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