Fiction: Epeimentll the following fetues e obseed to be tue of the foce of fiction: ) Fiction lws opposes the motion. The foce is dissiptie nd its diection is pllel to the sufce of the object in motion. ) The mgnitude of the fiction foce is popotionl to the objects noml foce fs µ s N fo sttic fiction nd N fk µ k fo kinetic fiction. µ s nd µ k e the coefficients of sttic fiction nd kinetic fiction espectiel. 3) µ s > µ k. This mkes sense since sttion object foms stonge contct welds. The kinetic foce of fiction is less thn the sttic foce of fiction. 4) f s nd f k e independent of the sufce e of contct nd object elocit. 5) When FPlel to the sufce eceeds f s m µ s N, the object beks fee. 6) The mgnitude of sttic fiction foce is equl to the mgnitude of Pllel F tht is pplied up until the object begins to slide.
Dg Foce nd Teminl Speed: Fo objects moing though fluid such s the tmosphee, fiction dg foce D esults t the fluid-sufce intefce. Fo low pticle elocities when flow is lmin o stemline, the dg foce on spheicl object of dius follows Stokes' Lw: D 6πη Heeη is the fluid iscosit, mesue of intenl fiction in the fluid les. ρ η Fo Renolds numbe N R 0 fluid flowing coss spheicl sufce begins to flow tubulentl nd dg foce depends on the sque of the elocit: D CρA ρ is fluid densit, A coss-sectionl e, elocit nd C dg coefficient. B setting D mg nd soling fo fee fll teminl elocit: mg t const. C ρ The object fee fll cceletion hs cesed. A Sk Die 5 mph Ping-Pong Bll 0 mph Bsebll 94 mph Pchutist mph
Unifom Cicul otion: Gien n object in fied cicul pth motion nd constnt elocit ecto mgnitude, thee eists cceletion since the elocit ecto diection is chnging continuousl duing this motion: V Y φ Object mss X The ngle mkes with hoizontl is φ The peiod of eolution is the cicumfeence diided b the elocit mgnitude: T π The position ecto is iˆ + ˆj with Cos(φ ) nd Sin(φ ) iˆ + ˆj Sin( φ )ˆ i + Cos( φ) ˆj iˆ + ˆj iˆ + ˆ j iˆ + ˆj
Cos( φ)ˆ i Sin( φ) ˆj + d 4π T The diection of is: Sin( φ) Tn( θ ) Cos( φ) Tn( φ) Angle θ Angle φ i.e, cceletion is centipetl o cente seeking. The ngle mkes with hoizontl is θ Fo non-unifom cicul motion, thee is both centipetl cceletion due to the chnging diection of the elocit ecto nd tngentil cceletion due to the chnging elocit ecto mgnitude. d tn d dt
Summ of Unifom Cicul otion: Pticles eecuting unifom cicul motion (constnt mgnitude) he centipetl cceletion centipetl foce F V V m. Accoding to Newton's nd Lw thee will be tht is diected towds the cente of the cicle. In soling poblems which he n object constined to moe in unifom cicul motion, we cn equte the component of the constining foce keeping the object moing long its cicul pth to the centipetl foce. Non-unifom Cicul otion: Fo non-unifom cicul motion ( mgnitude is no longe constnt), thee is centipetl cceletion due to the chnging diection of the elocit ecto nd tngentil cceletion due to the chnging elocit ecto mgnitude. d tn d dt d + tn
Gittion nd Keple's Lws In ddition to thee lws of motion, Isc Newton lso discoeed lw of gittion tht went unchnged fo oughl 50 es nd equied Einstein fo its eision. Git is the wekest mong the fou fundmentl foces. Git is esponsible fo pocesses nging fom pticle-pticle ttctions, to glctic scle eents like the fomtion of gl clustes nd supeclustes. Newtonin git ws the leding theo of git until the el 900's when the pedictions of the Genel Theo of Reltiit, subsequentl eified, dmticll chnged the iew of git s well s futue ppoches to undestnding the ntue of the othe known foces. Newton's Lw of Gittion Two msses nd septed b distnce e ttcted ccoding to: F G ˆ Along line joining the centes of nd. Note the following: * ) Popotionl to the poduct ) Inesel popotionl to the sque of 3) Diection is ttctie fo both nd 4) Newton's 3 d lw F F G is Newton's Gittionl Constnt: G 6.670 - Nm /kg
Notice the ction t distnce poblem ssocited with this model. Looking t n eth-pple sstem, the question of how # 4 boe esults in ou seeing the pple flling to eth nd not the eth cceleting up to the pple is nsweed: F Eth F Apple Apple g Eth F Eth Eth Apple Eth g The pple's cceletion is g, but eth's cceletion is onl smll fction of g Also, n object of mss on the sufce of eth epeiences the gittionl pull of the eth s g. Equting to Newton's Gittionl Lw: g Gies g dependent onl on the pmetes tht chcteize eth nd G G G Eth Eth g
Shell Theoems A unifom spheicl shell of mtte ttcts pticle tht is outside s if ll the shells mss wee concentted t its cente. We he ssumed this is tue of the eth, nd this is close to the ctul elit: ) g Is not constnt oe the sufce of the eth ) ρ Of eth in non-unifom with depth (cust, mntle, oute coe, inne coe) 3) Eth is not spheicl: R pole < Requto 4) Eth's ngul ottion ω mkes n object lighte t the equto: At the poles, mg 0 N such thtn (which is the weight) is equl to mg : At the equto, centipetl foces poduce sum of foces tht is not identicll zeo: N mg m R equto N m{ g R equto } The tem in pentheses is n effectie lue ofg t the equto. The lue is bout thid of pecent less thn the g 9. 80 eth ege.
A unifom shell of mtte eets no net gittionl foce on pticle locted inside the shell. Fo unifoml dense objects, then buowing downwd two opposing effects on the gittionl foce: ) Decesing R Incesing F ) oing into shell Decesing F Fo unifoml dense objects, fcto two is moe ponounced nd the object moes unifoml to egion of zeo git. Fo the eth no such unifomit eists nd wht is ctull found is: ) Fcto one is initill lge thn fcto two so g inceses (light cust/mntle) ) Fcto eentull wins out nd git diminishes upon futhe descent. Sstems of sses Fo sstems of mn msses, supeposition mens we cn use Newton's Lw of Git to detemine the foces pesent on indiidul msses b poceeding with ecto ithmetic Fm F + F 3 + F 4 +... + F n In two dimensions: Fm F + F3 + F4 +... + Fn And Fm F + F3 + F4 +... + F n m F F m + F
Keple's Lws Johnnes Keple using plnet dt collected b Tcho Bhe deduced the following thee lws of plnet motion: Fist Lw: Pth of ech plnet bout the sun is n ellipse with the sun t one focus: The ellipse hs SP+S'P Constnt. The hoizontl etent of the ellipse is its mjo is. Hlf of this is semi-mjo is. The eticl dimension is the mino is. Hlf of this is the semi-mino is. Second Lw: Plnets moe so n imgin line dwn fom the sun to the plnet sweeps out equl es in equl time intels.
oe kinetic eneg t peihelion fste obitl speed. Newton found tht this lw deies fom consetion of ngul momentum. A secto hs e ΔA Δt constnt Δ Δt θ ω θ. Constnt whee 'omeg' the ngul elocit is just the ngul momentum L m ω diided b the mss. Thid lw: T 3 Hee T is obitl peiod nd the semi-mjo is of the obit. Fo objects in obit constined to cicul pth b git, we equte centipetl foce to gittionl foce: V m V m G G π T 4π T G T π G 4 3 Note tht if T is in units of es nd in units of AU, this lw simplifies: T 3 E.g., Fo s,.5 AU T.88 es.
Geosnchonous Stellites: Stellites tht emin in obit boe fied point on the eth sufce he seel impotnt pplictions in communictions, wethe, nigtion, milit etc. Finding the obitl distnce boe the eth of such stellite equies onl tht we set its obitl peiod equl to eth d 86,400 s: Fom the 3 d lw of Keple T G 4π 3 4 86,400 6.670 5.970 3 4, 00km 4 π H height boe eth ~ 35,800 Km o bout si Eth dii.