8.4 oque oque otational Dynamics Poblem-Solving We began this couse with chaptes on kinematics, the desciption of motion without asking about its causes. We then found that foces cause motion, and used Newton s laws to study dynamics, the study of foces and motion. In chapte 8 we have a deceptively bief section on angula kinematics. It is bief because we aleady leaned how to solve such poblems in chaptes 2 and 3. What s net? kinematics dynamics otational kinematics otational dynamics he otational analog of foce is toque. Conside two equal and opposite foces acting at the cente of mass of a stationay mete stick. Conside two equal and opposite foces acting on a stationay mete stick. Does the mete stick move? Σ et = ma cm = 0 so a cm = 0. Does the mete stick move? Σ et = ma cm = 0 so a cm = 0. he cente of mass of the mete stick does not acceleate, so it does not undego tanslational motion. Howeve, the mete stick would begin to otate about its cente of mass.
A toque is poduced by a foce acting on an etended (not pointlike) object. he toque depends on how stong the foce is, and whee it acts on the object. You must always specify you efeence ais fo calculation of toque. By convention, we indicate that ais with the lette and a dot. oques cause changes in otational motion. oque is a vecto. It is not a foce,* but is elated to foce. Let s apply a foce to a od and see how we get a toque. ist apply the foce. You need to choose an ais of otation. Usually thee will be a smat choice. Label it with a point (o line) and an. Choose the diection of otation that you want to coespond to positive toque.* Label the positive diection with a cuved aow and a sign! Do this aound the point labeled. *So neve set a foce equal to a toque! *aditionally, the counteclockwise diection is chosen to be positive. You ae fee to choose othewise. Daw a vecto fom the oigin to the tail of the foce vecto. Give it a name (typically ). Label the angle between and (you may have to slide the vectos aound to see this angle). he symbol fo toque is the Geek lette tau (τ). he magnitude of the toque due to is sin. Look at you diagam and detemine if the toque would cause a o a otation (accoding to you choice of ). In this case, the otation would be -, so τ z =- sin. Impotant: is the angle between and. Slide and aound until thei tails touch. is the angle between them. Between means go fom to. Don t be fooled by a poblem which gives an angle not between the vectos! (Eample coming soon.) his is the coect angle between. Watch out fo diagams containing some othe angle!
In this diagam, which is the angle between and? hee ae othe ways to find the toque. ften it is easy to visualize, the component of which is pependicula to.? N! hee ae two choices fo the angle between and. Because sin()=-sin(-), eithe choice will give you the coect answe (switch diection of otation and switch sign on sine gives no net switch in sign). he magnitude of the toque due to is, and in this case τ z =-. (Note = sin.) Sometimes it is easie to visualize, the component of which is pependicula to. he magnitude of the toque due to is, and in this case τ z =-. (Note = sin.) Summaizing: SE: τ z = = = sin is called the leve am o moment am. he line along which is diected is its line of action. he z ais passes though the point and is pependicula to the plane of the pape. o find the diection of the toque, cul you finges aound the diection of otation fom into. he thumb of you ight hand points in the diection of the toque. You don t need to know this fo the eam! Impotant eminde: label the point about which you toques ae calculated and daw a cuved aow aound it with a sign to show what you have chosen fo a positive sense of diection. Daw the cuved aow aound the point, not somewhee else! A toque poducing a otation is. A toque poducing a - otation is -.
Eample 8-8. he biceps muscle eets a vetical foce of 700 N on the lowe am, as shown in the figue. Calculate the toque about the ais of otation though the elbow joint. hee is no new litany fo toques. You should adapt the litany fo foce poblems. When you wok with toques, the fist thing you need to do is daw an etended fee-body diagam. Befoe that, we need to have a diagam of the thing we ae investigating. We ae not inteested in the uppe am! =5 cm 30 We have ou diagam. Now we must do a fee-body diagam. o otational motion, we must do an etended fee-body diagam, which shows whee the foces ae applied. Label the otation ais. Choose a diection fo otation. How about this fo an SE? τ z = sin No! No! No! om the etended fee-body diagam, I see that the angle between and is 90, so τ z = sin(90) would wok. I think it is bette to look fo o. In this case, is easy to see. om the diagam = cos. SE: τ z = = cos. Done! (Ecept fo plugging in numbes.) hat was a lot of wok fo something that took 2 lines in the tet! No. I showed you a geneal appoach to toque poblems. he tet just solved one simple poblem. If moe than one toque acts on an object, the net toque is the algebaic sum of the two toques ( algebaic means thee may be signs involved). Eample 8-10. Calculate the net toque on the compound wheel shown in the dawing. he diagam will seve as an etended fee-body diagam. No need fo a sepaate one. τ z,net = τ z = τ z,1 τ z,2 τ z,net = 1 1 2 (- 2 cos) τ z,net = 1 1-2 2 cos 1 1 2 2
8.5 otational Dynamics; oque and otational Inetia We saw in ou study of dynamics that foces cause acceleation: Σ = ma. oques poduce angula acceleation, and the otational equivalent of mass is the moment of inetia, I: SE: Στ z = Iα z. his is eally a vecto equation, but ou poblems will all have a unique ais of otation, which is like a onedimensional poblem, so that the only vestiges of the vecto natue of τ z will be the sign. What is this moment of inetia, I? It is the otational analog of mass. I depends on the mass of the object. It also depends on how the mass is distibuted elative to the ais of otation.* igue 8-20 gives I fo vaious objects of unifom composition. You will be given this figue (o its equivalent, o appopiate potions of it) on an eam o quiz. Solid cylinde, mass M, adius I=½M 2 It doesn t matte how thick the cylinde is! *his means a single object can have diffeent I s fo diffeent aes of otation! 8.6 Solving Poblems in otational Dynamics Eample 8-13 (modified). A cod of negligible mass is wapped aound a fictionless pulley of mass M and adius. he pulley otates about a fied ale which passes though its cente. A bucket of mass m hangs fom the cod. Calculate the angula acceleation of the pulley and the linea acceleation of the bucket. he litany fo foce poblems still woks! Step 1. Daw a basic sketch. M m
Step 2. Daw fee-body diagams. o objects that otate, the fee-body diagam must be etended; it must show the actual points of application of foces. a bucket w=mg Step 3. Label each vecto (done). Step 4. Daw aes (done). α P (foce due to ale) W=Mg pulley Step 5. Daw pojections of foces not along aes (done). o avoid etaneous minus signs, make sue you a and α have the same sense. Step 6. SE bucket: Σ = ma pulley: Στ z = Iα z (We don t need the sum of foces equation fo the pulley in this eample.) a bucket w=mg α P (foce due to ale) W=Mg pulley Step 7. Wite out sum of foces/toques equation eplicitly, then eplace geneic quantities with labeled quantities. bucket pulley W = ma τ,z τ W,z τ P,z = Iα z - mg = ma 0 0 = Iα Step 8. Solve. You need to use the SE a = α to connect a and α.