energy density of the EM field, ρ, whose energy corresponds to the energy difference of the two states, hνij. The number of particles leaving this state, per unit time, by this mechanism is NjBjiρ(νij). iii) The transition rate for absorption from the lower level to the upper level also is proportional to the energy density of the electromagnetic field of en- ergy hνij. The number of particles entering the upper state from the lower state, per unit time, is NiBijρ(νij). iv) Remember that ρ is the radiation field energy density which depends upon the integral of the ra- diation intensity. As such, determining the values of the Einstein-B terms requires cross-talk with the radiative transfer equation. p) This is as far as we need to go here. I just wanted to demonstrate that carrying out non-equilibrium physics is very complicated. This is the main reason why scientists usually work on the equilibrium cases first, they are much simpler to solve. I–35