Lab 10: Newton s Second Law in Rotation

Transcription

1 Lab 10: Newton s Second Law in Rotation We can descibe the motion of objects that otate (i.e. spin on an axis, like a popelle o a doo) using the same definitions, adapted fo otational motion, that we have used fo objects that move in a staight line. That is, fo linea motion we defined position, velocity, and acceleation; fo otational motion we define position, angula velocity, and angula acceleation. Fo objects in linea motion, Newton s Second Law tells us that foces acting on the object cause its acceleation; fo an object in otational motion, Newton s Second Law tells us that toque acting on the object causes its angula acceleation. n Lab 4 we demonstated Newton s Second Law fo an object in linea motion: we attached a sting to a cat on an aitack and a hanging mass and allowed the system to acceleate. n Lab 10, we will attach a sting to a otating appaatus and a hanging mass; as the hanging mass falls, the tension of the sting will ceate a toque on the otating appaatus that will cause it to otate. Newton s Second Law in otation has exactly the same fom as fo linea motion. That is: Linea: Foce = mass acceleation Rotation: Toque = moment of inetia angula acceleation n symbolic notation, the latte equation is τ = α whee τ and α ae the Geek lettes tau and alpha. We use to epesent the moment of inetia, which plays the same ole in otational motion that mass does fo linea motion. The toque applied to a otating object is defined as: τ = F sinφ whee F is the foce applied to the object, is the distance fom the point of otation to the point whee the foce is applied, and φ is the angle between and F. Note that if a foce is applied at the point (o axis) of otation, the toque will be zeo (because is zeo.) Think about opening a doo: if you push on the side

2 with the hinge, will be zeo, and the doo will not move. This is why we push on the side of the doo opposite the hinge: bigge esults in bigge toque, making it easie to open the doo. Fo Lab 10, we can conside thee foces acting on the otating appaatus: gavity, nomal foce and the tension of the sting. Gavity and nomal foce each act at the cente of the appaatus, so is zeo fo each of these two foces; they exet no toque (i.e. gavity and nomal do not cause the appaatus to otate.) The tension acts at the edge of the clea hoizontal pulley at a distance equal to the adius of the pulley. The tension pulls along the edge of the pulley (i.e. in a tangential diection), so the angle between and F is 90 degees. Fom these obsevations, we can claim that the toque acting on the appaatus is: τ = T whee is the adius of the clea hoizontal pulley and T is the tension of the sting. Just as the toque acting on the object depends on the foces and whee those foces ae applied, the moment of inetia of a otating object is a measue of the mass of the object and how it is distibuted. Mass that is fathe fom the cente of otation (i.e. the axis of otation) is moe difficult to otate, because it moves in a lage cicle, than mass that is close to the axis. Fo this eason, the moment of inetia of an object is defined by its mass and its linea dimensions. The geate the mass and the geate the size of the object, the geate its moment of inetia. Lab 10 has two pats: Pat 1: Calculate the expected value of the moment of inetia of the otating appaatus. Pat 2: Measue the moment of inetia of the otating appaatus by applying a toque to the system (by the tension in the sting) and measuing the coesponding angula acceleation. Afte measuing the data fo Pat 2, we will ceate a gaph and fom the gaph we will be able to detemine the moment of inetia of the appaatus (i.e. how fast it actually acceleated elative to how much toque was applied... Newton s Second Law!) To accomplish this, we have to do a bit of algeba: Rotating appaatus: T = α (this is a esult of the definitions pesented above.) Hanging mass: mg - T = ma The sting and hanging mass have the same linea acceleation, and the sting has the same linea acceleation as a point on the edge of the clea pulley. The clea pulley also has the same angula acceleation as the otating appaatus. We can algebaically connect the linea acceleation and angula acceleation of a point on the edge of the clea pulley: a = α whee is the adius of the clea pulley

3 We can eliminate a fom these thee expessions: α T = mg T = mα And then eliminate T: mg α = mα We will measue seveal values of m and the coesponding value of α fo each. We would like to be able to gaph these two vaiables as a staight line, but the equation above is fa too complicated to even hope fo a staight line with α and m. Fotunately a little bit of algeba can help. Fist, we ll eaange the equation to solve fo α: mg = mα + α α = mg m + / This will not wok fo a staight line gaph, as m appeas in the numeato and denominato on the ight side. So we use a cleve algebaic tick: invet both sides. m + / O: α = α = + mg g mg α = m g + g n this final expession, we have α -1 on the left and m -1 on the ight. f we gaph α -1 vs m -1, we expect to get a staight line with a slope of / g. Pat 1 We will neglect the contibution of the clea plastic pulley to the moment of inetia of the system. The mass of the pulley is elatively small and it is elatively close to the axis of otation, so the effect of the pulley on the inetia of the otating system should be negligible. The moment of inetia of the otating system should include only the long, thin metal od and the two bass weights (at eithe end of the od.)

4 To find the expected value of the moment of inetia: Caefully measue M od and L, the mass and length of the od. Measue M b, the mass of both bass weights togethe. Measue d, the distance fom the cente of the od to the cente of one of the bass weights. Calculate the moment of inetia fo the od and bass weights sepaately, and then add the esults to get the total moment of inetia of the appaatus: od = 1 12 M od L 2 bass = M d b 2 Show these calculations in you Excel sheet. Expess the esult of each calculation to thee significant figues. You total moment of inetia should be aound 50,000 g-cm 2. Pat 2 Using the calipe, measue the diamete of the clea pulley that will hold the black sting. Measue the diamete with and without the sting and aveage these two measuements. Recod the adius of the pulley by taking half of the aveage diamete. Set up the otating appaatus so that the sting between the appaatus and the black pulley is hoizontal and passes ove the black pulley staight. You will have to set up the pulley at an angle to the appaatus; see the example. Place 30 gams on the mass hange. The hange itself is 5 gams, so the total hanging mass is 35 gams. Afte connecting the Rotay Motion Senso to the black box and the black box to the compute, open Data Studio. Select the Rotay Motion Senso fom the input menu. Choose only angula velocity fom the Measuements menu and change the units to ad/sec. Fom the display menu choose Gaph.

5 Make sue the sting is wapped aound the coect clea plastic pulley. Release the appaatus, allowing it to otate. Click START on Data Studio; the softwae will display points on a gaph of angula velocity vs time. Click STOP befoe the sting is completely unwound; stop the otating appaatus. Use the FT menu at the top of the Data Studio gaph to display the stats fo the line. The slope of the line on the Data Studio gaph is the angula acceleation; ecod it in you data table. Ceate a data table of fou columns: m (in gams) α (in ad/s 2 ) m -1 (in g -1 ) α -1 (in s 2 /ad) Repeat the above pocedue fo m = 35, 45, 55, 65, 75, 85, 95. Ceate a gaph of α -1 vs m -1 and find the equation of the best fit line. Using the slope of the best fit line, calculate the measued value of the moment of inetia. Fom the equation deived above, the slope should be equal to / g. So: meas = g( slope) =... Note that all of ou measuements ae in gams and centimetes, so you should use 980 cm/s 2 fo g. Compae you measued value to the expected value fom Pat 1 (i.e. calculate the pecent diffeence.) f you measued caefully, you should find a diffeence of less than 1%. We will look fo evidence of andom and systematic eo in the class esults (i.e. does eveyone have a small positive diffeence, o ae thee some positive and some negative diffeences.)