PHYSICS 151 Notes for Online Lecture 2.6

Size: px

Start display at page:

Download "PHYSICS 151 Notes for Online Lecture 2.6"

Malcolm Adams
5 years ago
Views:

1 PHYSICS 151 Note fo Online Lectue.6 Toque: The whole eaon that we want to woy about cente of ma i that we ae limited to lookin at point mae unle we know how to deal with otation. Let eviit the metetick. Say I apply a foce of 10 N on one end and 10 N on the othe end a hown 10 N 10 N We can fit attack thi poblem uin what we leaned when we wee tudyin tatic. Σ Since thee ae only foce in the y diection, we can add them up diectly. Takin up a poitive: -10 N + 10 N So accodin to ou old definition of equilibium, thi ytem i in equilibium. Howeve, we know that the metetick i oin to move, even thouh the net foce i zeo. Thi tell u that thee i omethin miin fom ou definition of equilibium. TRANSLATIONAL EQUILIBRIUM mean that the vecto um of the foce i zeo. Σ Line of Action The eaon ou definition ha failed u i that we now ae allowin object to have patial extent and o they can otate. Thee ae two thin that detemine the chaacteitic of the otation: the manitude of the foce and how fa fom the pivot the foce act. Axi of otation Applied oce Lectue.6 Pae 1

2 Impotant Toque Definition AXIS O ROTATION: The axi about which the object move (o would move if the object i in equilibium.) LINE O ORCE o LINE O ACTION: A taiht line unnin diectly thouh the applied foce. LEVER ARM: The leve am i the pependicula ditance fom the axi of otation to the line of action. indin the leve am i uually one of the hadet pat of toque poblem. Stat by findin, which i the ditance fom the axi of otation to the point whee the foce i applied. Note that i alway oin to lie alon the object that the foce i actin upon. We can then daw the leve am by dawin a line that i pependicula to the line of foce and pae thouh the axi of otation. We can call the leve am. The anle between and i called θ. Note, then that = b in θ I aid that the detail of the otation wee detemined by the manitude of the foce and the ditance fom the CM that the foce act. The poduct of thee two quantitie i called the toque. Toque decibe otation. τ = τ = inbθ Note: The unit of toque ae N-m even thouh we called that a Joule when we dealt with eney and wok, we don t call it a Joule when we e uin it to decibe toque. Axi of otation θ Line of Action θ Lectue.6 Pae

3 Execie: Identify the foce, how the line of foce and the leve am fo each of the followin ituation in the box next to the dawin. The axi of otation i denoted by a dot. Lectue.6 Pae 3

4 Diection: We ue the iht hand ule to detemine the Counteclockwie = poitive diection of the toque. If you cul you fine fom to, Clockwie =neative you thumb will point in the diection of the toque. In the left pictue, the toque will point into the pae. In the iht-hand pictue, the toque will point out of the pae. Ex 1: A civil eninee need to han a taffic liht ove an inteection fom a pole, a hown. The liht weih 70 N. 5 m 60 a) Daw the leve am. b) What i the manitude of the leve am aumin the bae of the uppot a the axi of otation? (in 60 = 3/, co 60 = 1/) c) What i the diection of the toque? d) What i the manitude of the toque? axi of e) Decibe diffeent way that the eninee could deceae the amount of toque on the uppot. Lectue.6 Pae 4

5 Ex. : An atit i deinin a mobile, a hown. Whee mut the mobile be hun to keep the od hoizontal and what i the tenion in the hanin tin? Aume that the od i male. An atit i deinin a mobile a hown. ind whee alon the od the mobile mut be hun to keep the od level. a) Aume the od i male and b) Aume the od ha a ma of 1.0 k m A.30 k 10.0m 8.0 m m B.50 k a) Aumin fit that the od ha no ma. Let aume that the point at which the mobile i to be hun i a ditance x fom the iht end of the od. it, daw a fee-body diaam to how all of the foce actin on the body. We can then tat by applyin the condition fo tanlational equilibium - the math fo tanlational equilibium i often eaie, o that' the fit one we'll ty applyin. Σy T ma mb T = ma+ mb T = m + m b A B b e j T 3. k+ 05. k 98. T = 784. N = 79. N m Thi tell u the tenion, but not whee the uppot mut be placed. o that, we have to apply the toque equation. We can take toque about any point. Let take them about the tin fom which the mobile balance. Thi ha the advantae that, if we couldn t fiue out the tenion in the tin, it won t ente the poblem. Lectue.6 Pae 5

6 8-x T x 8.0 m m A m B 10.0m Στ mbx+ mab8 x= 0 mbx = mab8 x mbx = 8mA xma bma + mbx = 8mA 8mA x = bma + mb 803 (. k) x = 03. k k b = 3m Note that we could alo take the toque about a diffeent point. Let calculate them about the iht end of the od. The um of the foce emain the ame. Lectue.6 Pae 6

7 Ex. 3: How doe the ditance chane if the od ha a ma of 1.0 k? We now have to include the weiht of the od. The weiht of the od act at the CM. Since the od i unifom, thi will be the eometic cente of the od. 8-x T x 5m m A m 8.0 m 10.0m m B Σ T m m m A B T = m + m + m T 3. k+ 05. k+ 10. k 98. T = 17. 6N = 18N So the tenion inceae, a we would expect. I m oin to take the axi of otation to be at the iht end aain. y Σ τ ihtend m(8) + m(5) Tx= 0 A m(8) + m(5) = Tx A b A B b e j ( ) ma m x T m [ 8(0.3 k) + 5(1.0 k) ]( 9.8 ) m x = = 4.0m 18 N Lectue.6 Pae 7

8 Ex. 3: A plank of ma m p = 1.0 k i placed on two cale. Mae ae placed a hown. M A = 5.0 k at a ditance 0.30 m and M B = 5.0 k at a ditance 0.90 m. What do the two cale ead? 1.00 m 0.30 m 0.90 m m A =5.0 k m B = 5.0 k it, daw the fee body diaam fo the plank, then apply the condition fo tanlational equilibium. Σy + m m m m 1.0 m 0.90 m 1 A B p + = m + m + m 1 A B p + = 50. k k k 1 d b Aain, we e tuck hee, o we need to do the toque equation. Let tat by takin the axi of otation about the left cale. Thi eliminate 1 fom the equation, allowin u to olve fo. Ccw otation will be conideed poitive. i m A m p m B Lectue.6 Pae 8

9 Σ τ 1 m(0.30 m) m(0.50 m) m(0.90 m) + (1.0 m) A p B ma(0.30 m) + mp(0.50 m) + mb(0.90 m) = (1.0 m) ma(0.30 m) + mp(0.50 m) + mb(0.90 m) = (1.00 m) [(5.0 k)(0.30) + (1.0 k)(0.50) + (5.0 k)(0.90) ] = (1.0) 6.5k = = m Now, let do the ame thin, but ue a the pivot point. Σ τ + m A (0.7 m) + m p (0.5 m) + m B (0.1 m) 1 (1.0 m) m A (0.70 m) + m p (0.50 m) + m B (0.10 m) = 1 (1.0 m) ma(0.70 m) + mp(0.50 m) + mb(0.10 m) 1 = (1.0 m) ma(0.70) + mp(0.50) + mb(0.10) 1 = (1.0) (5.0 k)(0.70) + (1.0 k)(0.50) + (5.0 k)(0. 10) 1 = (1.0) 1 4.5k = Thee eult can be checked by puttin the eult into the foce equation Σy + m m m 1 A B p 1 d A B pi + = m + m + m b 45. k( ) k( ) = 50. k k k 11 k = 11 k Lectue.6 Pae 9

10 Ex. 4: Cantileve. A tatue i to be placed at the end of the plank a hown. If the tatue ha a ma of 1.5 x 10 3 k and the plank ha ma =.00 x 10 3 k, what foce mut the two uppot exet? 10.0 m 0.0 m Daw a fee-body diaam: 15.0 m m 10.0 m m p it, wite that the um of the foce i zeo Σy + m m 1 p 1 d pi + = m + m Now, wite the toque equation takin the toque about the leftmot pt. Takin toque that caue ccw otation to be poitive, Σ τ 1 m(30.0 m) + m(15.0 m) (10.0 m) m m ( ) (30.0 ) (15.0 ) x N p m(30.0 m) + mp(15.0 m) = (10.0 m) x k m + x k m (10.0 m) = To et 1, we can eithe ue the equation fom foce, o take toque about anothe axi. I m oin to ue the foce equation. = Lectue.6 Pae 10

11 + = m + m 1 = m + m 1 p m = 15. x10 k+. 00x10 k x10 N 1 = 55. x10 N 1 d pi d i d ie j Note that the expeion fo 1 came out neative, which how u that we aumed the won diection fo the 1. Lectue.6 Pae 11

PHYSICS 151 Notes for Online Lecture #20

PHYSICS 151 Notes for Online Lecture #20 PHYSICS 151 Notes fo Online Lectue #20 Toque: The whole eason that we want to woy about centes of mass is that we ae limited to looking at point masses unless we know how to deal with otations. Let s evisit