Study Guide Final Exam. Part A: Kinetic Theory, First Law of Thermodynamics, Heat Engines

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1 Msschusetts Institute of Technology Deprtment of Physics 8.0T Fll 004 Study Guide Finl Exm The finl exm will consist of two sections. Section : multiple choice concept questions. There my be few concept questions on time nd specil reltivity but no nlytic questions. Section : nlytic problems with some concept questions requiring written responses. The nlytic questions will be divided into prt A covering the mteril since the third test. Prt B will cover problems from the entire yer. Exmples of section questions re given below. Test questions will not use numbers so you do not need clcultor. Prt A: Kinetic Theory, First Lw of Thermodynmics, Het Engines Problem Energy Trnsformtion, Specific Het nd Temperture Suppose person of mss m = kg is running t speed v = 3.8 m s nd is expending W of power during.0 0 km workout. Suppose the runner converts 0% of the internl energy chnge into mechnicl work. The rest of the energy goes into het. If the specific het of the runner is c = J kg K, how much would the body temperture rise fter running the0 km? Problem Kinetic Theory An idel gs hs density of.78 kg/m 3 is contined in volume of 44.8 x 0-3 m 3. The temperture of the gs is 73 K. The pressure of the gs is.0 x 0 5 P. The gs constnt - R = 8.3 J K - mole. ) Wht is the root men squre velocity of the ir molecules? b) How mny moles of gs re present? c) Wht is the gs? d) Wht is the internl energy of the gs?

2 Problem 3: Crnot Cycle of n Idel Gs In this problem, the strting pressure P nd volume V of n idel gs in stte, re given. The rtio R V = V c /V > of the volumes of the sttes c nd is given. Finlly constnt γ = 5/3 is given. You do not know how mny moles of the gs re present. ) Red over steps ()- (4) below nd sketch the pth of the cycle on P V plot on the grph below. Lbel ll pproprite points. () In the first of four steps, to b, n idel gs is compressed from V to V b while no het is llowed to flow into or out of the system. The compression of the gs rises the temperture from n initil temperture T nd to finl temperture T. During this process the quntity PV γ = constnt, where γ = 5/3. ) Wht is the pressure P nd volume V b of the stte b of the gs fter the b compression is finished? b) Wht is the chnge in internl energy of the gs during this chnge of stte? c) Wht is the work done by the gs during this compression? () The gs is now llowed to expnd isothermlly from b to c, from volume V to b volume V. c d) Express the work done by the gs in this process W cb nd the mount of het Q cb tht must be dded from the het source t T in terms of P, V, T, T, nd V. c

3 Is this het positive or negtive? Explin whether it is dded to the system or removed. e) Wht is the pressure P of the gs fter the expnsion is finished? c (3) When the gs hs reched point c is expnds from V to V d while no het is llowed c to flow into or out of the system. The expnsion of the gs lowers the temperture nd pressure from n initil temperture T to finl temperture T. During this process the quntity PV γ = constnt. f) Wht is the pressure P nd volume V d of the stte d of the gs fter the d expnsion is finished? g) Wht is the chnge in internl energy of the gs during this chnge of stte? h) Wht is the work done by the gs during this expnsion? (4) The gs is now compressed isothermlly from d to t constnt T from volume V d bck to V. i) Find the work done by the system on the surroundings W d nd the mount of het Qd tht flows between the system nd the surroundings. Are these quntities positive or negtive? Explin whether het is dded to the system or removed from the het source t T. Totl Cycle: j) Wht is the totl work W cycle done by the gs during this cycle? k) Wht is the totl het Q cycle ( from T ) drwn from the higher temperture het source during this cycle? l) Wht is the efficiency of this cycle ε mx = W cycle / Q cycle ( from T )? Problem 4 Het pump A reversible het engine cn be run in the other direction, in which cse it does negtive work W cycle on the world while pumping het Q cycle (into T ) into reservoir t n upper temperture, T, from lower temperture, T. The het gin of this cycle, defined to be

4 g Q cycle (into T ) /W cycle = (/ ε ) mx where ε = (T mx T ) / T is the mximum thermodynmic efficiency of het engine. The refrigertor performnce is defined to be K Q cycle ( from T ) / W cycle = T /(T T ) Consider tht you hve lrge swimming pool nd pln to het your house with het pump tht pumps het from the pool into your house. A lrge plte in the wter will remin t 0 o C due to the formtion of ice. You pick T to be 50 o C, which will be the temperture of the (lrge) rditors used to het your house. Assume tht your het pump hs the mximum efficiency llowed by thermodynmics. ) Wht is the het gin nd the refrigertor performnce for this cycle? Be creful to use units of Kelvin for temperture. b) If your house formerly burned 00 gllons of oil in winter (t $.00/gllon), how much will the electricity cost (t $0.0 per kilowtt-hour) to replce this het using 8 the het pump? A gllon of oil hs mss 3.4 kg nd contins.4 0 J gl -. c) The ice cube tht ppers in your pool over the winter will be how mny meters on 6 ech side? (It tkes J to melt one kg of ice; it tkes up this much het when 3 freezing.) The density of ice is kg m -3. This would be gret for cooling your house in the summer even if the pool wrmed up enough to swim in it, you could still cool your house by running the het pump in reverse s n ir conditioner! More prcticlly, you might be ble to use ground wter (nd the dirt round it) s the het sink.

5 Prt Two: Erlier Mteril Problem : (Momentum nd Impulse) A superbll of m = 0.08kg, strting t rest, is dropped from height flls h 0 = 3.0m bove the ground nd bounces bck up to height of h f =.0m. The collision with the ground occurs over =5.0ms. t c ) Wht is the momentum of the bll immeditely before the collision? b) Wht is the momentum of the bll immeditely fter the collision? c) Wht is the verge force of the tble on the bll? d) Wht impulse is imprted to the bll? e) Wht is the chnge in the kinetic energy during the collision? f) Assume tht the rubber hs specific het cpcity of c = 0.48cl g 0 r C nd tht ll the lost mechnicl energy goes into heting up the rubber. Wht is the chnge in temperture of the superbll? Problem : (Conservtion of Energy nd Momentum) An object of mss m =.5kg is initilly moving with velocity v 0. It collides completely inelsticlly with block of mss m =.0kg. The second block is ttched 3 to spring with constnt k = N m. The block nd spring lie on frictionless horizontl surfce. The spring compresses distnce d =.0 0 m. ) Wht is the velocity of the object of mss m nd the block immeditely fter the collision?

6 b) Wht is the initil velocity of the object of mss m immeditely before the collision? c) If the block were ttched to very long string nd hung s pendulum, how high would the block nd object of mss m rise fter the collision? Let g = 9.8m s. Problem 3: (Angulr Dynmics) F =.5 0 N t the rim of the merry- A plyground merry-go-round hs rdius of R = 4.0m nd hs moment of go-round for time t =.0 0 s. inerti I = cm kg m bout n xis pssing through the center of mss. There is negligible friction bout its verticl xis. Two children ech of mss m = 5kg were stnding on opposite sides distnce r 0 = 3.0m from the centrl xis. The merry-go-round is initilly t rest. A person on the ground pplied constnt tngentil force of ) Wht ws the ngulr ccelertion of the merry-go-round? b) Wht ws the ngulr velocity of the merry-go-round when the person stopped pplying the force? c) Wht verge power did the person put out while pushing the merry-go-round? d) Wht ws the rottionl kinetic energy of the merry-go-round when the person stopped pplying the force? The two children then wlked inwrd nd stop distnce of r =.0m from the centrl xis of the merry-go-round. e) Wht ws the ngulr velocity of the merry-go-round when the children reched their finl position? f) Wht ws the chnge in rottionl kinetic energy of the merry-go-round when the children reched their finl position?

7 Problem 4: (Energy, Force, nd Kinemtics) A child s plyground slide is d = 5.0m in length nd is t n ngle of θ =.0 0 deg with respect to the ground. A child of mss m b =.0 0 kg strts from rest t the top of the slide. The coefficient of sliding friction for the slide is µ k = 0.. ) Wht is the totl work done by the friction force on the child? b) Wht is the speed of the child t the bottom of the slide? c) How long does the child tke to slide down the rmp? Problem 5: (Plnetry Orbits) Comet Encke ws discovered in 786 by Pierre Mechin nd in 8 Johnn Encke determined tht its period ws 3.3 yers. It ws photogrphed in 93 t the phelion distnce, r = 6. 0 m, (furthest distnce from the sun) by the telescope t Mt. Wilson. The distnce of closest pproch to the sun, perihelion, is r p = m. The universl grvittion constnt G = N m kg. The mss of the sun is m = s kg. ) Explin why ngulr momentum is conserved bout the focl point nd then write down n eqution for the conservtion of ngulr momentum between phelion nd perihelion. b) Explin why mechnicl energy is conserved nd then write down n eqution for conservtion of energy between phelion nd perihelion. c) Find the velocities t perihelion nd phelion.

Problem 6: escpe speed of moon Find the escpe speed of rocket from the moon. Ignore the rottionl motion of the moon. The mss of the moon is m = 7.36 0 kg. The rdius of the moon is R =.74 0 6 m.

8 Problem 6: escpe speed of moon Find the escpe speed of rocket from the moon. Ignore the rottionl motion of the moon. The mss of the moon is m = kg. The rdius of the moon is R = m. Problem 7: (Torque nd ngulr ccelertion) A pulley of mss m p, rdius R, nd moment of inerti I cm = (/ ) mp R bout the center of mss is hung from ceiling with mssless string. A mssless inextensible rope is wrpped round the pulley n ttched on one side to n object of mss m nd on the other side to n object mss m > m. At time t = 0, the objects re relesed from rest. ) Drw the free body digrm on the pulley nd the two objects. b) Write down Newton s Second Lw for the pulley nd the two objects. c) Write down the rottionl eqution of motion for the pulley. d) Find the direction nd mgnitude of the trnsltionl ccelertion of the two objects. e) How long does it tke for the object of mss m to fll distnce d? f) Wht is the tension on the two sides of the rope? Problem 8: Projectile Motion A bt hits bsebll into the ir with n initil speed, v 0 = m / s, nd mkes n ngle θ = deg with respect to the horizontl. How high does it go from the point where it ws hit? How fr does the bll trvel if it is cught t exctly the sme height tht it is hit from? When the bll is in flight, ignore ll forces cting on the bll except for grvittion.

Study Guide Final Exam Solutions. Part A: Kinetic Theory, First Law of Thermodynamics, Heat Engines

Massachusetts Institute of Technology Department of Physics 8.0T Fall 004 Study Guide Final Exam Solutions Part A: Kinetic Theory, First Law of Thermodynamics, Heat Engines Problem Energy Transformation,