Lecture 9
• Beats
• Quiz
• Chapters 14, 15 (except 15.6), 20 and 21
• diverse topics: start preparing now
Reminder:1st mid-term (Tuesday, March 9)
• Introduction to thermodynamics
Beats• superposition of waves of slightly different f
(so far, same f): e.g. 2 tones with single tone with intensity modulated: loud-soft-loud
• Simplify:
D1 = a1 sin (k1x! !1t + "10)D2 = a2 sin (k2x! !2t + "20)
a1 = a2 = a; x = 0; !10 = !20 = "
D = D1 + D2 =!2a cos !mod.t
"sin!avg.t
!avg. = 12 (!1 + !2): average frequency
!mod. = 12 (!1 ! !2): modulation frequency
!f ! 1 Hz
Conditions for beats
• Note heard is , but loud-soft-loud: modulation (periodic variation of variation of amplitude)
• waves alternately in and out of phase: initially crests overlap A = 2 a, but after a while out of step (A = 0) due to unequal f ’s
Each loud-soft-loud is 1 beat
Beats frequency
!1 ! !2 " !avg. ! !1 or 2
!mod. # !avg.
2a cos !mod.t is slowly changing amplitudefor rapid oscillations sin!avg.t
fbeat = 2fmod. = f1 ! f2
I ! A2 ! cos2 !mod.t = 12 (1 + cos 2!mod.t)
favg. = !avg./(2") ! f1, 2
Thermodynamics (Chs. 16-19)• Next few weeks:
• This week
• Today:
bulk properties (T, p)
macro/micro connection: T related to motion of atoms...2nd law...
macroscopic systems instead of particles; temperature (T), pressure (p)...instead of position, velocity...
3 phases of matter; mass...
transformation of energy from one form to another: e.g. engines convert fuel energy into mechanical; living organisms convert chemical energy of food into kinetic...
phases of matter; phase changesstart micro/macro connection
How do pressure, temperature change or work done by engines related to fuel energy used?
ultimate goal: how heat engines convert heat energy to work
Solids, Liquids and Gases
• Solids
• Liquids
• Gases
rigid: atoms vibrate, but not free to move; incompressible (atoms as close as can be); crystal (arranged in periodic arrray) or amorphous (random)
change between liquild/solid or liquid/gas: phase change
incompressible, but flows/fits shape of container: molecules can slide around each other, but can’t get far apart
freely moving molecules...till collide with each other/wall compressible (lot of space...)
Atoms and Moles (I)
• parameters describing macroscopic systems e.g. mass (M), density ( ), p, T...related: e.g.,
• change state by changing variables:
• Thermal equilibrium: state variable not changing
• mass depends on total number of atoms, N, typically
State variables
: larger in solid than gas, independent of volume for uniform system
!X = Xf !Xi
! 1025, i.e., 1024 to 1026 " 105 or 1015
! = MV!
Atoms and Moles (II)• mass of atom determined by atomic mass number (number of
protons and neutrons)
• atomic mass unit: mass of atom (6 p + 6 n) defined to be 12 u
• molecular mass: sum of atomic masses
• 1 mole contains Avogadro’s number: of basic particles (as in12 g of ): e.g. number of moles,
• molar mass ( ): mass in grams of 1 mole (12 g/mol. for ) = atomic mass in u
1 atom of 12C has mass 12 g6.02!1023 = 1.993! 10"26 kg
" 1 u =m
“12C
”
12 = 1.661! 10"27 kgm
!20Ne
"= 20 u = 3.32! 10!26 kg
m!1H
"/m
!12C
"= 1.0078
12 ! m!1H
"= 1.00078u
m!1H
"" A (neglect small di!erence)
6.02! 1023
12C
12C
12CMmol.
1 mole of O2: 2! 6.02! 1023 O atoms