Physics 2A: Additional Equations for the Final Celebration

Chapter 10: Energy and Work ⇒ the total energy of a system is the sum of all of the different kinds of energy in the system: E = K + Ug + Us + Eth + Echem + … ⇒ energy within a system can be transformed from one form of energy into another; energy can be transferred to or from a system by doing work on it Work: cosW Fd θ=

⇒ The unit of work is the Joule (J). 1 J = 1 Nm = 1 kg m2/s2

⇒ Work can be +, -, or 0. Work Energy Theorem: The total energy of a system changes by the amount of work done on it:

...g s th chemE K U U E E W∆ = ∆ + ∆ + ∆ + ∆ + ∆ + = Law of conservation of energy: The total energy of an isolated system (no work is done on the system) remains constant:

... 0g s th chemE K U U E E∆ = ∆ + ∆ + ∆ + ∆ + ∆ + =

Kinetic Energy: 212K mv=

Gravitational Potential Energy: gU mgy=

Elastic Potential Energy: 212sU kx=

Thermal energy: th kE f x∆ = ∆ Conservation of Mechanical Energy: The mechanical energy of an isolated system without friction is conserved:

( ) ( ) ( ) ( )f g f s f i g i s iK U U K U U+ + = + + ⇒ If there is friction, the total energy is conserved:

( ) ( ) ( ) ( )f g f s f th i g i s iK U U E K U U+ + + ∆ = + +

Power: The rate at which energy is transformed or work is done: EP t

WP t

∆= ∆

= ∆

Chapter 11: Using Energy

∆E = ∆K + ∆Ug + ∆Us + ∆Eth + ∆Echem + … = W ⇒ energy is never created or destroyed, only transferred into different forms ⇒ when we say that energy is “lost”, is isn’t actually lost but converted into a non-useful form, usually thermal energy

Efficiency: what you getwhat you had to payfor

out

in

EeE

= =

⇒ the efficiency of the human body is ~25% 1 Cal = 1000 cal = 4186 J

EP t∆= ∆ ⇒ the human body uses energy at a rate of ~100W when at rest

Temperature: ⇒ thermal energy Eth for an ideal gas equals the total energy of all moving atoms and molecules in the gas ⇒ the temperature of an ideal gas depends upon the average KE of the atoms and molecules in the gas

ice point steam point Celsius scale 0°C 100°C Kelvin scale 273K 373K Fahrenheit scale 32°F 212°F

273.15K CT T= +

⇒ for an ideal gas:

Kavg = 3/2 kBT Boltzmann’s constant: kB = 1.38 × 10-23 J/K

Eth = N Kavg = 3/2 NkBT

Chapter 12: Thermal Properties of Matter Atomic Model of Matter: ⇒ a special notation is used to indicate the composition of a nucleus: A

Z X Atomic number Z: # of protons in nucleus Atomic mass number A: # of protons + # of neutrons (N) A = Z + N X: chemical symbol for element Atomic mass unit: 1 u = 1.6605 × 10-27 kg n → # of moles N → # of particles NA → 6.022 × 1023 mol-1

(in grams)

mol

MnM

= where Mmol is the molar mass

⇒ the mass per mole (g/mol) of a substance has the same numerical value as the

atomic or molecular mass of the substance (in u) Ideal Gas Law: pressure: p = F/A unit of p is the Pascal (Pa); 1 Pa = 1 N/m2

8.31 JPV nRT R mol K= =

221.38 10 JPV NkT k K

−= = ×

⇒ if the number of moles is constant: f f i i

f i

p V pVT T

=

Heat: heat (Q): energy transferred between objects because of a temperature difference

Q > 0 if object absorbs heat Q < 0 if object loses heat

Q Mc T= ∆ where c = specific heat capacity

cwater=4186 J/kg•K

⇒ during a phase change, heat is absorbed or lost but the temperature doesn’t change solid to liquid / liquid to solid Q = +MLf (Lf = latent heat of fusion) liquid to gas / gas to liquid Q = +MLv (Lv = latent heat of vaporization)

for water: Lf = 3.33 × 105 J/kg Lv = 22.6 × 105 J/kg ⇒ for two objects that are placed in thermal contact (assuming no heat is lost to the surroundings): Qnet = Q1 + Q2 = 0 Heat Transfer: ⇒ heat is transferred by three processes: convection, conduction, and radiation

conduction: Q kAP Tt L

= = ∆ ∆

k = thermal conductivity (units are W/m•K)

radiation: 4QP e T At

σ= =∆

(radiation emitted)

40

QP e T At

σ= =∆

(radiation absorbed)

Stefan-Boltzmann’s constant 8

2 45.67 10 Wm Kσ −= ×

net power radiated: Pnet = Pemitted - Pabsorbed

Chapter 13: Fluids Density: m

Vρ = 331.000 10water

kgmρ = ×

Pressure:

Pressure: FpA

= patm=1.013 × 105 Pa = 1 atm

Pressure in a Fluid: 0p p gdρ= + Buoyancy: Archimede’s Principle: the magnitude of the buoyant force on an object partially or completely immersed in a fluid equals the weight of the fluid displaced Buoyant Force: B fluid subF V gρ= ⇒ if an object is completely submerged, Vsub = Vobj

⇒ if an object is floating, FB = w = mg Fluids in Motion: Equation of Continuity: 1 1 1 2 2 2A v A vρ ρ= ⇒ if the fluid is incompressible (ρ1=ρ2): 1 1 2 2A v A v=

⇒ the volume flow rate Q = Av Bernoulli’s Equation: 2 2

1 1 1 2 2 21 1

2 2p v gy p v gyρ ρ ρ ρ+ + = + +

Bernoulli’s Principle: where the speed of a fluid increases, the pressure in the fluid decreases

Chapter 14: Oscillations Frequency: 1f T= (units of f = Hz)

Period: 1T f= (units of T = s)

Hooke’s law: F kx= − (units of k = N/m) ⇒ the – sign indicates that the direction of the force is opposite the displacement Simple Harmonic Motion: ⇒ for an object oscillating in simple harmonic motion:

max

max2 2

max

cos(2 )(2 )sin(2 ) (2 )

(2 ) cos(2 ) (2 )

x A ft x Av A ft ft v A ft

a A ft ft a A ft

ππ π π

π π π

= == − =

= − =

Note: v = 0 at x = +A; v = vmax at x = 0 a = 0 when x = 0; a=amax at x = +A

Energy in SHM: 2 2

2 2max

1 12 2

1 12 2

E mv kx

E mv kA

= +

= =

⇒ for a mass m oscillating on a spring with spring constant k:

1 22

k mf Tm k

ππ

= =

Pendulums:

⇒ for a simple pendulum (for small angles):

1 22

g Lf TL g

ππ

= =

Chapter 15: Traveling Waves and Sound Waves: Transverse Wave: the disturbance occurs perpendicular to the direction of travel of the wave Longitudinal Wave: the disturbance occurs parallel to the direction of travel of the wave

1 1f TT f

= = v v fTλ λ= =

⇒ if the frequency is increased, the wavelength is decreased but wave speed doesn’t change

Speed of waves on a string: sT mvL

µµ

= =

Doppler Effect: 0

0

1

1s

s

vvf f vv

± =

Chapter 16: Superposition and Traveling Waves Linear Superposition: The Principle of Linear Superposition: when two or more waves are present at the same place at the same time, the resultant disturbance is the sum of the disturbances from the individual waves ⇒ for 2 wave sources vibrating in phase:

0,1,2,... constructive interference

1( ) 0,1,2,... destructive interference2

r m m

r m m

λ

λ

∆ = =

∆ = + =

Beats: 1 2beatsf f f= − Standing Waves: ⇒ for standing waves on a string fixed at both ends:

2 1,2,3,...2m m

L vf m mm L

λ = = =