Picking Coordinate Axes

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Sandra Chambers
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1 Picing Coodinte Axes If the object you e inteested in Is cceleting Choose one xis long the cceletion Su of Foce coponents long tht xis equls Su of Foce coponents long ny othe xis equls 0 Clcultions e esie no coponents of cceletion to del with Exple: C going ound bned cone. Cone hs nown ngle nd dius o fiction, hoizontl pth ht is c s speed? Top iew Font iew Fee body Dig of c Rdil cceletion gies c s speed. Get cceletion fo foces Foce Dig of c Hoizontl Diection F x = x Fy = Fny F x = x F y = y x = Veticl Diection x = x y = y Coponents of x F φ β = sin φ φ + β = 90 o = φ y F Tget quntity is = cos φ + β = 90 o y = 0 = g y g= 0 PLA Find Fo cicul otion get speed fo dil cceletion = Find x = Find x Fnx = Fn sin Find Fny = Fn cos Find y y g = 5 equtions, 6 0 5, x y chec units y = g g = Fn cos g = cos g x = sin cos g sin = cos g sin = cos g sin = cos sin g = cos If od wee not bned, could c go ound cue? o fiction, hoizontl pth Top iew Font iew = Foces (no fiction cse) Fee body Dig of c o foce in desied diection dil inwd Theefoe no cceletion dil inwd C cnnot go ound the cue!!

2 Using diffeent coodinte syste Foce Dig of c Tget quntity is F x F y = x = x = y = y = cos = x sin = y x + y = PLA Find = Find cos = x Find x x = x Find x x = cos Find y = sin Find y y = y g Find y sin = y 7 equtions, 8 x, x y y ow dd fiction Don t wnt c to slide sttic fiction hich diection? Fo sttic fiction you CAOT use diection of otion to gie diection of fictionl foce Cse I: Flt od hoizontl pth Top iew Font iew Fee body Dig of c Fo c to go ound cue Sttic fictionl foce ust be dil = Cse II: Bned cue C going ound bned cone hoizontl pth Foces Top iew Font iew = To help, the sttic fictionl foce ust he dil coponent At high speeds the c will slide off the top of the bned cue If this is not to hppen Sttic fictionl foce is down the bn Fee body Dig of c F x F y = x + x Foce Dig of c = y y This will gie the c s xiu speed to sty in hoizontl pth (no sidding) At low speeds the c will slide down the bned cue If this is not to hppen Sttic fictionl foce is up the bn Fee body Dig of c F x F y = x x Foce Dig of c = y + y This will gie the c s iniu speed to sty in hoizontl pth (no sidding

3 Exple : pendulu T on constnt Foces in t diection: Σ Ft = t t = t, t 0 sin = t g sin = t g sin = t The gittionl foce,, cuses the tngentil cceletion. The tngentil cceletion chnges becuse the coponent of in the tngentil diection chnges ΣF t = sin T T + +t in diection: Σ F = T - = 0 T - cos = T - g cos = Fo definition of cceletion nd elocity fo cicul otion = The cobintion of the gittionl foce,, nd the foce of the sting on the bll, T, cuse the dil cceletion. The dil cceletion chnges becuse the coponent of in the dil diection chnges, nd T chnges ΣF = T - cos T = (g cos + ) FORCES HICH CHAGE AS OBJECT MOVES Object on Sping o otion (especilly no cceletion) Unstetched sping Fee-body dig y=0 since y = 0, Σ Fy = 0 Σ Fy = - Σ Fy = y = 0 = ²y Object t est This tells us the lue of I THIS CASE OLY Foce dig How does the sping foce behe: If you incese the weight of the object y inceses Double the weight doubles y Theoy of the sping foce y esued fo y unstetched position As the object gets futhe wy fo the unstetched position, the foce inceses. Position chnge could be y o x Cll it x x if x is esued fo the unstetched position Diection of tht foce is lwys opposite to the diection of the displceent fo the unstetched position The foce tht sping exets on n object inceses with its displceent fo its unstetched position Object in otion on eticl Sping Assue sping is ssless Deteine the object s cceletion sping Object t est Fee-body dig y 0 y y Pull object down nd let go Use eltionship between foce nd ccel. ewton s nd lw Tget quntity: y Σ Fy = y y Foce dig Pln Find y Σ Fy = y Find Σ Fy Σ Fy = - Find F s (y -y0) Σ Fy (y -y0) - (y -y0) - = y (y -y0) - = y Suppose we choose the oigin t y 0 y 0 = 0 y y y - = y

4 Suppose we choose the oigin t y (y -y0) - = y y 0 (y+ (y-y) - y0) - = y y = 0 y (y- y) + (y- y0) - = y ht is (y- y0)? If the object hngs t est on the sping (yi -y0) Σ Fy = - since y = 0, Σ Fy = 0 = (yi -y0) (y- y) + - = y (y- y) = y The cceletion of the object does not depend on its weight Supises fo the fundentl concepts of Foce nd cceletion Fo n object hnging eticlly on sping If you set it into otion y y' y is displceent fo equilibiu position of the object The cceletion of the object does not depend on g The gittionl foce does not ffect the object's otion Except, of couse, fo displcing the equilibiu point. Position fo which displceent is esued Sping Useful to esue the foce on n object Hng n object on sping Equilibiu position gies object s weight Useful to esue the ss of n object Let sping oscillte Acceletion gies object s ss Useful to esue the cceletion of syste Stetched sping exets foce on n object x = 0 (equilibiu position) hethe foce is + o - depends on you coodinte syste Copessed sping exets foce on n object = 0 (equilibiu position) Exple You ssignent is to design siple, hnd held deice to esue cceletion. You design is sping which you hold on one end so tht it hngs eticlly with.0 object on the othe end. To test the deice you te it to the eletos in the IDS building whee, you he been told, the eletos xiu cceletion is 0.0g. Befoe the eleto stts, you hng the object on the sping nd it stetches fo inch to 6 inches. ht is the length of the sping fo the eleto s xiu cceletion? o sping =0 6 in L 0.g Object t est in 6 in y 0 ht is length oping fo cceleting object? y y Object cceleting x hethe foce is + o - depends on you coodinte syste Use eltionship between foce nd ccel. ewton s nd lw Use foce lw fo sping Foce is lwys in opposite diection fo the displceent fo equilibiu position

5 Fee body dig object t est Fee body dig object cceleting =.0 = 0. g Find y o yo = 0 y = -6 in Foce dig object t est o Foce dig object cceleting Use F y = y y Pln Find y cceleting object = -(y-yo) = -y Find - = Find =g Find object t est o = - (y-yo) o = - y Find o o - = 0 5 5, 5 equtions y, o Fo 5 o = into = - y y into Fo into F s = y y F s = g F s = g + y y = g + y = ( g + )y. ( 6in) = y into independent os y = in You e sed to choose eplceent spings fo pinbll chine. The sping is used to lunch sll 50 g steel bll to begin the ge. In ode fo the ge to be fun, the bll should lee the sping t speed of 0 ft/sec. At the beginning of the ge, the bll is t est t the end of sping which hs been copessed.0 inches fo its unstetched length. hen you elese the sping, it lunches the bll hoizontlly. Assuing tht fiction cn be neglected, wht should be the sping constnt of the sping you choose?.0 in 0 ft/s Question: ht is sping constnt? Appoch: = 0 50 g Use sping foce to elte the sping constnt to the sping's foce on the bll Relte tht foce to the cceletion of the bll using ewton's nd lw. Use the definition of cceletion to elte the cceletion of the bll to it's chnge of elocity. ill I need the definition of elocity lso? eglect fiction. Fee body dig of the bll dig of the bll o = 0 xo to o xo = -.0 in to = 0 o = 0 o =? f = 0 xf tf xf = 0 to =? f = 0 ft/s f =0 Tget quntity: Quntittie eltionships: = -x x = d x ΣF x = x Foce dig of the bll f x = 50 g =

6 PLA Find = -x Find Fo foce dig, foces in x diection = x x x = = x x = d x into x = x + c Find c t x = xo, x = 0 0 = x o + c c = x o Find x x = d x Find t x = t x = d x x x = x d x F s = x d x x = x d x into x = x d x into x d x = x x x = x + x o ( x + x o ) x x + x o Elute t xf f x f + x o f x o ( 50 g ) (0 ft / s ) ( in) 5000 ( g )( ft / s ) ( in) in 5000 ( g )( ft / s ) ( ) ft (in ). 8 x 0 5 g / s Ae units of coect? [F] = [] [x] [ss] [/s ] = [] [] [ss/s ] = [] o Is this esonble? If you hung 00 g object on this sping, how f would it stetch 0 y - g y g (/) = y 5 (0 /s )(00 g)/(x0 g/s ) = y - (/) x 0 = y (/) c = y esonble length Exple You te hs just copleted n inexpensie pototype of n inetil guidnce syste fo use in cs. hile building the pototype, you collegues used thee sll spings to hold pt which is hnging eticlly ttched only to the spings. The thee spings he the se length nd ech he one end ttched to igid b nd the othe end ttched to the pt. Pecise djustents he been de to the otion of the pt by using diffeent sping constnt fo ech sping. These sping constnts e gien in the design epot. ow to e the finl design less expensie nd oe elible, you nge tells you to eplce the thee spings with single sping, with the specifictions you e to deteine, without chnging the design of the syste.

7 Find the sping constnt of single sping which hs the se behio s the spings Use sping foce lw Single sping should he the se displceent nd cceletion s the spings when the object is hnging on it Use eltionship between foce nd ccel Use ewton s nd lw Fee body digs spings sping Gien :,,, Tget quntity: Use Foce digs F y = y y Find single sping y Find F y = F s, y ΣFY Find ΣFY F y = y, y Find = g Find y thee spings F' y = y 5 ΣF Y Find ΣF Y F' y = F s + F s + F s Find,, y y y ,, Is the poble soled? 0, 9 equtions hich unnown is issing? Do we now nything else useful? Cn t thin of nything. Ty fo solution nywy ill ny (especilly y) cncel? Chec pln Put 7, 8, 9 into 6 F' y y + y + y into 5 y + y + y = y into F y y + y + y F s y + y + y F s y + y + y y y + y + y + + into into yes! y cncels out Anothe pln bsed on nothe ppoch The only influence on the otion of the object Foces exeted by othe objects Eth, Spings If the su of the foces on the object is the se fo spings nd sping the otion will be the se Pln Find single sping y Find F y = F s Find ΣFY F y = F s + F s + F s Find,, y y y 6 5, y ΣFY,, Put, 5, 6 into F y y + y + y into F s y + y + y F s y + y + y into y y + y + y y cncels out + +

1. A man pulls himself up the 15 incline by the method shown. If the combined mass of the man and cart is 100 kg, determine the acceleration of the

1. A man pulls himself up the 15 incline by the method shown. If the combined mass of the man and cart is 100 kg, determine the acceleration of the 1. n pulls hiself up the 15 incline b the ethod shown. If the cobined ss of the n nd ct is 100 g deteine the cceletion of the ct if the n eets pull of 50 on the ope. eglect ll fiction nd the ss of the