Reading Assignment: Lab #9: The Kinematics & Dynamics of Cicula Motion & Rotational Motion Chapte 6 Section 4 Chapte 11 Section 1 though Section 5 Intoduction: When discussing motion, it is impotant to be awae that thee ae seveal diffeent types and that each type is dependent upon a paticula fame of efeence. Linea motion is the most basic fom, involving the tanslation of a paticle fom one point to anothe in one dimension. Pojectile motion, although two-dimensional, is easily analyzed by sepaating it into its linea components. Cicula motion, is a special case of two-dimensional motion in which an object tanslates in a cicula path. Rotational motion is obseved when an object itself otates about some intenal axis. Conside the following examples: A ca taveling along a staight oad exhibits linea motion fom the pespective of a peson standing on the side of the oad. The wheels of the ca exhibit otational motion about the axle. A pebble embedded in the tead of the ca exhibits cicula motion about the axle. A baseball spins towads a batte at home plate. Fom the pespective of a spectato, the ball is undegoing tanslational motion acoss the infield. Howeve, the ball itself is otating about its own axis. The stitches of thead on the outside of the ball ae moving in a cicle elative to the spinning axis of the ball. (These cicles diffe depending upon the location of the paticula stitch.) The eath otates on its own axis as it also moves in a cicle fom the pespective of a stationay sun. In addition, people located at the suface of the eath undego a diffeent cicula motion each day aound the axis of otation of the eath. (These cicles diffe depending upon the latitude of each peson.) It is impotant to notice that cicula motion connects the concepts of linea and otational motion. Fo any object that is otating, a paticula point on that object is moving in a cicle. One of the goals of this lab activity is to exploe and undestand this connection. The tanslational motion of any point paticle can be descibed in tems of standad Catesian coodinates. In othe wods, Catesian coodinates can descibe both linea and cicula motion. Howeve, in the case of cicula motion, the paticle moves an ac length, s, aound a cicle with a (constant) adius,. Theefoe, in this case it is simple to use Pola coodinates whee the position of the paticle can be specified by and the angula position, θ, athe than x and y. Y (x,y) = ( cosθ, sinθ) S θ X Lab #9 Kinematics & Dynamics of Cicula & Rotational Motion
Notice that the ight tiangle defines the elationship between the location of the paticle in Catesian and Pola coodinates. The following equations descibe how to tansfom between Pola and Catesian coodinates. x = cosθ y = sin θ = x + y θ = tan x = y 1 s The adius of a paticle undegoing cicula motion is always a constant. The angula position, howeve, will change with time depending on the motion of the paticle. Since only the angula position changes with time its behavio is exactly analogous to the behavio of the position in one-dimensional motion that was studied peviously. Thus, the angula equivalent of the kinematic quantities fo onedimensional motion can be defined as follows: Ac distance taveled s s = θ Angula position Linea (tangential) velocity Linea (tangential) acceleation ds dθ v v = = = dt dt a dv = dt ω Angula velocity dω = dt t = at α Angula acceleation The elationships between the angula position, velocity, and acceleation ae exactly the same as the elationships peviously detemined fo one-dimensional motion. Fo example, fo motion with constant angula velocity, ω: θ = ωt + θ o Fo motion with constant angula acceleation, α : ω = αt + ω o θ = 1 αt + ω + θ o t The Net Foce that causes an object of mass, m, to move in a cicula path is called the centipetal foce, F c. At a paticula constant linea speed, v, the following equation (Newton s Second Law) descibes the dynamics of the object s cicula motion: F = m c a c whee the magnitude of the centipetal acceleation is given by: v a c = Fo cicula motion that is not constant, the total acceleation (magnitude and diection) of the object, at evey moment, is detemined by adding the centipetal (adial component) and linea (tangential component) acceleations togethe. Note: a c and a t ae vectos that fom a ight tiangle when added togethe because they ae pependicula to each othe. a total = a c + a t o Lab #9 Kinematics & Dynamics of Cicula & Rotational Motion
Remembe that tangential acceleation is a consequence of any change in the linea (and theefoe, angula) speed of the object. Centipetal acceleation is a consequence of the ate at which the diection of the object changes at evey moment. The centipetal foce, the component of the net foce diected towads the cente of the cicle, is caused by diffeent types of foces. Hee ae some examples fo the case of a single foce causing cicula motion: Tension in a sting (such as when a ball is whiled in a hoizontal cicle at the end of a ope) The nomal foce (such as on the Roto ide found at many amusement paks) Gavity (such as the obits of planets aound the sun) Static fiction (such as a ca taveling aound a level cuve) Hee ae some examples fo the case of two foces in combination causing cicula motion: Tension and gavity (such as when a ball is whiled in a vetical cicle at the end of a ope) The nomal foce and gavity (such as when a peson ides a vetical loop on a olle coaste) The nomal foce and static fiction (such as a ca taveling aound a banked cuve) In this lab, you will exploe the case of a penny moving in a cicle on a level suface due to the foce of static fiction. Recall that static fiction is a vaiable foce, able to povide esistance up to a paticula maximum value. At this point, the object is said to be on the vege of slipping. The following equations descibe the elationship between static fiction and the dynamics of the non-constant cicula motion at this moment. F net = f static ma total = µ F s nomal ma c + a t = µ smg whee, on a level suface, the nomal foce is equal in magnitude to the foce of gavity. Remembe that the magnitude of a total is detemined by taking a vecto sum. Lab #9 Kinematics & Dynamics of Cicula & Rotational Motion
Goals: Lab #9: The Kinematics & Dynamics of Cicula Motion Compae the gaphs of cicula motion fo a otating tuntable. Detemine the coefficient of static fiction between the tuntable suface and a penny. (Optional) Pedict the magnitude of the angula velocity that causes a penny at a given adius to slip. Compae the linea and cicula motion of diffeent points on a otating tie. Equipment List: Rotating Platfom with attached Rotay Motion Sensos Stickes located at two diffeent adii Pulley Sting Hanging mass and hange Penny Rule Compute & Equipment Set Up: 1. Stat by making cetain that the sting used to tun the tuntable is not attached to any hanging mass.. Set up Science Wokshop to ead the data collected fom the Rotay Motion Senso located at the base of the tuntable. Do not change any of the default settings of the senso. 3. Ceate a gaphing window to display Angula Position (q) vs. Time. 4. Check the calibation of the senso: Pess Recod; otate the tuntable exactly once; Pess Stop; Look at the Angula Displacement values ecoded on you gaph. (They ae measued in adians.) Decide whethe o not the gaph veifies that you tuntable is coectly calibated. (If it is not, see you TA immediately.) Impot you gaph to the Wod template and clealy explain you decision. Activity 1: Kinematics of Cicula Motion The pupose of Activity 1 is to compae the gaphs of otational motion fo an object, the tuntable, which stats fom est and otates with constant angula acceleation. 1. Using the Add Plot Menu on the Angula Position (q) vs. Time gaphing window, ceate additional gaphing windows fo Angula Velocity (w) vs. Time and Angula Acceleation (a) vs. Time.. Caefully wind the sting aound the base of the tuntable. Place the sting ove the pulley and attach the hanging mass (use 100 gams o 150 gams) to the othe end of it as shown in the pictue above. 3. Pess Recod and, eleasing the tuntable fom est, gathe data descibing the otational motion of the tuntable as the mass falls. The angula acceleation of the tuntable should be elatively constant. 4. Using the Statistics capabilities of Science Wokshop, calculate the Angula Acceleation, α, of the tuntable using two diffeent methods. Explain each of you methods and state you esults. 5. Copy the gaphing window (including the statistics infomation that you calculated) into the Wod template by using Paste Special. Paste each as if it wee a pictue. Lab #9 Kinematics & Dynamics of Cicula & Rotational Motion
6. Compae you thee gaphs and explain, mathematically, how the Angula Position vs. Time gaph & the Angula Velocity vs. Time gaph ae elated to each othe. the Angula Velocity vs. Time gaph & the Angula Acceleation vs. Time gaph ae elated to each othe. Activity : The Dynamics of Cicula Motion Thee ae two puposes fo Activity. The fist is to use the esults of a penny slipping off of the tuntable at a paticula adius to detemine the coefficient of static fiction between the suface of the tuntable and a penny. The second is to use this value of the coefficient to pedict the angula velocity at which the penny will slip when placed at a diffeent adius. Pat I: Detemine the Coefficient of Static Fiction on the Tuntable 1. Measue the adius of the cicle ceated by the oute blue sticke, R o. Recod this value in the table below.. Caefully ewind the sting aound the base of the tuntable and place the sting ove the pulley with the hanging mass attached. (Note: The size of the hanging mass must be lage enough to cause a penny to slip at both the blue and yellow positions, but small enough to keep it fom slipping too quickly. Recommended: 150 gams) 3. Place a penny at a distance, R o, fom the cente of the tuntable. [Note: Do not place the penny diectly on top of the blue sticke.] 4. Pess Recod and, eleasing the tuntable fom est, gathe data descibing the otational motion of the tuntable as the mass falls. Using you hand, stop the tuntable at the vey moment the penny slips fom the suface. Then, Pess Stop to end the collection of data. 5. Fom eading you gaph of Angula Velocity vs. Time, detemine the magnitude of the Angula Velocity of the tuntable at the moment just pio to when the penny slipped. Recod this value in the table below. 6. Recod the value of the Angula Acceleation, α, of the tuntable (detemined in Activity 1) in the table below. (Note: This value should agee with you cuent data.) 7. Fo the moment just pio to when the penny slipped, calculate the linea (tangential) velocity, the tangential acceleation, and the centipetal acceleation of the penny. (Note: The tangential acceleation will likely be much smalle in magnitude than the centipetal acceleation.) Recod you esults and explain you calculations in the table below. 8. Detemine the coefficient of static fiction between the tuntable suface and the penny. Recod you esults and explain you calculations in the table below. Clealy and completely explain you method of calculating µ s. Quantity Result Explanation of how Result was obtained R o = Radius (metes) This adius was measued using a ule. ω = Angula Velocity (adians/s) α = Angula Acceleation (adians/s ) V = Linea Velocity (m/s) a t = Tangential Acceleation (m/s ) a c = Centipetal Acceleation (m/s ) a total = Total Acceleation (m/s ) µ s = Coefficient of Static Fiction Lab #9 Kinematics & Dynamics of Cicula & Rotational Motion
Pat II: Pedict the Angula Velocity at which the Penny will Slip at a Diffeent Radius 9. Measue the adius of the cicle ceated by the inne yellow sticke, R i. Recod this value in the table below. 10. Recod the value of the coefficient of static fiction (calculated in Pat I) in the table below. 11. Pedict the Angula Velocity of the tuntable that will cause the penny to slip when placed at a distance, R i, fom the cente of the tuntable. Clealy and completely explain you method of calculating ω. Quantity Result Explanation of how Result was obtained R i = Radius (metes) This adius was measued using a ule. µ s = Coefficient of Static Fiction See Table in Pat I. ω = Pedicted Angula Velocity (ad/s) 1. Test you pediction: Place a penny at a distance, R i, fom the cente of the tuntable. [Note: Do not place the penny diectly on top of the yellow sticke.] Pess Recod and, eleasing the tuntable fom est, gathe data descibing the otational motion of the tuntable as the mass falls. Using you hand, stop the tuntable at the vey moment the penny slips fom its position. Then, Pess Stop to end the collection of data. 13. Fom eading you gaph of Angula Velocity vs. Time, detemine the magnitude of the Actual Angula Velocity of the tuntable at the moment just pio to when the penny slipped. Recod this value in the table below. ω = Actual Angula Velocity (ad/s) 14. By what % does you Pedicted value diffe fom the Actual value? Show you calculation in addition to you final answe. Does the % diffeence seem easonable? Can you account fo this diffeence in tems of the inaccuacy of you measuements? Explain. Lab #9 Kinematics & Dynamics of Cicula & Rotational Motion
Post Lab #9: The Kinematics & Dynamics of Name: Section #: Cicula Motion Compaing Rotational and Cicula Motion y 4.88 4.44 16 x diection of ca s motion A ca with 4.88 diamete ties dives 1 mile down a staight, level oad with a constant speed of 58 miles pe hou. Note the location of thee points on the tie: (1) the cente; () the im of the metal wheel; (3) the edge of the tie. Use the cente point as the oigin, (x, y) = (0,0). Assume that the oigin moves along with the ca as it tavels down the oad. The positions shown above epesent the location of each of these points at time t = 0. 1. Fom the pespective of the pictue above, in what diection does the tie otate? (Clockwise o Counte Clockwise?) 4.44 oad suface. Fill in the following chat by calculating the magnitudes of each quantity. Attach all calculations to you wite-up. Quantity (units) The Wheel s Rim The Tie s Edge : Radius (metes) Θ: Angula Displacement (adians) ω: Angula Velocity (adians/s) α: Angula Acceleation (ad/s ) s: Ac Distance Taveled (m) v: Tangential Velocity (m/s) a t : Tangential Acceleation (m/s ) a c : Centipetal Acceleation (m/s ) 3. Consideing only the point at the Tie s Edge, sketch each of the following gaphs fo the fist 3 evolutions of the wheel: Attach each of these gaphs to you wite-up. Clealy label all axes and the coodinates of all maximums, minimums, and intecepts. The Tie s Edge Radius vs. Angula Displacement X vs. Angula Displacement Y vs. Angula Displacement Y vs. X Lab #9 Kinematics & Dynamics of Cicula & Rotational Motion
4. A bicyclist tavels in a cicle of adius 5.0 m at a constant speed of 9.00 m/s. The combined mass of the bicycle and ide is 85.0 kg. Calculate the magnitudes of: a) the foce of fiction exeted by the oad on the bicycle and b) the total foce exeted by the oad on the bicycle (fiction and nomal foce combined) 5. A model aiplane of mass 0.75 kg is flying at constant speed in a hoizontal cicle at one end of a 30 m cod and at a height of 18 m above level gound. The othe end of the cod is tetheed to the gound. The aiplane cicles 4.4 times pe minute and has its wings hoizontal so that the ai is pushing it vetically upwad. a) What is the acceleation of the plane? b) What is the tension in the cod? c) What is the total upwad (lift) foce on the plane s wings? 6. A cicula space station otates about its cente so as to give an appaent weight to the cew when the cew stands on the inne suface of the oute wall. Suppose that the space station has a adius of 500 m. If a cew membe weighs 600 N on Eath, then a) What must the linea (tangential) speed of the oute wall of the station be if this peson is to have an appaent weight of 300 N when standing still on the oute wall? b) What does the appaent weight of this same peson become if they spint along the oute wall at 10 m/s (elative to the oute wall) in the same diection that the wheel is otating? Lab #9 Kinematics & Dynamics of Cicula & Rotational Motion