Physics 2A Chapter 10 - Moment of Inertia Fall 2018

Transcription

1 Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion. Essentially, eveywhee we used mass in ou expessions fo linea motion (i.e. Newton s Second Law, kinetic enegy, momentum...) we will use moment of inetia in the coesponding expessions fo otational motion. See the Chapte 0 notes on otational otion fo the details on how we use moment of inetia. The moment of inetia of an object depends not only on the mass of the object, but also on how fa fom the cente of otation (i.e. the pivot ) that mass is distibuted: the futhe the mass is fom the cente, the faste that mass is actually moving as the object otates. f we conside the motion of a single point on a otating object, we can wite the kinetic enegy of that point in tems of eithe its linea motion o its motion as pat of the otating object: K ½ m v o K ½ ω whee m is the mass of the single point and is the moment of inetia of that single point. We can also use the fact that the linea motion of that point is elated to the otational motion of the object: v ω whee is the distance fom the pivot to the point in question. f we set the two expessions fo K equal to each othe, and we eplace v with its equivalent as given by the thid equation, the esult is: ½ m (ω ) ½ ω o m moment of inetia fo a single point To ceate an expession fo the moment of inetia fo any object, we conside the moments of inetia of evey point on the object and we add the moments of inetia of all the points. ecause a point, by definition, is infinitesimally small and thee ae an infinite numbe of them fo any object, ceating an expession fo the moment of inetia of any object inheently equies an integal. Long Thin od Fo this deivation we define a long thin od of mass and length L. Page of 8

2 Physics Chapte 0 - oment of netia Fall 08 The point of otation is at the cente of the od. We can define an x-axis along the od with x 0 at the cente. The ends of the od ae at x L/ and X -L/. nd we can define a single point on the od which has an infinitesimally small mass, dm, and coespondingly small size dx. Using ou definition of the x-axis, we note that ou defined point is at some x coodinate, which we will efe to as simply x. The x-coodinates fo all of the points that make up the od go fom x -L/ to x L/. So we can now wite the moment of inetia of a single point: dm x This expession epesents evey point along the od, and we will need to add them by using an integal. ut fist we have to wite ou dm in tems of dx. To accomplish this, we use the fact that the od is unifom, i.e. the mass of the od is popotional to its length. Theefoe we can wite: mass of single point mass of entie od length of single point length of entie od O dm / dx / L We can use this expession to eplace dm in ou expession fo the moment of inetia: Page of 8

3 Physics Chapte 0 - oment of netia Fall 08 (dx / L) x ( / L) x dx We can now integate this expession to add all of the fo the points that make up the od. The esult will be the expession fo the moment of inetia of ou long, thin od: L L / L / L L x dx x L / L L / L 8 8 L long thin od otated about its cente We can also conside the moment of inetia fo the same long thin od when it is otated about one end. The definitions pesented above ae exactly the same. The only diffeence is now we define x 0 at the left end of the od, so that the points on the od ae fom x 0 to x L. The integal is exactly the same; we only change the limits of the integation: L L x dx x L L 0 L L 0 ( 0) L long thin od otated about one end Page of 8

4 Physics Chapte 0 - oment of netia Fall 08 ectangula Plate We can now use ou long thin ods to constuct a ectangula plate. magine laying an infinite numbe of infinitesimally thin ods side-by-side to fom a ectangle like this: We can define the dimensions of ou ectangle as and, and the mass of ou ectangle is. We can also define an x-axis so that x 0 is at the cente of the ectangle. nd we can conside one of the thin ods, the cente of which is at some coodinate x : Page 4 of 8

5 Physics Chapte 0 - oment of netia Fall 08 The length of the thin od is and the mass of the thin od can be defined by dm (i.e. since the od is infinitesimally thin, it has an infinitesimally small mass.) We know the moment of inetia of the od, if it is otated about its cente, is given by the expession we deived above. ut this od is not being otated about its cente; the pivot fo the otation is the cente of the ectangle. Which means to wite the moment of inetia of the od, as it otates about the cente of the ectangle, we must invoke the paallel axis theoem. The paallel axis theoem states that if an object is otated about an axis that is paallel to an axis though the cente of mass of the object, then the moment of inetia can be found by: pivot cm md paallel axis theoem whee cm is the moment of inetia if the object wee otated about its cente of mass, m is the mass of the object, and d is the distance fom the cente of mass of the object to the pivot (o axis) in question. We can make use of the paallel axis theoem to wite the moment of inetia fo ou long thin od when it is otated about the cente of the ectangle. The distance d fo ou long thin od is simply its x- coodinate (because the cente of the ectangle is at x 0) so the moment of inetia of ou long thin od is: dm dm x We can apply the same logic to dm in this situation that we did with the deivation fo the long thin od in the pevious section: mass of one thin od mass of entie ectangle aea of one thin od aea of entie ectangle dm dx so: dm dx Now we can wite the expession fo the moment of a single long thin od that is otated about the cente of the ectangle: dx dx x Page 5 of 8

6 Physics Chapte 0 - oment of netia Fall 08 Page 6 of 8 dx x We can now add the moments of inetia fo all of the long thin ods that make up the ectangle. To do this, we integate the expession above. The limits fo x ae those of the left side and ight side of the ectangle, i.e. x -/ and x /. / / / / x x dx x Thin ing The moment of inetia of a thin ing is vey simple and does not even equie an integal. We define a ing of mass and adius. We can define the mass of a single point on the ing as dm. f the cente of otation is at the cente of the ing, then the distance fom the point of otation to evey point on the ing is simply the adius of the ing. We can then wite the moment of inetia fo a single point on the ing as: dm f we add the moments of inetia fo all points on the ing, we have the integal of dm on the ight (because is constant), and the moment of inetia fo a thing ing is: ) ( ectangula plate otated about its cente unifom thin ing otated about its cente

7 Physics Chapte 0 - oment of netia Fall 08 Solid Disk We can now constuct a solid disk fom thin ings in exactly the same way that we constucted ou ectangula plate fom long thin ods. The disk will have adius and mass and is essentially an infinite numbe of concentic infinitesimally thin ings. We can define one of these thin ings to have adius. We can also define 0 at the cente of the disk: The moment of inetia of the one thin ing can be witten as: dm We can apply the same logic to dm in this situation that we did fo the pevious deivations: mass of thin ing mass of entie disk aea of thin ing aea of entie disk What is the aea of the ing? The thickness of the ing is infinitesimally thin. nd since the dimension fom the cente of the ing (and the disk) to the edge is defined as the adial diection, we can say that the thickness of the ing is an infinitesimally thin bit in the adial diection... o d. f we wee to lay out the thin ing in a staight line, it would have a length equal to its cicumfeence and a width equal to its thickness. So the aea of the thin ing is simply π d Page 7 of 8

8 Physics Chapte 0 - oment of netia Fall 08 nd: dm d dm d so: π π Now we can wite the moment of inetia of one thin ing as: d d We can now integate this expession to add all of the fo the thin ings that make up the disk. The esult will be the expession fo the moment of inetia of ou disk: 0 d unifom solid disk otated about its cente This is just a small sample of the simple objects fo which we can deive expessions fo moments of inetia. y using expessions aleady deived fo some objects (e.g. fo the long thin od and thin ing), as well as the paallel axis theoem, we can build othe objects and constuct thei coesponding moments of inetia. Page 8 of 8