Physics 24A Class Notes Section 26 The Laws of otational Motion What do objects do and why do they do it? They otate and we have established the quantities needed to descibe this motion. We now need to find the laws that explain it. Since we could find the otational vaiables by analogy to the tanslational vaiables, we ll ty to find the laws of otational motion in analogy to the laws of tanslational motion. Section Outline 1. The ist Law of otational Motion 2. The Second Law of otational Motion 1. The ist Law of otational Motion Newton s Laws of Motion wok so well fo tanslational motion that we should use them as a model of the laws fo otational motion. The idea hee is to change the laws as little as possible, but still popely undestand otational motion. Newton s ist Law Evey object will move with a constant velocity unless a foce acts on it. The ist Law explained the natual state of motion and that foce was the agent of change of motion. Just as the natual state of tanslational motion is any constant velocity, the natual state of otational motion is any constant angula velocity. Think about a spinning top. It will keep spinning at a constant ate as long as fictional foces don t cause it to slow down. So it looks like we just need to add the wod angula in font of velocity. Howeve, some foces don t cause changes in otation. Think about the font wheel on you bike. Lift it off the gound so that it is fee to spin. Thee ae two foces acting on it, gavity downwad and the foce the axle exets upwad at the cente. Neithe causes otation. You can even exet additional foces on the tie that won t cause it to otate if you push diectly towad the axle. If you push in any diection that doesn t point at the axle, the tie will begin to spin. This otating effect of a foce is called toque. So, we must eplace foce in the ist Law with toque. oces that act diectly towad o away fom the axle cause no otation. oces that don t act diectly towad the axle do cause otation. Newton s ist Law fo otation Evey object will move with a constant angula velocity unless a foce toque acts on it. We need to get a bit moe quantitative with toque. Think about a foce that acts tangentially on the bike wheel, as shown at the ight. The lage the foce, the geate the change in angula velocity. Howeve, the location of the foce also mattes. If the same foce wee exeted at half the distance fom the cente, it would cause half the change in angula velocity. So, we can wite the toque as the poduct of the foce and the adius at which it acts, τ. 26-1
θ Physics 24A Class Notes The foce doesn t have to be applied pependicula to the adius vecto. Suppose the foce acts at an angle θ to the adius vecto as shown at the left. Now, only the component of the foce that is pependicula to the adius vecto ceates toque. This pependicula component can be witten in tems of the angle, τ ( sinθ). We could associate the sinθ with the instead of the, θ τ ( sinθ) ( sinθ). This is shown at the ight. The line along which the foce acts is efeed to as the line of action. The moment am o leve am is the pependicula distance fom the pivot point to the line of action of the foce,. In summay, foces with lines of action not passing though the axis of otation (pivot) ceate toque. the toque gows as the foce gows. the toque is popotional to the distance between the foce and the pivot. any foce can be esolved a component towad the pivot which ceates no toque and a component pependicula to the line to the pivot which causes toque. We can encapsulate all of this using the mathematical idea of a coss poduct to define toque, The Definition of Toque τ. We ll woy about the diection of the toque vecto a bit latte. o now we ll only deal with the magnitude. Example 26.1: To open a paint can, a foce of 1N is exeted downwad on a 25.cm long scewdive. ind (a)the toque poduced and (b)the foce needed if it is exeted at an angle of 3. to the vetical. Given: 1N,.25m, and θ 3. ind: τ? and /? paint ' 3 (a)using the definition of toque and the fact that the moment am and foce ae pependicula, τ τ (.25)(1) τ 25.N m. (b)when the foce and moment am ae not pependicula, it is only the pependicula pat of the foce that counts, τ τ cos3 τ cos3 line of action 25..25cos3 115N. 26-2
Physics 24A Class Notes 2. The Second Law of otational Motion The Second Law told us how much foce was equied to change the velocity at a given ate and defined the idea of inetia. Newton s Second Law Acceleation of an object is diectly popotional to the net foce acting on it and invesely popotional to its mass (inetia). We definitely need to eplace the acceleation with angula acceleation and foce with toque. Instead of mass we need to undestand the inetia associated with otation. It is called otational inetia. Newton s Second Law fo otation Angula Acceleation of an object is diectly popotional to the net foce toque acting on it and invesely popotional to its mass otational inetia. Mathematically we just need to change Σ ma to Στ Iα whee I is the otational inetia. The Second Law fo otation Στ Iα What is the otational inetia? What does it depend on? How do you calculate it? pivot α Conside a solid object pivoted at a cetain point having an angula acceleation, α, as shown at the ight. o any small chunk of its mass, dm, the foce, d, on this chunk must be pependicula to the vecto fom the pivot. Othewise, this chunk would acceleate towad the pivot. The toque, dτ, on this chunk of mass is given by the definition, τ dτ d. Using the Second Law fo Tanslation and the elationship between tangential acceleation and angula acceleation, dτ d adm α 2 dm. Integating ove the entie object, dτ α 2 dm τ α 2 dm. Compaing this esult with the Second Law fo otation gives the expession that will seve as the definition of otational inetia. The Definition of otational Inetia I egula inetia, the esistance objects offe to changes in tanslation, depends only on the mass of an object. otational inetia, the esistance objects offe to changes in otation, depends not only on the mass of the object, but also on the shape. 2 dm dm d 26-3
Physics 24A Class Notes Example 26.2: A dumb bell consisting of two 1.kg masses 5.cm apat sepaated by a light stick. It is otating about its cente of mass. ind the toque equied to slow it unifomly fom 5.pm to est in 3.s. Given: m 1 m 2 1.kg, 1 2.25m, ω o, ω 5.pm, and t 3.s ind: τ? cm m1 m 2! 1 2 In ode to apply the Second Law fo otation, we need the otational inetia and the angula acceleation. Since the masses ae small compaed to thei distance fom the axis of otation, the integal in the definition of otational inetia becomes a sum, I 2 2 dm I m i i. Putting in the numbes,i m 1 2 1 + m 2 2 2 (1)(.25) 2 + (1)(.25) 2 1.25kg m 2. Now we can use the definition of angula acceleation, α dω dt α Δω Δt ω ω ev min 2πad o (5. min )( 6 s )( ) ev.175ad / s 2. Δt 3. inally, using the Second Law, Στ Iα τ Iα (1.25)(.175) τ.219n m. An extended object is one whee the distance fom the otation axis to the mass elements is not lage compaed to the size of the object. In this case, the otational inetia must be found by integation. Example 26.3: (a)ind the otational inetia of a unifom ing of adius,, and mass, M. (b)epeat fo a disk. Given: M and ind: I ing and I disk Use the definition of otational inetia keeping in mind that all the dm s have the same, I 2 dm 2 dm 2 dm 2 M. The sum of all the dm s is M. The otational inetia fo a ing is then, I ing M 2. o a disk, the dm is an infinitely thin ing of adius. The otational inetia of this ing is di given by the expession we just found, di 2 dm. Adding these up ove all the ings will give the otational inetia of the disk, I disk di 2 dm I disk 2 dm. The mass, dm, in each ing depends on the adius. Assuming the disk is unifom, the mass is distibuted in popotion to the i M d y dm x 26-4
Physics 24A Class Notes suface aea so, dm da. M A The aea of the disk is A π 2. The da is the aea of the thin ing. At ight is a sketch of this ing laid out in a staight line of thickness d and length 2π. It has an aea of da 2πd. So, dm d 2! dm 2πd 2M d. 2 dm M π 2 Now the integation can be done, 2M 3 2M 4 2 2 2M I disk d d I disk 12 M. 2 2 2 4 The otational inetia of objects depends on thei shape. Hee is a table listing the equation fo the otational inetia of some objects: 26-5
Physics 24A Class Notes Section Summay We now have some undestanding of why objects otate the way they do. We built the laws of otational motion in analogy to Newton s Laws of Motion fo tanslation, Newton s ist Law fo otation Evey object will move with a constant angula velocity unless a toque acts on it. Newton s Second Law fo otation Angula acceleation of an object is diectly popotional to the net toque acting on it and invesely popotional to its otational inetia. The ist Law led us to define toque as the otating effect of a foce, The Second Law was ewitten mathematically as, whee we defined the otational inetia as, The Definition of Toque τ (τ ). The Second Law fo otation Στ Iα, The Definition of otational Inetia I 2 dm. 26-6