Torques, Atwood Machines, Angular Momentum. Click to add text

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1 orques, Atwood Mchines, Anulr Moentu Click to dd text

2 orque So fr we hve nlyzed trnsltionl otion in ters of its nulr quntities. But we hve relly only focused on the kinetics nd enery. We hve yet to dd dynics (Newton's Lws) to the eqution.. Since Newton's Lws overns how forces ct on n object we need to look t how force is lied under nulr conditions. OQUE is the ANGULA counterrt to FOCE. orque is defined s the Force tht is lied ANGEN to the circle rottin round secific oint of rottion.

3 orque WO HNGS NEED O BE UNDESOOD: ) he dislceent fro oint of rottion is necessry. Cn you unscrew bolt without wrench? Mybe but it isn't esy. ht extr distnce AWAY fro the oint of rottion ives you the extr levere you need. HUS we cll this distnce the LEVE (EFFO) AM (r). ) he Force MUS be erendiculr to the dislceent. herefore, if the force is t n nle, sin is needed to eet the erendiculr requireent.

4 orque is COSS PODUC f the force is truly erendiculr, then the sine of 90 derees will equl to. When the force is lied, the bolt itself oves in or out of the e. n other words, the FOCE nd DSPLACEMEN (lever r) re in the X/Y lne, but the ctul dislceent of the BOL is on the "Z xis. We therefore hve wht is clled, COSS PODUC. Counterclockwise rottion is considered to be POSVE nd OU OF HE PAGE Clockwise rottion is considered to be NEGAVE nd NO HE PAGE.

5 Sttic Equilibriu Accordin to Newton's first lw, if n object is t rest it cn be sid to be in stte of sttic equilibriu. n other words, ll of the FOCES cncel out to tht the net force is equl to zero. Since torque is the nulr nlo to force we cn sy tht if syste is t rest, ll of the OQUES cncel out. r r

6 Sttic Equilibriu Exle r r Suose 4600 k elehnt were lced on see-sw with 0.05 k ouse. he elehnt is lced 0 eters fro the oint of rottion. How fr fro the oint of rottion would the ouse need to be lced so tht the syste exists in stte of sttic equilibriu? τ Fr sin, τccw τ F r F eleh eleh r ouse cw r ouse 90, sin 90 r (4600)(9.8)(0) r eleh r ouse r (0.05)(9.8) r.84 x 0 6 or 433 iles (certinly not rcticl)

7 Wht did we foret to include in the lst exle? r r HE PLANK SELF! f the lever itself hs ss, you ust include it in the clcultions. t s force( or weiht in this cse) will ct t the rods CENE OF MASS. f the lnk ws unifor nd its COM ws in the iddle the eqution would hve looked like this. r 3 COM lnk F 3 τ Fr sin, 90, sin 90 τccw τcw F r F r + F r eleh eleh r ouse ouse r lnk + 3 lnk r 3

8 Not in sttic equilibriu? f n object is NO t equilibriu, then it ust be ccelertin. t is then looked t ccordin to Newton s Second Lw. Under trnsltionl conditions NE FOCE roduces n ACCELEAON. Under Anulr Conditions NE OQUE roduces n ANGULA ACCELEAON. his NEW eqution for OQUE is the ottionl Anlo to Newton's second Lw.

9 Exle Consider be of Lenth L, ss, nd oent of inerti (COM) of /L. t is inned to hine on one end. Deterine the be's nulr ccelertion. Let s first look t the be s F.B.D. here re lwys verticl nd horizontl forces on the inned end inst the hine holdin it to the wll. However, those forces AC t the oint of rottion. F H F v

10 Exle Consider be of Lenth L, ss, nd oent of inerti (COM) of /L. t is inned to hine on one end. Deterine the be's nulr ccelertion. cos Fr sin τ ( cos )( L ( cos )( L ( cos )( L α α )() ) ) 3 cos L ( c in ( α + d L ) α + ( L ) n this cse, it ws the verticl coonent of the weiht tht ws erendiculr to the lever r. Also, we hd to use the rllel xis theore to deterine the oent of inerti bout the END of the be. ) α

11 Exle Consider hnin ss wred round MASSVE ulley. he hnin ss hs weiht,, the ss of the ulley is, the rdius is, nd the oent of inerti bout its center of ss c /. (ssuin the ulley is unifor disk). Deterine the ccelertion of the hnin ss. Let s first look t the F.B.D.s for both the ulley nd hnin ss F N h

12 Exle cont F h h h h Net r Fr CM disk ) (, α α α τ h F N h h h h h h + +

13 Exle A trickier roble: Clculte the ccelertion of the syste: Assue is ore ssive thn Wht you hve to understnd is tht when the PULLEY is ssive you cnnot ssue the tension is the se on both sides. Let s first look t the F.B.D.s for both the ulley nd the hnin sses. F N

14 Exle cont F Net F Net + F N Fr sin τ α α, αr ( ) CM

15 Exle + ) (

16 Exle Consider bll rollin down r. Clculte the trnsltionl ccelertion of the bll's center of ss s the bll rolls down. Find the nulr ccelertion s well. Assue the bll is solid shere. Let s first look t the bll s F.B.D F n F f he key word here is rollin. U to this oint we hve lwys delt with objects slidin down inclined lnes. he ter rollin tells us tht FCON is cusin the object to rotte (by lyin torque to the bll).

17 Exle cont F F F f f net sin sin F F r F Fr f f CM shere f 3 ) ( 3 3, α α α τ F n F f cos sin F f 5 sin 3, 5 sin sin 3 sin 3 sin α α +

18 Anulr Moentu rnsltionl oentu is defined s inerti in otion. t too hs n nulr counterrt. As you cn see we substituted our new nulr vribles for the trnsltionl ones. τ F L We cn look t this nother wy usin the MPULSE-MOMENUM theore Settin ulse equl to the chne in oentu r r Fr sin r sin vr sin Or we could look t this fro the oint of view of torque nd its direct reltionshi with nulr oentu.

19 wys to find the nulr oentu ottionl reltionshi L n the cse for ss ovin in circle. L ss rnsltionl reltionshi v L r, 90 L L v v n both cses the nulr oentu is the se.

20 Anulr Moentu is lso conserved Here is wht this sys: F HE NE OQUE is equl to ZEO the CHANGE ANGULA MOMENUM is equl to ZEO nd thus the ANGULA MOMENUM is CONSEVED. Here is coon exle. An ice skter beins sin with his rs out. His nulr velocity t the beinnin of the sin is.0 rd/s nd his oent of inerti is 6 k. As the sin roceeds he ulls in his rs decresin his oent of inerti to 4.5 k. Wht is the nulr velocity fter ullin in his rs? Lo L o (6)() (4.5).67 rd/s

21 Don t foret Just like OQUE, nulr oentu is cross roduct. ht ens the direction is lwys on serte xis fro the vribles you re crossin. n other words, if you cross vribles in the X/Y lne the cross roduct s direction will be on the Z xis

22 Soe interestin Clculus reltionshis K ottionl W d W dt d d dt d W dt d d d Frd W ds ds F r F Fdr W o α α τ 0 ) ( rd ds r s sll rc lenth, tnent

23 More interestin clculus reltionshis W F r W W τ τ, t t t P τ P Fv