Midterm 3 - overview

=I (compare to F=ma)

Moment of inertia I: I=(miri2)

: angular accelerationI depends on the choice of rotation axis!!

Rotational Kin. Energy KEr=½I2

Conservation of energy for rotating object:[PE+KEt+KEr]initial= [PE+KEt+KEr]final

[mgh+0.5mv2+0.5I2]I= [mgh+0.5mv2+0.5I2]F

=v/r I=xMr2 with x: depending on the object

Rolling of a slope:[mgh]top= [0.5mv2+0.5I2]bottom

[mgh]top= [mgh+0.5mv2+0.5xmv2]bottom

The smaller I (and thus x), the larger the linearspeed at the bottom.

Conservation of angular momentumIf the net torque equals zero, theangular momentum L does not change Li=Lf

Iii=Iff

Rotational Kin. Energy KEr=½I2=½L

Solids: LA

FL

LL

AFY

0

0/

/ Young’s modulus

xA

Fh

hx

AFS

/

/Shear modulus

pressureP

VV

P

VV

AFB

00 //

/ Bulk modulusAlso fluids

P=F/A (N/m2=Pa) Fpressure-difference=PA=M/V (kg/m3)

General:

Pascal’s principle: a change in pressure applied to a fluid that is enclosed is transmitted to the wholefluid and all the walls of the container that hold the fluid.

Bouyant Force B:

weight of the water in the volume displaced by the object:

B=mwater,displacedg = waterVdisplacedg

If object is fully submerged: Vdisplaced=Vobject

If floating: Vdisplaced=Vpart of object under water

Gravitational force acting on object in/under water:Fg=mobjectg= objectVobjectg

If floating: B=Fg so waterVdisplacedg= objectVobjectg

P = P0+ fluidghh: distance between liquid surface and the point where you measure P

P0

P

h

B = fluidVobjectg = Mfluidg = wfluid

The buoyant force equals the weight of the amount of water that can be put in the volume taken by the object.If object is not moving: B=wobject object= fluid

Pressure at depth h

Buoyant force for submerged object

Buoyant force for floating objecthB

w

The buoyant force equals the weight of the amount of water that can be put in the part of the volume of the object that is under water.objectVobject= waterVdisplaced h= objectVobject/(waterA)

Bernoulli’s equationP1+½v1

2+gy1= P2+½v2

2+gy2

P+½v2+gy=constant

The sum of the pressure (P), the kinetic energy per unit volume (½v2) and the potential energy per unit volume (gy)is constant at all points along a path of flow.

Note that for an incompressible fluid:A1v1=A2v2

This is called the equation ofcontinuity.

Contact surface A

moving Viscous flowF=Av/d=coefficient of viscosityunit: Ns/m2

or poise=0.1 Ns/m2

Rate of flow Q= v/t=R4(P1-P2)

8L(unit: m3/s)

Poiseuille’s Law

Temperature scales

ConversionsTcelsius=Tkelvin-273.5Tfahrenheit=9/5*Tcelcius+32

We will use Tkelvin.

If Tkelvin=0, the atoms/moleculeshave no kinetic energy and everysubstance is a solid; it is called theAbsolute zero-point.

Kelvin

Celsius Fahrenheit

Thermal expansionL=LoT

L0

L

T=T0T=T0+T

A=AoT =2

V=VoT =3

length

surface

volume

: coefficient of linear expansion different for each material

Boyle & Charles & Gay-LussacIDEAL GAS LAW

PV/T = nR

n: number of particles in the gas (mol)R: universal gas constant 8.31 J/mol·K

If no molecules are extracted from or added to a system:

2

22

1

11 constant T

VP

T

VP

T

PV

M

RT

m

Tkvv

nRTTNkE

Tkvm

vmk

T

TNkPV

vmNPV

brms

Bkin

B

B

B

33

2

3

2

32

3

2

1

)2

1(

3

2

2

1

3

2

2

2

2

2

Microscopic

Macroscopic

Temperature ~ average molecular kinetic energy

Average molecular kinetic energy

Total kinetic energy

rms speed of a moleculeM=Molar mass (kg/mol)

Calorimetry

If we connect two objects with different temperatureenergy will transferred from the hotter to the coolerone until their temperatures are the same. If the system is isolated:

Qcold=-Qhot

mcoldccold(Tfinal-Tcold)=-mhotchot(Tfinal-Thot)

the final temperature is: Tfinal=

mcoldccoldTcold+mhotchotThot

mcoldccold+mhotchot

Phase Change

GAS(high T)

liquid (medium T)

Solid (low T)Q=cgasmT

Q=cliquidmT

Q=csolidmT

Gas liquid

liquid solid

Q=mLf

Q=mLv

Heat transfer via conductionRate of energy transfer PP=Q/t (unit Watt)P=kA(Th-Tc)/x=kAT/x

k: thermal conductivityUnit:J/(msoC)

iii

ch

kL

TTA

t

QP

)/(

)(multiple layers:

Li=thickness of layer iki=thermal conductivity of layer i

RadiationP=AeT4 : Stefan’s law (J/s)

=5.6696x10-8 W/m2K4

A: surface areae: object dependent constant emissivity (0-1)T: temperature (K)

P: energy radiated per second.

P=Ae(T4-T04) where

T: temperature of object

T0: temperature of surroundings.

Isobaric compressionLet’s assume that the pressure does notchange while lowering the piston (isobariccompression).

W=-Fy=-PAy (P=F/A)W=-PV=-P(Vf-Vi) (in Joule)

W: work done on the gas+ if V<0- if V>0

This corresponds to the area underthe curve in a P-V diagram

Work done on gas: signs.

If the arrow goes from right to left, positive work isdone on the gas. If the arrow goes from left to right, negative work is done on the gas (the gas has done positive work on the piston) Not mentioned in the book!

First Law of thermodynamics

U=Uf-Ui=Q+W

U=change in internal energyQ=energy transfer through heat (+ if heat is

transferred to the system)W=energy transfer through work (+ if work is

done on the system)

This law is a general rule for conservation of energy

Types of processes

A: Isovolumetric V=0B: Adiabatic Q=0C: Isothermal T=0D: Isobaric P=0

PV/T=constant