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Midterm 3 - overview =I (compare to F=ma) Moment of inertia I: I=(miri2) : angular acceleration I 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 linear speed at the bottom. Conservation of angular momentum If the net torque equals zero, the angular momentum L does not change Li=Lf Iii=Iff Rotational Kin. Energy KEr=½I2=½L Solids: General: FL0 F/A Y L / L0 AL Young’s modulus F/A Fh Shear modulus S x / h Ax F / A P Bulk modulus B V / V0 V / V0 Also fluids P pressure P=F/A (N/m2=Pa) =M/V (kg/m3) Fpressure-difference=PA 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=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 P0 Pressure at depth h P = P0+ fluidgh h: distance between liquid surface and the point where you measure P h P Buoyant force for submerged object 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 Buoyant force for floating object 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) B h w Bernoulli’s equation P1+½v12+gy1= P2+½v22+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 of continuity. Contact surface A moving Viscous flow F=Av/d =coefficient of viscosity unit: Ns/m2 or poise=0.1 Ns/m2 Poiseuille’s Law Rate of flow Q= v/t= R4(P1-P2) 8L (unit: m3/s) Temperature scales Conversions Tcelsius=Tkelvin-273.5 Tfahrenheit=9/5*Tcelcius+32 We will use Tkelvin. If Tkelvin=0, the atoms/molecules have no kinetic energy and every substance is a solid; it is called the Absolute zero-point. Celsius Kelvin Fahrenheit Thermal expansion length L L=LoT surface A=AoT =2 volume V=VoT =3 L0 : coefficient of linear expansion different for each material T=T0 T=T0+T 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: PV constant T P1V1 P2V2 T1 T2 2 1 2 PV N mv Microscopic 3 2 Macroscopic PV Nk B T 2 1 2 T ( mv ) 3k B 2 Temperature ~ average molecular kinetic energy 1 2 3 mv k B T Average molecular kinetic energy 2 2 3 3 E kin Nk B T nRT Total kinetic energy 2 2 3k bT 3RT rms speed of a molecule 2 v rms v M=Molar mass (kg/mol) m M 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: Qcold=-Qhot mcoldccold(Tfinal-Tcold)=-mhotchot(Tfinal-Thot) the final temperature is: Tfinal= mcoldccoldTcold+mhotchotThot mcoldccold+mhotchot Phase Change GAS(high T) Q=cgasmT Gas liquid Q=mLv Q=csolidmT Solid (low T) liquid (medium T) Q=cliquidmT liquid solid Q=mLf Heat transfer via conduction Rate of energy transfer P P=Q/t (unit Watt) P=kA(Th-Tc)/x=kAT/x k: thermal conductivity Unit:J/(msoC) multiple layers: Q A(Th Tc ) P t ( Li / ki ) i Li=thickness of layer i ki=thermal conductivity of layer i Radiation P=AeT4 : Stefan’s law (J/s) =5.6696x10-8 W/m2K4 A: surface area e: 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 compression Let’s assume that the pressure does not change while lowering the piston (isobaric compression). 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 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=Uf-Ui=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