Final Review Prof. WAN, Xin

Transcription

1 General Physics I Final Review Prof. WAN, Xin xinwan@zju.edu.cn

2 About the Final Exam Total 6 questions. 40% mechanics, 30% wave and relativity, 30% thermal physics. Pick easier ones to solve first. You are allowed to bring a calculator and a dictionary (without communication function). Intermediate results may be have credits. So don't be lazy.

3 The Scope Classical Mechanics (or Newtonian Mechanics) Einstein's Theory of Relativity Thermal Physics When to use what?

4 The Scope Classical Mechanics when the speed is small (compared to the speed of light) when we do not need super-high precision (e.g., of time), and when the number of objects is small (unless we have a computer to help) Einstein's Theory of Relativity When there is no classical counterpart (such as fission) when the speed is close to the speed of light, or when we do need super-high precision (e.g., for GPS) Thermal Physics when we have many microscopic particles (e.g. in gases), and when we are interested in macroscopic properties

5 Calculate, Estimate, or Guesstimate Based on exact or perturbatively exact formulas, you can calculate velocity, acceleration, work, heat, entropy, etc. You can often determine the dependence of a physical quantity (e.g., the wave velocity for the wave on a string) on a few other parameters (e.g., tension and linear density) by dimension analysis. The prefactor may not be fixed, but you can estimate the value of the quantity. Guesstimate requires you to come up with a simple model or reasoning to estimate the approximate order of magnitude of a certain quantity, often with guessing.

6 Dimension Analysis Dimension analysis often helps us solve unknown problems in an easy way. Consider an object moving at velocity v relative to a fluid. Turbulent flow can occur if v is too large. But how large is too large? It should depend on characteristic length scale, the viscosity and density of the fluid. We expect [L/T ] [?] v c η a ρ b D c [ML 3 ] [L]

7 Dimension Analysis You then need an independent formula for viscosity. A y F, v A F = η A dv dy [η] = [ML 1 T 1 ]

8 Reynolds Number We then obtain a = 1, b = -1, and c = -1 vc ~ h / (rd) vc = R c h / (rd) Reynolds number: R = ρ D η v R < 2000 R > 4000

9 Energy: Various Forms 1 F Δ x 2 mv2 GMm r W =p ΔV k B T γ mc I ω2 mgh Q=T Δ S 1 2 k(x x 0) 2

10 Conservation of Energy Broader sense: The first law of thermodynamics Narrower sense: The conservation of mechanical energy ΔU = Q P Δ V 1 2 mv I ω2 + mgh + = constant

11 Linear vs Angular Momentum Symmetry & conservation law Translational invariance conservation of linear momentum Rotational invariance conservation of angular momentum

12 Linear vs Angular Momentum Symmetry & conservation law Translational invariance conservation of linear momentum Rotational invariance conservation of angular momentum

13 Path Dependence Conservative forces A force is conservative if the work it does on a particle moving between any two points is independent of the path taken by the particle. F x = du dx State functions: internal energy, entropy U (T,V ) = f 2 RT

14 Equilibrium Mechanical equilibrium: An object is in equilibrium means that both the linear acceleration and the angular acceleration are zero. Thermal equilibrium: After two objects have been in contact long enough (such that their macroscopic properties no longer change), we say that they are in thermal equilibrium (microscopic properties still change).

15 Example: Standing on a Beam

16 Motion Figure out whether you have a problem of translation, rotation, vibration, or combined motion, such as pure rolling motion. Whenever you can use the conservation laws, use them first. Write down Newton's law if necessary (use torque, moment of inertia, and angular acceleration in a rotation problem). Integrate the acceleration, then the velocity, to obtain the motion of the object(s).

17 Continuous Medium The philosophy is to pick a piece of the continuous medium, such as a string segment or a liquid cube, and treat it as a particle like in the Newtonian mechanics. A 1 v 1 = A 2 v 2 P ρ v2 + ρgh = constant v = T μ v = ω/k = λ f y = Asin(kx ωt + φ)

18 Summary on Relativity The two basic postulates of the special theory of relativity are The laws of physics must be the same in all inertial reference frames. The speed of light in vacuum has the same value c = m/ s in all inertial frames, regardless of the velocity of the observer or the velocity of the source emitting the light.

19 Summary on Relativity To satisfy the postulates of special relativity, the Galilean transformation equations must be replaced by the Lorentz transformation equations: t ' = γ(t vx/c 2 ) x ' = γ(x vt) y ' = y where z ' = z γ = 1 1 v 2 /c 2

20 Summary on Relativity Three consequences of the special theory of relativity are Events that are simultaneous for one observer are not simultaneous for another observer who is in motion relative to the first. Clocks in motion relative to an observer appear to be slowed down by a factor γ = (1 v 2 /c 2 ) 1/2. This phenomenon is known as time dilation. The length of objects in motion appears to be contracted in the direction of motion by a factor 1/ γ = (1 v 2 /c 2 ) 1/2. This phenomenon is known as length contraction.

21 Summary on Relativity The relativistic form of the velocity transformation equation is where γ = 1 1 v 2 /c 2 u x is the speed of an object as measured in the K frame and u' x is its speed measured in the K' frame.

22 Summary on Relativity The relativistic expression for the linear momentum of a particle moving with a velocity u is Not relative v of K & K' The relativistic expression for the kinetic energy of a particle is Mass is an invariant

23 Example u = v -v v m m m m 2v 1 + v 2 /c 2 M v M v γ u = 1 1 u 2 /c 2 γ u mu = = 1 + v 2 /c 2 1 v 2 /c 2 Mv 1 v 2 /c 2 M = 2m 1 v 2 /c 2

24 Thermal Physics: Formulas N( v) 4 N m 2 kt 3/ 2 v 2 e mv 2 / 2 kt 3kT v rms v 2 m PV nrt U = 3 2 NkT = 3 2 nrt = 3 2 PV Adiabatic process PV const

25 Thermal Physics: Formulas Δ U = Q W W V V i f PdV C V = dq dt fi xed V C P = dq dt fixed P (C P C V ) per mole = R e efficiency cost W Q in Δ S = Q T reversible

26 Internal Energy, Work, Heat & Entropy Let's work them out on the blackboard for the following processes: isovolumetric, isobaric, isothermal, adiabatic, and adiabatic free expansion.