Midterm 3 - overview

 =I  (compare to  F=ma ) Moment of inertia I: I=(  m i r i 2 )  : angular acceleration I depends on the choice of rotation axis!! Rotational Kin. Energy KE r =½I  2 Conservation of energy for rotating object: [PE+KE t +KE r ] initial = [PE+KE t +KE r ] final [mgh+0.5mv I  2 ] I = [mgh+0.5mv I  2 ] F  =v/r I=xMr 2 with x: depending on the object Rolling of a slope: [mgh] top = [0.5mv I  2 ] bottom [mgh] top = [mgh+0.5mv xmv 2 ] bottom The smaller I (and thus x), the larger the linear speed at the bottom.

Conservation of angular momentum If the net torque equals zero, the angular momentum L does not change L i =L f I i  i =I f  f Rotational Kin. Energy KE r =½I  2 =½L 

Solids: Young’s modulus Shear modulus Bulk modulus Also fluids P=F/A (N/m 2 =Pa) F pressure-difference =  PA  =M/V (kg/m 3 ) General: Pascal’s principle: a change in pressure applied to a fluid that is enclosed is transmitted to the whole fluid 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=m water,displaced g =  water V displaced g If object is fully submerged: V displaced =V object If floating: V displaced =V part of object under water Gravitational force acting on object in/under water: F g =m object g=  object V object g If floating: B=F g so  water V displaced g=  object V object g

P=P 0 +  fluid gh h: distance between liquid surface and the point where you measure P P0P0 P h B =  fluid V object g = M fluid g = w fluid 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=w object  object =  fluid Pressure at depth h Buoyant force for submerged object Buoyant force for floating object h B 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.  object V object =  water V displaced h=  object V object /(  water A)

Bernoulli’s equation P 1 +½  v  gy 1 = P 2 +½  v  gy 2 P+½  v 2 +  gy=constant The sum of the pressure (P), the kinetic energy per unit volume (½  v 2 ) and the potential energy per unit volume (  gy) is constant at all points along a path of flow. Note that for an incompressible fluid: A 1 v 1 =A 2 v 2 This is called the equation of continuity.

Contact surface A moving Viscous flow F=  Av/d  =coefficient of viscosity unit: Ns/m 2 or poise=0.1 Ns/m 2 Rate of flow Q=  v/  t=  R 4 (P 1 -P 2 ) 8L8L (unit: m 3 /s) Poiseuille’s Law

Temperature scales Conversions T celsius =T kelvin T fahrenheit =9/5*T celcius +32 We will use T kelvin. If T kelvin =0, the atoms/molecules have no kinetic energy and every substance is a solid; it is called the Absolute zero-point. Kelvin Celsius Fahrenheit

Thermal expansion  L=  L o  T L0L0 LL T=T 0 T=T 0 +  T  A=  A o  T  =2   V=  V o  T  =3  length surface volume  : coefficient of linear expansion different for each material

Boyle & Charles & Gay-Lussac IDEAL 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:

Microscopic Macroscopic Temperature ~ average molecular kinetic energy Average molecular kinetic energy Total kinetic energy rms speed of a molecule M=Molar mass (kg/mol)

Calorimetry If we connect two objects with different temperature energy will transferred from the hotter to the cooler one until their temperatures are the same. If the system is isolated: Q cold =-Q hot m cold c cold (T final -T cold )=-m hot c hot (T final -T hot ) the final temperature is: T final = m cold c cold T cold +m hot c hot T hot m cold c cold +m hot c hot

Phase Change GAS(high T) liquid (medium T) Solid (low T) Q=c gas m  T Q=c liquid m  T Q=c solid m  T Gas  liquid liquid  solid Q=mL f Q=mL v

Heat transfer via conduction Rate of energy transfer P P=Q/  t (unit Watt) P=kA(T h -T c )/  x=kA  T/  x k: thermal conductivity Unit:J/(ms o C) multiple layers: L i =thickness of layer i k i =thermal conductivity of layer i

Radiation P=  AeT 4 : Stefan’s law (J/s)  =5.6696x10 -8 W/m 2 K 4 A: surface area e: object dependent constant emissivity (0-1) T: temperature (K) P: energy radiated per second. P=  Ae(T 4 -T 0 4 ) where T: temperature of object T 0 : temperature of surroundings.

Isobaric compression Let’s assume that the pressure does not change while lowering the piston (isobaric compression). W=-F  y=-PA  y (P=F/A) W=-P  V=-P(V f -V i ) (in Joule) W: work done on the gas + if  V<0 - if  V>0 This corresponds to the area under the curve in a P-V diagram

Work done on gas: signs. If the arrow goes from right to left, positive work is done 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=U f -U i =Q+W  U=change in internal energy Q=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=0 B: Adiabatic Q=0 C: Isothermal  T=0 D: Isobaric  P=0 PV/T=constant