The Dulong-Petit law proposed in 1819 states that the heat capacity of many solid elements is approximately 3R per mole, where R is the universal gas constant. Dulong and Petit experimentally found the heat capacity of various elements was close to a constant value when multiplied by the assumed atomic weight. In modern terms, they discovered the heat capacity per mole of many elements is about 25 joules per kelvin. Later, Einstein derived that the heat capacity of solids is due to lattice vibrations, providing a theoretical basis for the Dulong-Petit law being expressed in terms of molar heat capacities. The law offers a good prediction for many simple crystalline solids at high temperatures but fails at low temperatures and
The document summarizes key concepts about spin algebra:
- Spin ("S") is the intrinsic angular momentum of fundamental particles. It is quantized and can only take on discrete values.
- For a spin-1/2 particle (e.g. electron), there are two eigenstates of the spin operators S2 and Sz: |0⟩ and |1⟩, representing spin up and down.
- The spin operators S2, Sx, Sy, Sz can be represented by 2x2 matrices in the basis of the |0⟩ and |1⟩ eigenstates. This allows spin states to be modeled as two-component spinors.
This document defines key terms and equations related to simple harmonic motion (SHM). It discusses oscillating systems that vibrate back and forth around an equilibrium point, like a mass on a spring or pendulum. The key parameters of SHM systems are defined, including amplitude, wavelength, period, frequency, displacement, velocity, acceleration. Equations are presented that relate the displacement, velocity, acceleration as sinusoidal functions of time. The concepts of kinetic, potential and total energy are also explained for oscillating systems undergoing SHM.
How to find moment of inertia of rigid bodiesAnaya Zafar
The document provides expressions for calculating the moment of inertia of various regularly shaped rigid bodies about different axes of rotation. It discusses:
1) Calculating moment of inertia using integral methods, considering small elements of the rigid body.
2) Examples of calculating moment of inertia for a rod, rectangular plate, circular ring, thin circular plate, hollow cylinder, solid cylinder, hollow sphere, and solid sphere.
3) Key steps involve identifying the small element, elemental mass, and integrating the expression for elemental moment of inertia over the body.
This document discusses rotational motion and provides definitions and equations for key angular quantities such as angular displacement (θ), angular velocity (ω), angular acceleration (α), torque (τ), moment of inertia (I), angular momentum (L), and rotational kinetic energy. It defines these quantities, gives their relationships to linear motion quantities, and provides examples of how to set up and solve problems involving rotational dynamics.
1. The Stern-Gerlach experiment discovered that silver atoms split into two beams, indicating the presence of an intrinsic "spin" angular momentum of 1/2 beyond orbital angular momentum.
2. Elementary particles are classified as fermions, with half-integer spin, and bosons, with integer spin. The spin of the electron is represented by a two-component spinor.
3. In a magnetic field, the spin precesses around the field direction at the Larmor frequency, independent of initial spin orientation. This principle underlies paramagnetic resonance and nuclear magnetic resonance spectroscopy.
Lecture 8: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
1. O documento apresenta 10 questões sobre física que abordam tópicos como campo elétrico de uma esfera carregada, campo eletromagnético em um capacitor de placas paralelas, mecânica quântica em poços de potencial e decaimento de múons.
2. As questões envolvem cálculos de campo elétrico, força, energia, probabilidade, momento angular, equações de Lagrange e Hamilton, e processos termodinâmicos em uma cavidade ressonante.
3. São abordados conce
This document discusses damped and forced harmonic motion. It explains that in damped harmonic motion, a damping force acts opposite to the velocity to dissipate energy and stop vibrations. The damping causes the amplitude to decay exponentially over time. A system can be under-damped, over-damped, or critically damped depending on how quickly it stops oscillating. Forced harmonic motion occurs when an external periodic force drives the system, like pushing a swing. At resonance, the driving frequency matches the natural frequency, causing large amplitude oscillations. While resonance can be dangerous if it causes collapse, it can also be useful in applications like radios and musical instruments.
The document discusses the derivation of the Navier-Stokes equations, which describe compressible viscous fluid flow. It derives the continuity, momentum, and energy equations using conservation principles. The equations contain terms for advection, pressure, and viscous forces. Viscous stresses are related to velocity gradients via Newton's law of viscosity. The Navier-Stokes equations, along with appropriate equations of state, form the governing equations for fluid dynamics problems.
This document provides an overview of magnetostatics, which is the study of magnetic fields in systems where charges are stationary or where currents are not varying with time. It discusses key concepts such as:
- The magnetic field produced by a current-carrying conductor based on Oersted's experiment.
- Biot-Savart law, which relates the magnetic field to the magnitude, direction, and proximity of an electric current.
- Magnetic fields produced by straight conductors and circular loops using Biot-Savart law.
- Lorentz force law, which describes the force on a moving charge in a magnetic field.
- Flux and divergence of magnetic fields.
- Bound
This document discusses density operators and their use in quantum information and computing. It begins by introducing density operators and how they can be used to describe quantum systems whose state is not precisely known or composite systems. The key properties of density operators are that they must have a trace of 1 and be positive operators. The document then covers reduced density operators which describe subsystems by taking the partial trace. Finally, it discusses how the reduced density operator gives the correct measurement statistics for observations on a subsystem.
1) The document discusses deriving the continuity and momentum equations in cylindrical coordinate systems (r, θ, z), which are needed to model fluid flow in pipes and veins.
2) It shows the conversions between Cartesian and cylindrical coordinates needed to express differential operators and velocity components in the cylindrical system.
3) The resulting continuity and momentum equations in cylindrical coordinates are presented, which replace the standard forms used in Cartesian coordinates.
The Heisenberg Uncertainty Principle[1]guestea12c43
The document discusses three quantum physics concepts:
1) The Heisenberg Uncertainty Principle, which states that certain pairs of measurable properties, such as position and momentum, cannot be known simultaneously due to the energy required to observe a system.
2) The Schrödinger Equation, which Erwin Schrödinger derived to describe electrons and their behavior under external potential fields using a 'wave function'.
3) Tunneling, a quantum effect where particles can transition through classically-forbidden energy barriers, rather than needing to pass over them.
The Dulong-Petit law proposed in 1819 states that the heat capacity of many solid elements is approximately 3R per mole, where R is the universal gas constant. Dulong and Petit experimentally found the heat capacity of various elements was close to a constant value when multiplied by the assumed atomic weight. In modern terms, they discovered the heat capacity per mole of many elements is about 25 joules per kelvin. Later, Einstein derived that the heat capacity of solids is due to lattice vibrations, providing a theoretical basis for the Dulong-Petit law being expressed in terms of molar heat capacities. The law offers a good prediction for many simple crystalline solids at high temperatures but fails at low temperatures and
The document summarizes key concepts about spin algebra:
- Spin ("S") is the intrinsic angular momentum of fundamental particles. It is quantized and can only take on discrete values.
- For a spin-1/2 particle (e.g. electron), there are two eigenstates of the spin operators S2 and Sz: |0⟩ and |1⟩, representing spin up and down.
- The spin operators S2, Sx, Sy, Sz can be represented by 2x2 matrices in the basis of the |0⟩ and |1⟩ eigenstates. This allows spin states to be modeled as two-component spinors.
This document defines key terms and equations related to simple harmonic motion (SHM). It discusses oscillating systems that vibrate back and forth around an equilibrium point, like a mass on a spring or pendulum. The key parameters of SHM systems are defined, including amplitude, wavelength, period, frequency, displacement, velocity, acceleration. Equations are presented that relate the displacement, velocity, acceleration as sinusoidal functions of time. The concepts of kinetic, potential and total energy are also explained for oscillating systems undergoing SHM.
How to find moment of inertia of rigid bodiesAnaya Zafar
The document provides expressions for calculating the moment of inertia of various regularly shaped rigid bodies about different axes of rotation. It discusses:
1) Calculating moment of inertia using integral methods, considering small elements of the rigid body.
2) Examples of calculating moment of inertia for a rod, rectangular plate, circular ring, thin circular plate, hollow cylinder, solid cylinder, hollow sphere, and solid sphere.
3) Key steps involve identifying the small element, elemental mass, and integrating the expression for elemental moment of inertia over the body.
This document discusses rotational motion and provides definitions and equations for key angular quantities such as angular displacement (θ), angular velocity (ω), angular acceleration (α), torque (τ), moment of inertia (I), angular momentum (L), and rotational kinetic energy. It defines these quantities, gives their relationships to linear motion quantities, and provides examples of how to set up and solve problems involving rotational dynamics.
1. The Stern-Gerlach experiment discovered that silver atoms split into two beams, indicating the presence of an intrinsic "spin" angular momentum of 1/2 beyond orbital angular momentum.
2. Elementary particles are classified as fermions, with half-integer spin, and bosons, with integer spin. The spin of the electron is represented by a two-component spinor.
3. In a magnetic field, the spin precesses around the field direction at the Larmor frequency, independent of initial spin orientation. This principle underlies paramagnetic resonance and nuclear magnetic resonance spectroscopy.
Lecture 8: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
1. O documento apresenta 10 questões sobre física que abordam tópicos como campo elétrico de uma esfera carregada, campo eletromagnético em um capacitor de placas paralelas, mecânica quântica em poços de potencial e decaimento de múons.
2. As questões envolvem cálculos de campo elétrico, força, energia, probabilidade, momento angular, equações de Lagrange e Hamilton, e processos termodinâmicos em uma cavidade ressonante.
3. São abordados conce
This document discusses damped and forced harmonic motion. It explains that in damped harmonic motion, a damping force acts opposite to the velocity to dissipate energy and stop vibrations. The damping causes the amplitude to decay exponentially over time. A system can be under-damped, over-damped, or critically damped depending on how quickly it stops oscillating. Forced harmonic motion occurs when an external periodic force drives the system, like pushing a swing. At resonance, the driving frequency matches the natural frequency, causing large amplitude oscillations. While resonance can be dangerous if it causes collapse, it can also be useful in applications like radios and musical instruments.
The document discusses the derivation of the Navier-Stokes equations, which describe compressible viscous fluid flow. It derives the continuity, momentum, and energy equations using conservation principles. The equations contain terms for advection, pressure, and viscous forces. Viscous stresses are related to velocity gradients via Newton's law of viscosity. The Navier-Stokes equations, along with appropriate equations of state, form the governing equations for fluid dynamics problems.
This document provides an overview of magnetostatics, which is the study of magnetic fields in systems where charges are stationary or where currents are not varying with time. It discusses key concepts such as:
- The magnetic field produced by a current-carrying conductor based on Oersted's experiment.
- Biot-Savart law, which relates the magnetic field to the magnitude, direction, and proximity of an electric current.
- Magnetic fields produced by straight conductors and circular loops using Biot-Savart law.
- Lorentz force law, which describes the force on a moving charge in a magnetic field.
- Flux and divergence of magnetic fields.
- Bound
This document discusses density operators and their use in quantum information and computing. It begins by introducing density operators and how they can be used to describe quantum systems whose state is not precisely known or composite systems. The key properties of density operators are that they must have a trace of 1 and be positive operators. The document then covers reduced density operators which describe subsystems by taking the partial trace. Finally, it discusses how the reduced density operator gives the correct measurement statistics for observations on a subsystem.
1) The document discusses deriving the continuity and momentum equations in cylindrical coordinate systems (r, θ, z), which are needed to model fluid flow in pipes and veins.
2) It shows the conversions between Cartesian and cylindrical coordinates needed to express differential operators and velocity components in the cylindrical system.
3) The resulting continuity and momentum equations in cylindrical coordinates are presented, which replace the standard forms used in Cartesian coordinates.
The Heisenberg Uncertainty Principle[1]guestea12c43
The document discusses three quantum physics concepts:
1) The Heisenberg Uncertainty Principle, which states that certain pairs of measurable properties, such as position and momentum, cannot be known simultaneously due to the energy required to observe a system.
2) The Schrödinger Equation, which Erwin Schrödinger derived to describe electrons and their behavior under external potential fields using a 'wave function'.
3) Tunneling, a quantum effect where particles can transition through classically-forbidden energy barriers, rather than needing to pass over them.
2. 1.Sarrera
Solido zurruna: GORPUTZ ZABAL DEFORMAEZINA.
Partikulen arteko distantzia kteaEz puntualaPartikula-sistema
Idealizazio oso erabilgarria!! (Idealizazioa da, edozein gorputz era batean edo bestean deformatu baitaiteke)
Adib: Barrak, hagatxoak, poleak, bolak, diskoak, zilindroak, … Simetria handikoak
Solido zurrunean bi higidura mota berezitu:
A. TRANSLAZIOA: Partikula guztiek ibilbide paraleloak deskribatu.
B. ERROTAZIOA: Partikula guztiek ibilbide zirkularrak errotazio ardatzaren inguruan.
TRANSLAZIOA
ERROTAZIOA
SOLIDOAREN HIGIDURA
Higidura definitzeko 6 koordenatu
3 posizioari dagozkionak
3 orientazioari dagozkionak
Momentu linealaren teorema: Translazio higidura ebazteko.
Momentu angeluarraren teorema: Errotazio higidura ebazteko.
3. 2. Solido zurrunaren estatika
Solido zurruna orekan
Pausagunean (oreka estatikoa)
Abiadura ktean (oreka dinamikoa)
Demagun F1, F2, … FN indarren eraginpean dagoen solido bat:
i
i 0FF
0FrMM
i
ii
i
OiO
OREKA !!
4. 3. Ardatz finko baten inguruko solido
zurrunaren dinamika. Inertzia-momentua.
Demagun solido zurrun bat biratzen, zeinen errotazio
ardatza solidoaren bi puntu finkoetatik pasatzen baita.
• dm-ren abiadura zirkunferentziarekiko tangentea:
dm
O
Y
X
r
Z
oLd
dm masako elementu infinitesimal arbitrarioa
r dm-ren posizio bektorea
ρ = r sinφ erradioko zirkunferentzia deskribatu dm-k
non ω solidoaren abiadura angeluarra den.
ω berdina da solidoaren puntu guztietan, baina ez v (ardatzaren distantziaren
menpekoa da).
• dm-ren momentu angeluarra: dmvrLd O
dmvrdLo r eta v perpendikularrak
z ardatzaren norabidean: sindmvrdLoz dmdL 2
Oz
ρ = r sinφ
direnez:
5. 3. Ardatz finko baten inguruko solido zurrunaren dinamika. Inertzia-momentua
Solidoaren puntu guztientzat integratuz…
2 2
Oz Oz
M M M
L dL dm dm
masa guztiarentzat integrala
Loz osagaiak ez du jatorriarekiko menpekotasunik, beraz: Loz = Lz hau da;
zz IL
M
2
z dmI non
Solidoaren inertzia momentua
errotazio ardatzarekiko
Indarren momentua:
dt
Ld
M O
O
dt
dL
M z
z
Solidoaren
azelerazio
angeluarraerrotazio
ardatzarekiko
0 0 ktez
z z
d I
M I
dt
Orokorrean Lx eta Ly ez dira zertan nuluak izan behar:
kˆLjˆLiˆLL zyx
L
y
zL
Ardatzaren norabidea finko mantentzeko, solidoari
nulua ez den indar-momentu bat egin behar zaio.
6. Gorputz baten inertzia momentuak
adierazten du, gorputz horren
erresistentzia azelerazio angeluar bat
hartzeko.
Dantzari batek inertzia
momentu handiagoa
izango du besoak
zabalduz, hauek itxiz
gero arinago biratuko
du.
3. Ardatz finko baten inguruko solido zurrunaren dinamika. Inertzia-momentua
7. 4. Inertzia ardatz nagusiak. Steiner-en
teorema.
-k biraketa-ardatzaren norabidearekin bat datorrenean INERTZIA-ARDATZ NAGUSIA
Hau da, // biraketa ardatza
Solidoa simetria zilindrikoa badauka:
• ARDATZ NAGUSIA = SIMETRIA-ARDATZA
• 3 ardatz nagusi cartesiar:
Y
X
Z
Y
X
Y
X
Z Z
Steiner-en teorema:
Masa zentrotik pasatzen den ardatz batekiko inertzia-momentua ezaguna bada, kalkula
daiteke horrekiko paraleloa den beste edozein ardatzen inertzia-momentua.
MZ
P
dm
2
2 2 2
' 2P
M M M M M
I dm d dm dm d dm d dm
d
'
M
IMZ
Ip = IMZ + Md2
8. 4. Inertzia ardatz nagusiak. Steiner-en teorema.
Solido simetriko batzuen inertzia momentuak ardatz nagusiarekiko:
Eraztuna
Zilindro (L 0)
edo
Diskoa (L 0)
(a)
(b)
Hagatxo mehea
Esfera
Paralelepipedoa
R
2
I MR
21
2
I MR
2 21 1
4 12
I MR ML
21
12
I ML
22
5
I MR
2 21
12
I M a b
R
(a)
(b)L
L
2
L
2
a
b
9. 5. Solido zurrunaren ardatz finko baten inguruko
biraketako energia zinetikoa. Energiaren teorema.
Demagun berriz ere
Solido zurrunarentzat (barne
indarren lana nulua denez)
energiaren teorema:
Ez t-rekiko deribatuz:
dm
O
Y
X
r
Z
oLd
2 2 21 1
2 2
zdE v dm dm
2 2 2 21 1
2 2
z z
M M M
E dE dm dm
ktea solidoaren puntu guztientzat
Iz
21
2
z zE I
kan
zW E
2
z
1
2
z
z z
dE d
I I M
dt dt
Kanpo indarrek egindako potentzia: zP M
10. 5. Solido zurrunaren ardatz finko baten inguruko biraketa energia zinetikoa.
Energiaren teorema
Energia potentzial grabitatorioa:
non
MZ
y
dm
X
Y
Z
z
MZz
MZ
MM
p gMzzdmggzdmE
M
MZ zdm
M
1
z
Solido zurrun baten Ep-a MZ-ren altuerarekiko menpekoa baino ez da:
Ep=MghMZ
Indar momentua:
O MZM R mg
11. 5. Solido zurrunaren ardatz finko baten inguruko biraketa energia zinetikoa.
Energiaren teorema
Translazio eta biraketa magnitudeen baliokidetasunak:
a
v
x
mI
Puntu materialaren higidura Simetria-ardatz finko baten inguruko
solido zurrunaren higidura
Momentu lineala, Ardatzarekiko momentu angeluarra,
Energia zinetikoa, Biraketako energia zinetikoa,
Lana, Lana,
Newton-en bigarren legea, edo
.
Biraketa-higiduraren oinarrizko ekuazioa,
edo .
vmp
L I
21
2
zE mv 21
2
zE I
dxFdW dMdW
amF
dt
pd
F
M I dL
M
dt
12. 6. Solido zurrunaren higidura laua
• Solidoaren partikula guztiak plano batekiko paraleloki higitzen direnean.
• Hiru askatasun gradu
2 translazio higidura deskribatzeko
1 planoarekiko perpendikularra den ardatzarekiko biraketa deskrib.
Adib: zilindroa, gurpila, kotxe baten motorraren parte asko, …
- MZ-aren higidura
- MZ-aren inguruko barne higidura
1
2
aztertuko ditugu.
Erraztasunerako: OXY planoa = MZ-ren higidura planoa
OZ ardatza = errotazio-ardatza
1 MZ-ren higidura ekuazioak:
2
2
dt
Rd
MAMF
2
2
xx
dt
Xd
MM AF
2
2
yy
dt
Yd
MM AF
MZ-ren azelerazioa
MZ-ren posizioa
2 MZ-ren inguruko higidura:
MZz IM
13. Energia zinetikoa:
6. Solidu zurrunaren higidura laua
2
z z
1
2
E M V E'
MZ-ren
energia
zinetikoa
MZ-rekiko higidurari dagokion Ez
2 21 1
2 2
z MZE M V I ω
2
2
1
'' MZMZz IEE
14. q
C
C
P
P’
r
P
7. Errodadura-higidura
Zilindro edo esfera batek gainazal lau batean errodatzen dutenean: lotura bat egongo da.
Biraketarekin
lotutakoa
Translazioa deskribatzeko: (1)
Bi askatasun gradu baino ez!
Translazio
zuzenarekin
lotutakoa
Errotazioa deskribatzeko: (2)
RM
dt
Pd
F
MZMZ
MZz
z I
dt
d
I
dt
Id
dt
dL
M
Suposa dezagun objektua plano inklinatu batean errodatzen duela:
RF
N
gM
q
r
q
s= r
gorputzak errodatu labaindu gabe
Bi egoera posible
Solidoak labaindu gabe errodatzen du
Solidoak labainketarekin errodatzen du
1
2
15. 7. Errodadura-higidura
(1) eta (2) berridatziz:
Solidoak labaindu gabe errodatzen du1
s = r φ
ds
V r
dt
dV
R A r
dt
Mg sin q - FR = MA
r
A
IIrF MZMZR
2
sin
1
R
MZ
Mg
F
Mr
I
q
2
sin
1MZ
g
A
I
Mr
q
Labainketarik gabe errodatzeko vzorua ukitzen duen solidoko puntuaren abiadura = 0
marruskadura estatikoa: cosR s sF N Mg q
2
sin
cos
1
R s
MZ
Mg
F Mg
Mr
I
q
q
2
tan 1s
MZ
Mr
I
q
16. 7. Errodadura-higidura
Plano inklinatuak malda handia badu
edo
s <<
2
Labainketarik gabeko errodaduraren kasuan ez da energiarik disipatzen:
Kontserbatzen da!
EZ+Ep= ktea
2 21 1
sin kte
2 2
MZMV I Mgs q
Objektuak labaindu
2
tan 1s
MZ
Mr
I
q
Ekuazioak:
Mg sin q - mdN = MA
d MZN r I
Labainketa dagoenean marruskadura indarrak lan negatiboa egin Em galtzen da
EZ DA KONTSERBATZEN!!