Leon : Equilibriu, Newton econd law, Rolling, Angular Moentu (Section 8.3- Lat tie we began dicuing rotational dynaic. We howed that the rotational inertia depend on the hape o the object and the location o the rotational ai. We alo learned that the torque change the tate o rotation. Rotational Equilibriu We reeber that or an object to reain at ret, the net orce acting on it ut be equal to zero. (Newton irt law.) However, that condition i not uicient or rotational equilibriu. What happen to the object to the right? The orce have the ae agnitude. Condition or equilibriu (both tranlational and rotational): 0 and 0 The obedient pool. and ake the pool roll to the let, 4 to the right, and 3 ake it lide. Proble-Solving Step in Equilibriu Proble (page 74). Identiy an object or yte in equilibriu. Draw a diagra howing all the orce acting on that object, each drawn at it point o application. Ue the center o gravity (CM) a the point o application o any gravitational orce.. To apply the orce condition, chooe a convenient coordinate yte and reolve each orce into it - and y-coponent. 3. To apply the torque condition, chooe a convenient rotation ai generally one that pae through the point o application o an unknown orce. Then ind the torque due to each orce. Ue whichever ethod i eaier: either the lever ar tie the agnitude o the orce or the ditance tie the perpendicular coponent o the orce. Deterine the direction o each torque; then either et the u o all torque (with their algebraic ign) Leon, page
Leon : Equilibriu, Newton econd law, Rolling, Angular Moentu (Section 8.3- equal to zero or et the agnitude o the CW torque equal to the agnitude o the CCW torque. 4. Not all proble require all three equilibriu equation (two orce coponent equation and one torque equation). Soetie it i eaier to ue ore than one torque equation, with a dierent ai. Beore diving in and writing down all the equation, think about which approach i the eaiet and ot direct. There are any good eaple worked out or you in the tet. See page 8-89. Eaple: What i the allet angle a ladder can ake o that it doe not lide? W g N Solution: We will ue the condition or rotational equilibriu 0 We can chooe any ai about which to take torque. The ai I chooe i where the ladder touche the loor. The lever ar or the noral orce and the rictional orce will be zero and their torque will alo be zero. Recall that the torque i r I the ladder ha length L, the lever ar or the weight i the hort horizontal line below the loor in the diagra. The lever ar i the perpendicular ditance ro the line o the orce to the point o rotation. Here it i L r co The lever ar or the orce o the wall puhing againt the ladder i r Lin Leon, page
Leon : Equilibriu, Newton econd law, Rolling, Angular Moentu (Section 8.3- Uing the condition or rotational equilibriu The condition or tranlational equilibriu are 0 g 0 L W Lin g co 0 0 and 0 y Reerring to the BD, the -coponent give W W 0 0 Again looking at the BD, the y-coponent N g 0 N g Oh, no! our equation: Ue the torque relation y 0 L W Lin g co 0 W W N g N L Lin g co 0 L W Lin g co g W in co Leon, page 3
Leon : Equilibriu, Newton econd law, Rolling, Angular Moentu (Section 8.3- tan g W Subtitute the two orce equation ro Newton econd law Since thi i a tatic riction orce, tan g N W N tan N N tan tan tan N I the coeicient o tatic riction i 0.4, the allet angle i 5 o. Equilibriu in the Huan Body orce act on the tructure in the body. Eaple 8.0: The deltoid ucle eert on the hueru a hown. The orce doe two thing. The vertical coponent upport the weight o the ar and the horizontal coponent tabilize the joint by pulling the hueru in againt the houlder. There are three orce acting on the ar: it weight (g), the orce due to the deltoid ucle () and the orce o the houlder joint () contraining the otion o the ar. Leon, page 4
Leon : Equilibriu, Newton econd law, Rolling, Angular Moentu (Section 8.3- Since the ar i in equilibriu, we ue the equilibriu condition. To ue the torque equation we ue a convenient rotation ai. We chooe the houlder joint a the rotation ai a that will eliinate ro conideration. (Why?) r g g in5r r g 0 g 0 g 0 r 0 g rg (30 N)(0.75) r in5 (0. )in5 66 N To upport the 30 N ar a 70 N orce i required. Highly ineicient!! The Iron Cro. Here i an intereting video: http://www.youtube.co/watch?v=sdljygi_4 Becaue o the yetry, hal o the gynat weight i upported by each ring. Conider the BD above. 0 0 0 w Wrw r Leon, page 5
Leon : Equilibriu, Newton econd law, Rolling, Angular Moentu (Section 8.3- Wr r w W (0.60 ) (0.045)in 45 9.4W The orce eerted by the latiiu dori and the pectorali ajor on one ide o the gynat body i ore than nine tie hi weight! The tructure o the huan body ake large ucular orce neceary. Are there any advantage to the tructure? Due to the all lever ar, the ucle orce are uch larger than they would otherwie be, but the huan body ha traded thi or a wide range o oveent o the bone. The bicep and tricep ucle can ove the lower ar through alot 80º while they change their length by only a ew centieter. (p. 9) A video o equilibriu and how eaily it can be dirupted: http://www.youtube.co/watch?v=k6rxaei57c Rotational or o Newton econd law Very iilar to the other econd law. I Motion o Rolling Object A rolling object ha rotational kinetic energy and tranlational kinetic energy. K K tran K rot v CM I CM Why doe the object roll (and not lide)? rictional orce eert a torque on the object. Leon, page 6
Leon : Equilibriu, Newton econd law, Rolling, Angular Moentu (Section 8.3- Eaple 8.3 The acceleration o a rolling ball. The rotational or o Newton econd law i The torque on the ball i due to riction So I r I r I I r We can ue Newton econd law to ind the linear acceleration o the ball. A we uually do, take the +-ai along the incline. gin a a Ue the epreion or the rictional orce to ind, gin a I gin a r But the acceleration o the ball i related to it angular acceleration, a = r. Leon, page 7
Leon : Equilibriu, Newton econd law, Rolling, Angular Moentu (Section 8.3- I gin a r Ia gin a r Ia gin a r gin a I r or a unior, olid phere, I = (/5)MR and or a thin ring, I = MR. Which ha the larger acceleration? Conider the eect o the rotational inertia on the acceleration. Eaple: A olid phere roll down a hill that ha a height h. What i it peed at the botto? Solution: Ue conervation o energy. Since the ball roll without lipping, the rictional orce doen t do any work. The diplaceent i zero in the deinition W = r co. U gy K U K 0 0 v I The tranlational peed o the ball i related to it rotational peed, v = r. gy gh v v v ( I gh I r I I( v / r) r ) v or the olid phere, I = (/5)MR gh I r gh ( 5) gh (7 5) 0 v 7 gh Thi i le than the anwer we ound when we ignored rolling, v = gh. Angular oentu We introduced the idea o linear oentu in chapter 7. We had dp dt Leon, page 8
Leon : Equilibriu, Newton econd law, Rolling, Angular Moentu (Section 8.3- A iilar epreion eit or rotational otion dl dt The net eternal torque acting on a yte i equal to the rate o change o the angular oentu o the yte. The angular oentu L I i the tendency o a rotating object to continue rotating with the ae angular peed about the ae ai. Angular oentu i eaured in kg /. I the net torque i zero, we have the conervation o angular oentu L 0 L i L I the rotational inertia o the yte change, it angular peed will change to copenate. Angular oentu i a vector. (So i the torque!) The direction i given by the righthand rule. Our eaon are a conequence o the conervation o angular oentu. Leon, page 9
Leon : Equilibriu, Newton econd law, Rolling, Angular Moentu (Section 8.3- We have copleted our tudy o rotational otion. Try to ee how it i analogou to (the ore ailiar) linear otion. Here i a uary: Decription Linear Rotational poition diplaceent Rate o change o poition v Rate o change o velocity a Average rate o change o poition v, av av t t Intantaneou rate o change o poition v li li t 0 t t 0 t Average rate o change o velocity v a, av av t t Intantaneou rate o change o velocity v a li li t 0 t t 0 t v v a t t Equation o unior acceleration v t i ( v i Leon, page 0 a ( t) v ) t v v a i Inertia I r i i i t i ( t) ( ) t Inluence that caue acceleration r r Newton econd law Work Kinetic energy y a a y i I W r co W v I i
Leon : Equilibriu, Newton econd law, Rolling, Angular Moentu (Section 8.3- Moentu p v L Iω dp dl Newton econd law dt dt Condition or conervation o oentu 0 0 Conervation o oentu p i p Li L Leon, page