τ TOT = r F Tang = r F sin φ Chapter 13 notes: Key issues for exam: The explicit formulas

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Ernest Parsons
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1 Key issues fo exa: Chapte 13 notes: 1. Right hand ule. Cente of ass 3. Toque 4. oent of netia 5. Rotational Enegy 6. Rotational oentu Physics 7: ectue 16, Pg 1 R C = N i=1 i i The explicit foulas = X C ˆ i + Y Cˆ j + Z C ˆ k τ TOT = F Tang = F sin φ N i= 1 i i K = 1 ω ω ΔK = 1 ( ω f ωi ) = WNET Physics 7: ectue 16, Pg 1

2 Exaple: Rotating Rod A unifo od of length =.5 and ass =1 kg is fee to otate on a fictionless pin passing though one end as shown below. The od is eleased fo est in the hoizontal position. What is (A) its angula speed when it eaches the lowest point? (B) its initial angula acceleation? (C) its initial linea acceleation of its fee end? Physics 7: ectue 16, Pg 3 Exaple: Rotating Rod A unifo od of length =.5 and ass =1 kg is fee to otate What is (A) its angula speed when it eaches the lowest point? 1. Fo foces you need to locate the Cente of ass Notice that the Cente of ass is at / ( halfway ) and use the Wok-Enegy Theoe o Consevation of echanical Enegy K i + U i = K f + U f + gh C = (1/) ω + / g C = (/1) end = (/3) (fo table) g g / = (1/) ( (/3) ) ω 9 g / = ω Physics 7: ectue 16, Pg 4

3 Exaple: Rotating Rod A unifo od of length =.5 and ass =1 kg is fee to otate What is (B) its initial angula acceleation? 1. Fo foces you need to locate the Cente of ass C is at / ( halfway ) and put in the Foce on a FBD Σ F = occus only at the hinge g τ z = α z = F sin 9 at the cente of ass (/3) α z = -(/) g (CW) α z = - 3g / () Physics 7: ectue 16, Pg 5 Exaple: Rotating Rod A unifo od of length =.5 and ass =1 kg is fee to otate What is (C) initial linea acceleation of its fee end? g a = α Physics 7: ectue 16, Pg 6 3

4 Two childen on a see-saw Two childen of ass 3 kg and 1 kg ae sitting at the ends of a assless see-saw of length = 4.. The seesaw is at an angle θ = 3 fo the hoizontal as shown. The heavy child is 1.5 fo the pivot point. What is the net toque about this pivot? =.5 R = 1.5 Physics 7: ectue 16, Pg 7 Two childen on a see-saw Two childen of ass 3 kg and 1 kg ae sitting at the ends of a assless see-saw of length = 4.. The seesaw is at an angle θ = 3 fo the hoizontal as shown. The heavy child is 1.5 fo the pivot point. What is the net toque about this pivot? τ z = α z = - R g sin g sin 6 τ z = -1.5 x 3 x 1 x x 1 x 1 x.83 τ z = -17 N Physics 7: ectue 16, Pg 8 4

5 Two childen on a see-saw Two childen of ass 3 kg and 1 kg ae sitting at the ends of a assless see-saw of length = 4.. The see-saw is at an angle θ = 3 fo the hoizontal as shown. The heavy child is 1.5 fo the pivot point. What is the net toque about the pivot using the cente of ass? f hoizontal: x C = 1 x + (3 x 4)/(1+3) = 3. fo left edge o.5 to the ight of the pivot τ z = α z = - (+) g sin 6 + τ z = -.5 x (3+1) x 1 x.83 τ z = -17 N C Physics 7: ectue 16, Pg 9 Exaple: Thowing ball fo stool A student sits on a stool, initially at est, but which is fee to otate. The oent of inetia of the student plus the stool is. They thow a heavy ball of ass with speed v such that its velocity vecto oves a distance d fo the axis of otation. What is the angula speed ω F of the student-stool syste afte they thow the ball? ω F d v Top view: befoe afte Physics 7: ectue 16, Pg 1 5

6 Exaple: Thowing ball fo stool What is the angula speed ω F of the student-stool syste afte they thow the ball? Pocess: (1) Define syste () dentify Conditions (1) Syste: student, stool and ball (No Ext. toque, is constant) () oentu is conseved init = = final = - v d + ω f ω F d v Top view: befoe afte Physics 7: ectue 16, Pg 11 6

7 SUARY The goal of Chapte 14 has been to undestand systes that oscillate with siple haonic otion. GENERA PRNCPES Dynaics SH occus when a linea estoing foce acts to etun a syste to an equilibiu position. k Hoizontal sping (F net ) x 5kx Vetical sping The oigin is at the equilibiu position D 5 g/k. (F net ) y 5ky v5 Å k Pendulu (F net ) t 51 g s v5 Å g T 5 p Å k T 5 p Å g y x k s Enegy f thee is no fiction o dissipation, kinetic and potential enegy ae altenately tansfoed into each othe, but the total echanical enegy E 5 K 1 U is conseved. E 5 1 v x 1 1 kx 5 1 (v ax) 5 1 ka n a daped syste, the enegy decays exponentially E 5 E e t/t whee t is the tie constant. A E E.37E All kinetic All potential t A x t PORTANT CONCEPTS Siple haonic otion (SH) is a sinusoidal oscillation with peiod T and aplitude A. Fequency f 5 1 x T A T Angula fequency v5pf 5 p T Position x(t) 5 A cos (vt 1f ) A 5 A cos 1 pt T 1f Velocity v x (t) 5v ax sin (vt 1f ) with axiu speed v ax 5vA t SH is the pojection onto the x-axis of unifo cicula otion. f5vt 1f is the phase The position at tie t is x(t) 5 A cos f 5 A cos (vt 1f ) The phase constant f deteines the initial conditions: x 5 A cos f v x 5vA sin f y A f f x x 5 A cos f x 5 A cos f Acceleation a x 5v x APPCATONS Resonance Aplitude When a syste is diven by a peiodic extenal foce, it esponds with a lage-aplitude oscillation if f ext < f whee f is the syste s natual oscillation fequency, o esonant fequency. f f ext Daping f thee is a dag foce D 5bv, whee b is the daping constant, then (fo lightly daped systes) x(t) 5 Ae bt/ cos(vt 1f ) The tie constant fo enegy loss is t 5/b. x A A t Copyight 4 Peason Education, nc., publishing as Addison Wesley

8 SUARY The goal of Chapte 15 has been to undestand acoscopic systes that flow o defo. GENERA PRNCPES Fluid Statics Gases iquids Fluid Dynaics deal-fluid odel Feely oving paticles oosely bound paticles ncopessible Copessible ncopessible Sooth, laina flow Pessue piaily theal Pessue constant in a laboatoy-size containe Pessue piaily gavitational Hydostatic pessue at depth d is p 5 p 1gd Nonviscous otational Density p v y PORTANT CONCEPTS Density 5/V, whee is ass and V is volue. Pessue p 5 F/A, whee F is the agnitude of the fluid foce and A is the aea on which the foce acts. Exists at all points in a fluid Pushes equally in all diections Constant along a hoizontal line Gauge pessue p g 5 p 1 at Equation of continuity v 1 A 1 5 v A Benoulli s equation p 1 v 1 y 1 Fluid paticles ove along stealines. p v 1 1gy 1 5 p 1 1 v 1gy A 1 Benoulli s equation is a stateent of enegy consevation. A APPCATONS Buoyancy is the upwad foce of a fluid on an object. Achiedes pinciple The agnitude of the buoyant foce equals the weight of the fluid displaced by the object. Sink avg. f F B, w o Rise to suface avg, f F B. w o Neutally buoyant avg 5 f F B 5 w o f F B w o Elasticity descibes the defoation of solids and liquids unde stess. inea stetch and copession: (F/A) 5 Y (D/) Stain Tensile stess Young s odulus Volue copession: p 5 B (DV/V ) Bulk odulus Volue stain A D F Copyight 4 Peason Education, nc., publishing as Addison Wesley

9 SUARY The goal of Chapte 16 has been to lean the chaacteistics of acoscopic systes. GENERA PRNCPES Thee Phases of atte Solid iquid Gas Rigid, definite shape. Nealy incopessible. olecules loosely held togethe by olecula bonds, but able to ove aound. Nealy incopessible. olecules ove feely though space. Copessible. The diffeent phases exist fo diffeent conditions of tepeatue T and pessue p. The boundaies sepaating the egions of a phase diaga ae lines of phase equilibiu. Any aounts of the two phases can coexist in equilibiu. The tiple point is the one value of tepeatue and pessue at which all thee phases can coexist in equilibiu. p SOD elting/ feezing point QUD Boiling/ condensation point Tiple point GAS T PORTANT CONCEPTS deal-gas odel Atos and olecules ae sall, had sphees that tavel feely though space except fo occasional collisions with each othe o the walls. The olecules have a distibution of speeds. The odel is valid when the density is low and the tepeatue well above the condensation point. deal-gas aw The state vaiables of an ideal gas ae elated by the ideal-gas law pv 5 nrt o pv 5 Nk B T whee R J/ol K is the univesal gas constant and k B J/K is Boltzann s constant. p, V, and T ust be in S units of Pa, 3, and K. Fo a gas in a sealed containe, with constant n: Counting atos and oles A acoscopic saple of atte consists of N atos (o olecules), each of ass (the atoic o olecula ass): N 5 Altenatively, we can state that Volue V the saple consists of n oles (in gas) n 5 N ass o N A ol N A ol 1 is Avogado s nube. The nueical value of the ola ass ol, in g/ol, equals the nueical value of the atoic o olecula ass in u. The atoic o olecula ass, in atoic ass units u, is well appoxiated by the atoic ass nube A. 1 u kg The nube density of the saple is N V. p V T 5 p 1V 1 T 1 APPCATONS Tepeatue scales T F T C 1 3 T K 5 T C 1 73 The Kelvin tepeatue scale is based on: Absolute zeo at T 5 K The tiple point of wate at T K Thee basic gas pocesses 1. sochoic, o constant volue. sobaic, o constant pessue 3. sotheal, o constant tepeatue pv diaga p 1 3 V Copyight 4 Peason Education, nc., publishing as Addison Wesley

10 SUARY The goal of Chapte 17 has been to expand ou undestanding of enegy and to develop the fist law of theodynaics as a geneal stateent of enegy consevation. GENERA PRNCPES Fist aw of Theodynaics DE th 5 W 1 Q The fist law is a geneal stateent of enegy consevation. Wok on W. Syste Wok by W, E th Wok W and heat Q depend Q. Q, on the pocess by which the Heat in Heat out syste is changed. The change in the syste depends only on the total enegy exchanged W 1 Q, not on the pocess. Enegy Theal enegy E th icoscopic enegy of oving olecules and stetched olecula bonds. DE th depends on the initial/final states but is independent of the pocess. Wok W Enegy tansfeed to the syste by foces in a echanical inteaction. Heat Q Enegy tansfeed to the syste via atoiclevel collisions when thee is a tepeatue diffeence. A theal inteaction. PORTANT CONCEPTS The wok done on a gas is V f W 5 3 pdv V i 54 5(aea unde the pv cuve) An adiabatic pocess is one fo which Q 5. Gases ove along an adiabat fo which pv g 5 constant, whee g5c P /C V is the specific heat atio. An adiabatic pocess changes the tepeatue of the gas without heating it. Caloiety When two o oe systes inteact theally, they coe to a coon final tepeatue deteined by Q net 5 Q 1 1 Q 1 c 5 p p i Adiabat f sothes V V The heat of tansfoation is the enegy needed to cause 1 kg of substance to undego a phase change Q 56 The specific heat c of a substance is the enegy needed to aise the tepeatue of 1 kg by 1 K. Q 5 cdt The ola specific heat C is the enegy needed to aise the tepeatue of 1 ol by 1 K. Q 5 ncdt The ola specific heat of gases depends on the pocess by which the tepeatue is changed: C V 5 ola specific heat at constant volue. C P 5 ola specific heat at constant pessue. C P 5 C V 1 R, whee R is the univesal gas constant. SUARY OF BASC GAS PROCESSES Pocess Definition Stays constant Wok Heat sochoic sobaic sotheal Adiabatic DV 5 Dp 5 DT 5 Q 5 V and p/t p and V/T T and pv pv g W 5 W 5pDV W 5nRT ln (V f /V i ) W 5DE th Q 5 nc V DT Q 5 nc P DT DE th 5 Q 5 All gas pocesses deal-gas law Fist law pv 5 nrt DE th 5 W 1 Q 5 nc V DT Copyight 4 Peason Education, nc., publishing as Addison Wesley

11 SUARY The goal of Chapte 18 has been to undestand the popeties of a acoscopic syste in tes of the icoscopic behavio of its olecules. GENERA PRNCPES Kinetic theoy, the ico/aco connection, elates the acoscopic popeties of a syste to the otion and collisions of its atos and olecules. The Equipatition Theoe Tells us how collisions distibute the enegy in the syste. The enegy stoed in each ode of the syste (each degee 1 of feedo) is Nk BT o, in tes of oles, 1 nrt. The Second aw of Theodynaics Tells us how collisions ove a syste towad equilibiu. The entopy of an isolated syste can only incease o, in equilibiu, stay the sae. Ode tuns into disode and andoness. Systes un down. Heat enegy is tansfeed spontaneously fo the hotte to the colde syste, neve fo colde to hotte. PORTANT CONCEPTS Pessue is due to the foce of the olecules colliding with the walls. p 5 1 N 3 V v s 5 N 3 V P avg The aveage tanslational kinetic enegy of a olecule is P avg 5 3 k B T. The tepeatue of the gas T 5 3k B P avg easues the aveage tanslational kinetic enegy. Entopy easues the pobability that a acoscopic state will occu o, equivalently, the aount of disode in a syste. nceasing entopy The theal enegy of a syste is E th 5 tanslational kinetic enegy 1 otational kinetic enegy 1 vibational enegy onatoic gas E th 5 3 Nk BT 5 3 nrt Diatoic gas E th 5 5 Nk BT 5 5 nrt Eleental solid E th 5 3Nk B T 5 3nRT Heat is enegy tansfeed via collisions fo oe-enegetic olecules on one side to lessenegetic olecules on the othe. Equilibiu is eached when (P 1 ) avg 5 (P ) avg, which iplies T 1f 5 T f. Q APPCATONS The oot-ean-squae speed v s is the squae oot of the aveage of the squaes of the olecula speeds: v s 5 "(v ) avg Fo olecules of ass at tepeatue T, 3k B T v s 5 Å ola specific heats can be pedicted fo the theal enegy because DE th 5 ncdt. onatoic gas C V 5 3 R Diatoic gas C V 5 5 R Eleental solid C 5 3R Copyight 4 Peason Education, nc., publishing as Addison Wesley

Tidal forces. m r. m 1 m 2. x r 2. r 1

Tidal forces. m r. m 1 m 2. x r 2. r 1 Tidal foces Befoe we look at fee waves on the eath, let s fist exaine one class of otion that is diectly foced: astonoic tides. Hee we will biefly conside soe of the tidal geneating foces fo -body systes.