Final Exam. covering the entire semester. Extra time granted about 1 hour about 5 Problems about 30 Multiple Choice

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Ursula McKenzie
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1 his week Applications o oces and oues hap. 12, sec. 1-5 onseation o anula momentum hap. 10, sec. 1-4 ast weeks Oscillations hap. 14 inal Exam coein the entie semeste Exta time anted about 1 hou about 5 Poblems about 30 Multiple hoice Ealy stat o those who want it - 6 pm Each poblem will typically inole seeal undamental physics concepts. Statin a Solution hee possible appoaches to solin physics poblems Kinematics -inea and otational Position ime Velocity Acceleation What the uantities mean. What the uantities ae eual to. Dynamics - inea and otational Add oces onseation -inea and otational X -X i = X tanse Eney Momentum Anula Momentum Anle ime Anula Velocity Anula Acceleation Add oues Newton s 2nd aw Newton s 3d aw Example You hae a pat time job with a company that desins loadin euipment o eiht. You team has desined a simple cane o litin heay cao. A 45.0 k ba 15 t lon is made out o lihtweiht aluminum and is suppoted at its base by a hine that allows the ba to piot etically. A suppot cable uns om the othe end o the ba that is in the ai to the ound. he team is woied that the hine miht ail. You task is to detemine the oce on the hine when a packae o 225 k is lited staiht up into the ai om the end o the ba. You hae been asked to conside the case whee the ba is at an anle o 45 to the ound and the suppot cable is at an anle o 30 to the ound. he packae is lited with an acceleation o /2 a W p What is the oce on the hine? = 30 = 45 W p = (225 k) = (45 k) a = 0.10 = 15 t =? Use dynamics on the ba Sum hoizontal oces = ma h = 0 Sum etical oces = ma = 0 Sum toues out o plane = Ia = 0 ake axis o otation as hine Geomety o anles to et = = 30 > W p because packae is acceleatin up ee body diaam o ba ee body diaam o packae W p S y = - W p oces oce diaam o ba S y = y - y - - = 0 S x = x - x = 0 x 2 + y 2 = 2 y = sin x = cos W = m

2 oue b m m St = t - t- (/2) t = 0 t = sinb t = sinm t = sinm Geomety + b = b = = 15 m o = 180 m = = 45 unknowns ind 2 x + 2 y = 2 [1] x, y ind x (oces) x - cos = 0 [2] ind (oue) sinb - sinm - (/2) m b sinm = 0 sinb - sinm - (1/2) m b sinm = 0 ind (oces on packae) [3] - = a [4] an sole o x [4] - = a Sole o put in [3] = (a + ) sinb - (a + ) sinm - (1/2) m b sinm = 0 Sole o put into [2] sinb = (a + ) sinm + (1/2) m b sinm = ( (a + ) sinm + (1/2) m b sinm) / sinb x = cos ( (a + ) sinm + (1/2) m b sinm) / sinb heck units Ok units o oce = units o ma Eeythin is known Just plu in the numbes x = cos30 ((225k) (9.8) (m/s 2 )(1.1) sin45 + (1/2)(45k) (9.8) (m/s 2 ) sin45 ) / sin15 ) ind y (oces) y - sin - m b - = 0 ind (om beoe) = ( (a + ) sinm + (1/2) m b sinm) / sinb ind (om beoe) = a + y =sin (( (a + ) sinm + (1/2)m b sinm ) / sinb ) +m b + (a + ) heck units Eeythin is known just plu in the numbes Deined the ba as the system used = ma x = 0 y = 0 t=0 t= t = t Reiew needed to ind anles took hine as axis o otation needed to ind anles Deined packae as the system = ma on packae to et Oanize the aleba Rotations and onseation An ice skate is spinnin in place on the ice when he bins his ams in close to his body. Detemine his anula speed as a unction o his initial and inal moment o inetia and initial anula speed. w initial Possible appoaches Dynamics (ecto) 1 2 I w = 0 w 2 = I 2 t = Ia onseation o Eney (scala) Use eney ---- system: skate inal I KE = 1 2 Iw2 inally x 2 + y 2 = 2 w = I i I < w > Is it tue?

3 Pediction and Expeiment check uantitatiely with expeiment pediction w = I i I inceases by a acto o 4 w deceases by a acto o 2 Do an expeiment like this in lab Inceased I system: disk, in his pediction does not wok out Detemine actual elationship in lab onseation o Eney oot the chane o intenal eney 1 2 I w DE int enal = 0 In case o skate What is won Intenal eney deceases to pull ams in In case o disk and in Intenal eney inceases in inelastic collision Don t know the chane o intenal eney onseation o eney is not useul. onseation o anula momentum = Iw initial system I w o o input system output - i = input - output input output Dtanse = input - output inal system Iw In case o skate o in dopped on disk No extenal inteaction No anula momentum tanse I w - = 0 w = I I I inceases by a acto o 4 o w w deceases by a acto o 4 Need an extenal inteaction: extenal toue x t = Ia Anula momentum tanse initial Use Dynamics t = I dw dt x inteaction tdt = I dw dw dt = I dt dt dt tdt = I( w -w i ) int eaction time x inal - i = D tanse w = Iw Anula momentum Anula momentum is conseed D system = D tanse D tanse = tdt inteaction time A 250 hockey puck taelin acoss the ice at 5.0 t/sec hits the end o a 1.0 k hockey stick that is layin at est on the ice. he puck hits the hockey stick 3.0 t om its cente o mass. he puck bounces staiht back at 1.0 t/sec. ow does the hockey stick moe just ate it is hit? he moment o inetia o the hockey stick otatin about its cente o mass is 0.10 k m 2. tdt = - i int eaction time tdt = D tanse inteaction time

4 m s =1.0 k =0.10 k m 2 =3.0 t =0.25 k o =5.0 t/s s Use conseation o momentum w =1.0 t/s What is the elocity o the cente o mass o the hockey stick and the anula elocity about the cente o mass? Use conseation onseation o eney is useless. Why? Use conseation o anula momentum system: puck + stick initial time: just beoe the collision inal time: just ate the collision Momentum: system: puck + stick Initial time p o inal time conseation o momentum: no momentum tanse ( m s s - )- o = 0 Anula momentum: system: puck + stick Is thee any anula momentum tanse to the system? Does the system hae anula momentum ate the collision? Does the system hae anula momentum beoe the collision? Gies the elocity o cente o mass! Need to et the anula elocity. conseation o anula momentum: - i = 0 No extenal toue ps p Beoe ollision o What is the system? w = t just beoe collision = o An object moin in a staiht line can hae an anula elocity Depends on axis o otation Initial anula momentum i = I p = I p o Diection: same as w out Does the hockey puck hae a moment o inetia? Depends on the axis o otation I = 2 Anula momentum o hockey puck just beoe the collision = ( 2 ) o = o Anula momentum o stick just beoe collision? Diection out hoose out as + What is the anula momentum o system just beoe the collision? Ate ollision ut system = puck + stick onseation o anula momentum - i = 0 inal diection same as initial diection: out z inal - z initial = w - 2 o = 0 Gies anula elocity about cente o mass Soles the poblem: Diection? Objects moin in staiht lines can hae anula momentum!!! onseation o Momentum ind s ( m s s - )- o = 0 m s s = o + s = mp ( m o + ) s s = 0.25k 1.0k s = 1.50 t s check units t t Ł s s ł note that thee was no need to conet to consistent units.

5 onseation o Anula Momentum ind w w - 2 o = 0 w = 2 o +m p 2 w = ( o + ) w = ( o + ) check units Just beoe the puck hits the stick it has an anula momentum with espect to the cente o mass o the stick. = Iw = ( 2 ) o = o What is the anula momentum o the puck with espect to the cente o mass o the stick some time beoe that? d t t d I = 2 w = t cos = d cos = t t = d put in numbes I = 2 Anula momentum o puck w = t t is the component o pependicula to Does the anula momentum o the puck chane as it moes towad the stick? = Iw = ( m 2 ) t = m t = m d = m d Does not chane!!! Anothe expession o anula momentum o a paticle = p X Anula Momentum as a oss Poduct = w I p Manitude o anula momentum diection Anula momentum is in diection o anula elocity. Out in this case. (p cos) iht hand ule: om to p out p p t p t onseation o anula momentum D system = Dtanse Any chane in the anula momentum o a system must come om Inteactions with objects outside the system System bicycle wheel You choose the system w : down tun it oe w : up Diection o Anula Momentum o system Initial: down inal: up tdt =Dtanse inteactiontime Anula Momentum tanse by? Anothe system System bicycle wheel + peson + stool hai is ee to tun (no extenal toue) No Anula Momentum tanse Initial inal w : down w w : up tun it oe w p : down Diection o Anula Momentum o system Initial: down inal: down

6 Anula Momentum anse balanced piot point Now apply a oce he oce causes a toue on the system t = oue diection: out he toue causes anula momentum tanse to the system tdt = D tanse int eaction time Anula momentum tanse diection: out D tanse is out i = 0 =? onseation o anula momentum - i = Dtanse = D tanse is out = Iw w is out System tuns up What happens i wheel is spinnin? he otatin wheel has a lae anula momentum Anula momentum tanse Balanced with otatin wheel apply a oce he oce causes a toue on the system t = oue diection: out he toue causes anula momentum tanse to the system tdt = Dtanse int eaction time Anula momentum tanse diection: out ow does the system moe? D tanse is out onseation o anula momentum -i =Dtanse = Dtanse+ i 3-D iew i +z D top iew i D +z When the ba tuns anula momentum tanse is still pependicula to the ba Ba tuns aain so that wheel anula momentum in diection o inal anula momentum he ba keeps tunin w ba is up top iew i D D Anula momentum om the wheel is always alon the ba D Ba tuns so that wheel anula momentum in diection o inal anula momentum

HW 7 Help. 60 s t. (4.0 rev/s)(1 min) 240 rev 1 min Solving for the distance traveled, we ll need to convert to radians:

HW 7 Help. 60 s t. (4.0 rev/s)(1 min) 240 rev 1 min Solving for the distance traveled, we ll need to convert to radians: HW 7 Help 30. ORGANIZE AND PLAN We ae given the angula velocity and the time, and we ae asked to ind the distance that is coveed. We can ist solve o the angula displacement using Equation 8.3: t. The distance