Chapter 10 Dynamics of Rotational Motion

2

1

2

1

tan

2 2

2 1

2 22 1

sin( )

1 12 2

( )

1 12 2

z z

cm cm

cm

ext cm

z cm z

z

z

tot z z

z z

Fl rF F r

r F

I

K Mv I

v R

F Ma

I

W d

W

W I d I I

P

θ

θ

ω

ω

τ θ

τ

τ α

ω

ω

τ α

τ θ

τ θ θ

ω ω ω ω

τ ω

= = =

= ×

=

= +

=

=

=

=

= −

= = −

=

∑

∑

∑

∫

∫

L r p r mv

L I

dLdt

ω

τ

= × = ×

=

=∑

Chapter 11 Equilibrium and Elasticity Conditions for equilibrium

0

0 0 0

0

x y z

F

F F F

τ

=

= = =

=

∑

∑ ∑ ∑

∑

1 1 2 2 3 3

1 2 3

0

0

0

Stress Elastic modulusStrain

Tensile stress /Tensile strain /

(pressure in a fluid)

Bulk stressBulk strain /

Shear stressSh

i ii

cmi

i

m rm r m r m rr

m m m m

F A F lYl l A l

FpA

pBV V

S

⊥ ⊥

⊥

+ + + ⋅⋅⋅= =

+ + + ⋅⋅

=

= = =Δ Δ

=

Δ= = −

Δ

=

∑∑

/ear strain /

F A F hx h A x

= =

Chapter 12 Gravitation

1 22g

Gm mFr

=

Weight of a body of mass m at the earth’s surface

2E

gE

Gm mw FR

= =

Acceleration due to gravity at the earth’s surface

2E

E

GmgR

=

3/2

2

(circular orbit)

2 22

2 (Schwarzchild radius)

E

E

E E

s

Gm mUr

Gmvr

r r rT rv Gm Gm

GMRc

π ππ

= −

=

= = =

=

Chapter 13 Periodic motion

2

2

1 1

(relationships between frequency and period)

22 (angular frequency)

(restoring force exerted by an ideal spring)

(simple harmonic motion)

(simple harmonic motion)

2

x

x

f TT f

fT

F kx

d x ka xdt m

km

f

πω π

ω

ω

= =

= =

= −

= = −

=

=

2 2 2

12

(simple harmonic motion)

1 2 2

(simple harmonic motion)

cos( ) (harmonic displacement)

1 1 1 constant2 2 2

(total mechanical energy)

Angular or rotational harmonic motion

= an

x

km

mTf k

x A t

E mv kx kA

I

π π

π πω

ω φ

κω

=

= = =

= +

= + = =

1d2

fIκ

π=

( )

( )

2

2

2

max22 2 2

Simple pendulum

12 2

2 1 2

Physical pendulum

=

2

Damping - small

cos( ' )

'4

Driven oscillator

b tm

d d

mgk gLm m L

gfL

LTf g

mgdI

ITmgd

x Ae t

k bm m

FAk m b

ω

ωπ π

π πω

ω

π

ω φ

ω

ω ω

−

= = =

= =

= = =

=

= +

= −

=− +

Chapter 14 Fluid mechanics

(definition of density)mV

ρ =

(definition of pressure)dFpdA

⊥=

( )2 1 2 1

(pressure in a fluid of uniform density)p p g y yρ− = − −

0

(pressure in a fluid of uniform density)p p ghρ= +

1 1 2 2

(continuity equation, incompressible fluid)Av A v=

(volume flow rate)dV Avdt

=

2 2

1 1 1 2 2 21 12 2

Bernoulli's equation

p gy v p gy vρ ρ ρ ρ+ + = + +

Chapter 17 Temperature and Heat

9 325F CT T= + °

( )5 329C FT T= − °

273.15K CT T= +

2 2

1 1

(constant-volume gas thermometer,T in kelvins)

T pT p

=

0

(linear thermal expansion)L L TαΔ = Δ

0

(volume thermal expansion)V V TβΔ = Δ

(thermal stress)F Y TA

α= − Δ

(heat required for a temperature change of mass m)

Q mc T

T

= Δ

Δ

(heat required for temprature change of n moles)

Q nC T= Δ

(heat transfer in a phase change)Q mL= ±

(heat curent in conduction)

H CdQ T TH kAdt L

−= =

4 (heat current in radiation)H Ae Tσ=

( )4 4 4 4

net s sH Ae T Ae T Ae T Tσ σ σ= − = − Chapter 18 Thermal Properties of Matter

(totalmass, numberofmoles,and molar mass)totalm nM=

(ideal-gasequation)

or B

pV nRTpV Nk T=

=

(molar mass,Avogardro's number,and mass of a molecule)

AM N m=

32

(average translational kineticenergy of moles of ideal gas)

trK nRT

n

=

( )21 32 2(average translational kinetic energy of a gas molecule)

Bavm v k T=

( )2 3 3

(root-mean-square speedof a gas molecule)

Brms av

k T RTv vm M

= = =

24 2(mean free path of a gas molecule)

meanVvt

r NπΛ = =

32

(molar heat capacity of anideal monoatomic gas, constant volume)

VC R=

52

(molar heat capacity of anideal diatomic gas, constant volume)

VC R=

3

(high temperature limit,solid heat capacity, N/V - atom density)

V BNC k TV

=

( ) 23/2

/242

(Maxwell-Boltzmann distribution)

Bmv k T

B

mf v v ek T

ππ

−⎛ ⎞= ⎜ ⎟

⎝ ⎠

Chap. 19 The First Law of Thermo 2

1

(work done by the systemin a volume change)

V

VW pdV= ∫

( )2 1

(work done by the system in avolume change at constant pressure)

W p V V= −

2 1

(first law of thermo)U U U Q W− = Δ = −

(first law of thermo, infinitessimal process)

dU dQ dW= −

(molar heat capacity of an ideal gas,constant pressure)

p VC C R= +

(ratio of heat capacities)

p

V

CC

γ =

( )2 1

(adiabatic process,ideal gas)

VW nC T T= −

( )1 1 2 2 1 1 2 21( )

1(adiabatic process, ideal gas)

VCW pV p V pV p VR γ

= − = −−

1

1

constant

constant

constant

(adiabatic process, ideal gas)

pV

TVT

P

γ

γ

γγ

−

−

=

=

=

Chap. 20. The Second Law of Thermo

1 1

(thermal efficiency of an engine)

C C

H H H

W Q QeQ Q Q

= = + = −

1

11

(thermal efficiency in Otto cycle)

erγ −

= −

(coefficient of performanceof a refrigerator)

C C

H C

Q QK

W Q Q= =

−

1

(efficiency of a Carnot engine)

C H CCarnot

H H

T T TeT T

−= − =

(coefficient of perfromance of aCarnot refrigerator)

CCarnot

H C

TKT T

=−

ln

(microscopic expression for entropy)BS k w=

2

1

(entropy change in a reversible process)

dQST

Δ = ∫

2 1

(reversible isothermal process)

QS S STΔ

Δ = − =

22 1

1

ln

(for an ideal gas in an isothermal process)

BVS S S NkV

⎛ ⎞Δ = − = ⎜ ⎟

⎝ ⎠