Centipetal Foce OBJECTIVE To veify that a mass moving in cicula motion expeiences a foce diected towad the cente of its cicula path. To detemine how the mass, velocity, and adius affect a paticle's centipetal foce. To explain why the centipetal foce is necessay fo cicula motion. INTRODUCTION When any mass m moves in a cicle of adius with a constant velocity v, it expeiences a centipetal foce diected towad the cente of its cicula path. Fo this laboatoy, we will study the case of unifom cicula motion, whee the speed (a scala quantity) of the moving mass is held constant. Yet, as the mass moves in a cicula path, the velocity (a vecto quantity) is constantly changing. By definition a velocity that changes ove time is acceleation. Thus, this change in velocity comes fom a centipetal acceleation due to a centipetal foce. The centipetal foce is that foce that pulls an object out of a staight-line path and into a cicula path. By measuing the peiod of otation, the object's mass, and the path adius, the centipetal foce can be expeimentally detemined. APPARATUS Compute with Logge Po Venie LabPo Compute Inteface Foce Senso with Clamp Set Two Testing Masses (Foam Ball & -hole ubbe stoppe) 500g Calibation Mass Sting Hollow Tube Mete Stick Balance THEORY As peviously stated, when a paticle moves in a cicula path, its velocity is constantly changing. The velocity of the paticle is tangent to the path of its motion at all times. Thus, as it is eve changing its path, it is eve changing its velocity. The magnitude of the velocity, howeve, will emain constant. Note, the speed (velocity magnitude) does not have to be constant but would pesent futhe study to pusue what amounts to a changing centipetal foce. Centipetal Foce - Page 1
The figue below illustates this motion. v v i m vf Figue 1 v Note, if the mass wee to beak fom the confines of its cicula path, it would move off in a diection that is tangent to the cicle at the point of elease. Two of the tangential velocities have been intentionally labeled vi and vf to illustate the diection of the esultant velocity. This is done by applying the elation fo change in velocity: Δv vf - vi. Fom the two labeled vectos in Figue 1 and the peviously cited equation, we get: D v F v c a c m vf -v i v Equation 1 a c Equation m v Figue Note that Δv points towad the cente of the cicula path. This vecto diection, fo the velocity, will always be the same as long as we take vi and vf to have a vey shot time inteval between them. This change in velocity esults fom a centipetal acceleation. It can be futhe shown that, the magnitude of this centipetal acceleation is given by: Whee, ac [m/s ] is the centipetal acceleation, v [m/s] is the tangential velocity, and [m] is the adius of the cicula path. By Newton's second law, F ma, the magnitude of the coesponding centipetal foce is given by: Whee, Fc [N] is the centipetal foce that acts on mass m [kg]. Thus, the magnitude of the coesponding centipetal foce, whethe applied by the tension in a sting o the foce due to a sping, equals that of Equation. Centipetal Foce - Page
The velocity v [m/s] may be detemined by the time it takes the mass to move aound its cicula path. The cicumfeence of a cicle, and thus the distance taveled by the mass, is π. If the mass does tavese this distance in a time inteval Δt, then the coesponding velocity is given by: m v v x t m ( Equation 3 T T Whee, [m] is the adius of the cicula path and T [s] is the peiod of otation (the time inteval it takes to complete one otation aound the cicle). Futhe, making the substitution of the velocity fom Equation 3 into Equation yields: ) 4 F c Equation 4 T This gives an expession fo the centipetal foce in tems of expeimentally deteminable quantities. m EXPERIMENTAL PROCEDURE 1. Mount the Foce Senso so that the hook points vetically upwad! 10N Switch Setting!. Connect the Foce Senso to CH1 of the LabPo. 3. Stat Logge Po. A Foce vs. Time (F vs. t) gaph is displayed. This is the only gaph you will need. Centipetal Foce - Page 3
4. To assue that the foce senso eads accuately, poceed to calibate it as follows: The foce senso hook must be pointed upwad fo pat a & b!! a. Choose Calibate fom the Expeiment menu. Select LabPo:1 CH1: Dual Range Foce. Click on the button. b. Ente a 0 (zeo) in the Value 1 field, and afte the eading shown fo Reading 1 is stable, click. This defines the zeo foce condition. c. NOW, otate the senso so the hook hangs vetically downwad and hang the 500g mass fom the senso; this applies a foce of 4.9 N. Ente 4.9 in the Value field, and afte the eading shown fo Reading is stable, then click. Click to complete the calibation of the fist Foce Senso. This calibation pocedue may need to be edone between tials. Monito you senso values in the uppe left of the sceen. 5. Remove the 500g mass and otate the foce senso so that the hook again points vetically upwad. Attach the sting to m1 (the ball), pass it though the glass tube, and attach the othe end to the hook on the Foce Senso. Centipetal Foce - Page 4
6. The glass tube is used as a hand-hold while you otate the mass in a hoizontal cicula path with constant speed. You hand MAY NOT touch the sting! Befoe collecting data, pactice swinging the mass in a hoizontal cicle while maintaining a constant speed. o The "tape flag" is used as a visual indication of the position of the tube with espect to the mass. If the tape moves up, you'e otating the mass too fast. If the tape moves down, you'e otating the mass too slow. Obviously, the idea is that you keep the "tape flag" steady and as such as keeping a constant adius of otation. DO NOT allow the "tape flag" to touch the bottom of the tube; this will esult in false eadings on the senso. Be caeful not to hit youself o anyone else in the head while otating the mass. 7. The adius distance is measued fom the tube's mouth to the cente of the mass; this value, while not equied to be any paticula length, is equied to be the same length duing each pat of the expeiment; geneally an R1 of appoximately 0.8-1.0m is sufficient. Choose an R1 fo you fist set of tials and ecod this length in you data table along with the mass of m1 (the ball). 8. When you'e eady, begin otating the mass and click to begin collecting the foce data fo the expeiment. Typically, a somewhat epeating/vaying foce patten is obseved on the Foce vs. Time gaph. See the sample data shown below: o You MAY need to autoscale the gaph to see the details you need! You will need to extact two pieces of infomation fom this gaph: 1) Peak-to- Peak Time Readings & ) The Expeimental Centipetal Foce. Centipetal Foce - Page 5
9. Fo the Peak-to-Peak Time Readings, you need to locate the time at the peak point fo FOUR consecutive peaks. You can use the examine button to accomplish this: Recod these values in the data table. Fo the Expeimental Centipetal Foce, with the cuso, scan an intege numbe of cycles of evolution (fixed numbe of peaks) which display a athe oveall unifom foce (elatively the same height) and click on the statistics button. Recod these MEAN value in the data table. Check the % diffeence: If it is >0%, conside edoing the tial. If it is >50%, definitely edo the tial!! A Peak Time Location A Selection of an Intege Numbe of Unifom Foce Peaks Pint a copy of this gaph, with the statistics box shown, fo inclusion into you laboatoy epot. Only one copy is needed fo all nine tials. 10. Repeat you obsevations and data collection fo TWO additional tials fo m1 (the ball) and R1. 11. Repeat the pocedue fo m1 with a shote R ( 1/ R1; 0.4-0.5m) so as to complete the middle pat of the data table. 1. Finally, epeat the pocedue with a diffeent mass m (the ubbe stoppe) and the length R in ode to complete bottom pat of the data table. Centipetal Foce - Page 6
COVER PAGE REPORT ITEMS (To be submitted and stapled in the ode indicated below) (-5 points if this is not done popely) Completed Laboatoy Responsibility and Cove Sheet DATA (woth up to 10 points) One Foce vs. Time gaph with the statistics box shown Data table available as a downloadable Excel file DATA ANALYSIS (woth up to 30 points) Requied sample calculations, to be shown on a sepaate sheet of pape in you laboatoy epot, ae highlighted in yellow on the downloadable Excel data table speadsheet. Useful Calculation Infomation To Know: Pecentage Diffeence: In othe expeimental settings whee a known value is not known, a compaison may still be useful to evaluate the effectiveness of the expeiment based on the compaison of two expeimental esults. This will not tell you anything as to the accuacy of the expeiment but will give you an indication of how epeatable (pecise) the expeiment is. Below is the fomula fo achieving a pecentage diffeence compaison: E 1 - E E 1 + E * 100% Whee, E1 is expeimental esult #1 (Expeimental Centipetal Foce), E is expeimental esult # (Theoetical Centipetal Foce), and the paallel lines in the numeato indicate an absolute value. Centipetal Foce - Page 7
GRAPHS (woth up to 0 points) None Requied GRAPH ANALYSIS (woth up to 5 points) On the Foce vs. Time gaph, discuss the fom of the data, what it epesents as a function of the lab, and what/how infomation was gatheed fom this data. CONCLUSION (woth up to 30 points) See the Physics Laboatoy Repot Expectations document fo detailed infomation elated to each of the fou questions indicated below. 1. What was the lab designed to show?. What wee you esults? 3. How do the esults suppot (o not suppot) what the lab was supposed to show? 4. What ae some easons that the esults wee not pefect? QUESTIONS (woth up to 15 points) DO NOT foget to include the answes to any questions that wee asked within the expeimental pocedue 1) If the sting beaks when an object is undegoing the cicula motion, descibe its esulting motion. ) Why, when appoaching a cuve on a highway, ae signs posted to educe speed? 3) In step #6 of the pocedue, you wee instucted that the hand holding the tube, otating the mass, may not touch the sting duing the pocedue. Why was this statement made? 4) Geneally it is difficult to make the mass tavel in a pefectly flat hoizontal cicle. Usually the motion is moe like the figue illustated below. What affect would a non-hoizontal cicula path have on the esults? Centipetal Foce - Page 8