Centripetal Force I. Introduction. In this experiment you will study the centripetal force required for a mass in uniform circular motion. You will determine the centripetal forces required for different masses, speeds, and radii and will see if your results agree with the theoretical predictions. For a mass M in uniform circular motion with tangential speed v at radius R, the required centripetal force is F c = Mv 2 /R. If the period of revolution is T, the linear speed of the mass is v= 2πR/T and the centripetal force is F c = 4π 2 MR/T 2. The apparatus used in this experiment has a rotating vertical spindle with a horizontal cross-arm at the top. From this cross-arm is suspended a plumb bob (to which additional mass can be added) which will tend to move outward as the spindle rotates unless a centripetal force is supplied. A horizontal spring supplies this centripetal force. The vertical spindle is driven by a small DC motor connected to a power supply. Timing information is recorded by a photogate connected to a computer operating the Science Workshop program which you have already used in an earlier experiment on Inertial Mass. 4-1
The required centripetal force, as calculated from the measured mass, radius, and period, will be compared with the measured force supplied by the spring. The spring force is determined by measuring the equivalent static force F s required to stretch the spring to the same length. II. Required Equipment. A. A Beck rotating-spindle assembly (illustrated). Protective enclosure around the apparatus. B. Accessory equipment: weights, a weight holder, string, two springs, photogate. C. DC motor and power supply. D. Macintosh computer with Pasco Signal Interface. Science Workshop program and Graphical Analysis program. III. Procedure. Initially make sure that the power supply for the DC motor is turned off and that the voltage controls are set to their full counterclockwise (off) position. Before you make any measurements, the rotating-spindle assembly must be level and balanced. The table tops are sufficiently level that no additional leveling is normally necessary. It is necessary, however, to balance the rotating cross-arm. Marks are provided on the base of the unit at fixed radii from 14 to 20 cm. The marked radius is to the outside edge of each bump. Set up for the first radial measurement at 16 cm. For uniform circular motion at a given radius, the mass suspended from the cross-arm should first hang vertically downward over the corresponding mark when the cross-arm is at rest and the spring is unhooked. (Unless the bob hangs vertically, the suspension cords will exert an additional unmeasured horizontal force on the mass during a measurement.) Use the weaker, more pliable spring for your first measurement. Whenever the cross-arm is moved to change the radius, the assembly must be balanced as follows. Unhook the spring. Adjust the cross-arm until the bob is hanging directly above the mark at the chosen radius but does not touch it. (When the bob is in this position the mounted spring should be horizontal so that it exerts only a horizontal force on the bob.) When the arm is balanced, it will not tend to rotate even if the base is tipped. Balance the assembly by adjusting the position of the counterweight and then lock it in place. If you find you cannot move the counterweight out far enough for exact balance at larger radii, move it out as far as possible and lock it in that position. Hook up the spring again, checking to see that it is horizontal when the bob is pulled out to hang vertically above the chosen mark, and be sure all the screws and locknuts are tight. Make sure that counterweight tightening screw passes freely through the photogate. Each revolution, the screw will cut the light beam in the photogate, providing a measurement of the period of the circular motion. Before turning on the DC power supply, be sure that the protective enclosure is in place. A. One measurement consists of determining the period of rotation for ~ 50 revolutions when the spindle is turning at such a rate that the bob passes directly over the mark at the chosen 4-2
radius on each revolution. The DC power supply voltage must be adjusted carefully to achieve the proper radius. The power supply has both coarse and fine voltage adjustments. Turn on the supply. Use the coarse adjustment to get the spindle rotating and adjust the control slowly in order to get close to the proper radius. Then use the fine adjustment for fine tuning. Observe that the system is running stably. B. To measure the period, T, start up the Science Workshop program, by double-clicking on the Science Workshop icon on the Desktop. You should see the familiar Control Window and Menu Bar. Click and drag the digital plug image to digital channel 1 to logically connect the photogate to that channel. From the pop-up timing menu, scroll down and select photogate and picket fence. Create the data table and graph by dragging each little image to the same channel as you did the plug, and selecting Time between bands (tdelta) to display for each. In the table heading, click on the [0.00...] button and select 4 digits to be displayed. Start recording data and record ~ 50 revolutions. You can watch your data in real time on the graph. Click on the stop button when you have reached 50 entries in the graph. C. Next run down the voltage on your power supply so that rotation stops. Make sure both coarse and fine voltage controls are full counterclockwise (the off position). Turn the supply off. The radius R should be checked and recorded, and the error in R estimated (i.e. how well you can determine the position of the bob over a marker). The mass of the plumb bob (M B ) must be measured using a laboratory balance and its error (δm B ) estimated. Weigh the bob once without additional weights and use this value to determine the mass of the bob for all cases. If additional mass is added, that value must be recorded so that the total mass M is known. D. Given these data, the centripetal force F c can be calculated from the formula given in Section I above. The error in F c can be calculated from the errors in period, radius, and mass by using the rules for error propagation which can be found in section 4.2 of "Measurement and Error". The detailed formulae are listed in Section V below. E. The equivalent force F s for the static case (no rotational motion) is measured by passing a string from the plumb bob over the pulley provided and by hanging weights from it until the 4-3
bob is again over the pointer. Make sure the string and the spring are both horizontal. Be sure to record your results. Remember that the weights are marked with their mass, not their weight, and make sure you include the mass of the weight holder. This total mass, on and including the weight holder, should be referred to as m, and should not be confused with the mass of the bob system which is referred to as M. Estimate the errors in your measurements of F s. This may be accomplished by adding (or removing) small amounts of mass from the weight hanger which do not significantly alter the static equilibrium (i.e. the bob stays directly over the radial position pointer). F. Any one set of data allows you to check the equality of the measured static force F s and the calculated centripetal force F c. It is desirable, however, to test the dependence of the centripetal force on mass, radius and speed separately. While agreement of F c and F s in a single case might be accidental, agreement for several cases under different conditions should tend to establish the correctness of the assumptions. Using two different springs will provide data for two different forces and hence for two different speeds with the same mass and radius. You can add 150 grams to the plumb bob to compare two different masses with the same radius and spring. (Be sure the slots in the added weights face outward, clamp them securely, and make sure the spring is horizontal when the bob is in uniform circular motion.) Two or more different radii can be used with the same mass and spring. G. Data should be obtained for four separate sets of conditions: (a) Spring 1, mass 1, radius 1; (b) Spring 2, mass 1, radius 1; (c) Spring 2, mass 2, radius 1; (d) Spring 2, mass 2, radius 2. Spring 1 should be the weaker of the two springs you are given. Mass 1 should be the bob (M B ) with no added mass; mass 2 should be the bob (M B ) plus 150 grams. Radius 1 should be 16 cm; radius 2 should be 20 cm. Remember to determine the static force in each case before changing anything, and don t forget to include the mass of the weight hanger when determining the static force. Record your data in Table 2; at this stage, you should have all the values except T and δt for each run. H. The period measurements of four Runs in Science Workshop will be used to find T and δt. We will copy these to Graphical Analysis for plotting and averaging. IV. Analysis in the Laboratory. A. For each of the four measurements, it is necessary to determine the period T and its error (δt) from your measurements. To do so, double-click on the Graphical Analysis icon on 4-4
the desktop. Then go back to Science Workshop, by selecting it in the Finder menu (icon at right end of the Menu bar). Select Run 1 in the data box; then in the data table select all data by clicking and holding on the first data entry and dragging the pointer to the last entry. From the Edit menu select Copy. Then go back to Graphical Analysis using the Finder menu. Select the data table and select the first box in the first open column. Then select Paste from the Edit menu. Create additional columns as needed in the Data menu, and copy and paste the remaining 3 data runs (one by one) into the next columns of your data table. Label the columns according to what they contain, e.g., twb16 (for time for weak spring, bob only, 16 cm radius), tsb16, tsmb16, tsmb20. Set the units of each column to sec. From the Menu Bar, pull down the Data menu and select Data Options. Set the number of significant figures to 5 and click on ok. Additionally from the Data Menu, select rename data set. In the dialog box, enter an identifying name, such as your lab bench and room number. This will then identify your data table, and will identify any printing that you will do on the laser printers. From the Graph Menu, select New Graph. Using the mouse, click and hold on the vertical axis label and select the first data column (e.g., twb16). This quantity is the period of the motion for a given revolution. Click and hold on the horizontal axis and select index. Index in the Graphical Analysis program is the equivalent of row in the Science Workshop program. From the Graph Menu, make sure that the error bars are not selected (no check mark in front). The data should look similar to that obtained during data recording. A regression line will also be displayed on the data, and with error in slope and intercept written below the plot. From the Graph Menu, select New Histogram. If the horizontal axis is not displaying the same label as the vertical axis above, click and hold on the axis label to select this quantity. B. Do a print screen for each group member. C. The average (mean) value of the period (T) and also the error in the period (δt) may be obtained directly from the data table itself. Select the first column, by clicking on the header in the data table. Next pull down the data menu, and select statistics. The mean value number of measurements and standard deviation for your ~ 50 values of this column will be listed in a dialog box, along with other quantities. Take the mean value to be your best estimate of T for this measurement: T = mean value. Additionally, take the standard deviation of the mean to be your best estimate of the error δt for this measurement: δt = σ(t)/ N. Write down these quantities in Table 1 (as it is not possible to print these 4-5
quantities directly). Check to see if these numerical values for T and δt are reasonable. To do so, refer to your histogram. The mean value which you have set equal to T should be consistent with the average value of the time in the histogram, and δt should be of the same numerical order as the half width at half height (the procedure used in the Inertial Mass experiment). For reference, the meaning of mean and standard deviation are described in your handout on Measurement and Error. D. Repeat this procedure separately for the other three data columns. E. You should now be able to complete Table 2 F. You have now completed all the required work in the laboratory. Exit from the Graphical Analysis program. G. If time permits, start up the Science Workshop program and run the apparatus at the last set up position (strong spring, radius = 20 cm, and M = M B + 0.15kg). Set up your graph and watch the period as a function of row number for ~300 rows. Make a print of the graph for each group member. Describe qualitatively what you observe. Does the period of the motion behave in a simple way or a complicated way? Do you observe any regularities? H. Finally, make sure all power supplies are off and all voltage controls are in the full counterclockwise (off) position. Then exit Science Workshop. I. Before leaving the laboratory, you should have four printed sheets, each containing a graph and a histogram, based on your four data sets. You should have filled in the above table. You should have recorded the names of lab partners for reference. You should have your teaching assistant initial all sheets. V. Analysis outside the laboratory. A. Calculate the centripetal force F c and its error. In each case, compare the calculated force with the measured static force. Calculate the difference between the calculated F c and the measured F s ; also calculate the error in the difference using the rules of error propagation. Insert these values in Table 3. For each measurement, decide whether the two values are consistent. Remember to include appropriate units on all quantities in all tables. Using units of meters, kilograms, and seconds allows us to express forces directly in Newtons. The quantities in Table 3 are: : F c = 4 π 2 M R / T 2 δf c = F c * SQRT [ (δm / M) 2 + (δr / R) 2 + 4 (δt / T) 2 ] F s = mg 4-6
δf s = g * δm δ (F c - F s ) = SQRT [ (δf c ) 2 + (δf s ) 2 ]. You should refer to Section 4.2 of Measurements and Errors to see how the above expressions are determined. B. Consider the results of your analysis. Are the values of F c and F s consistent in all of the cases? Are your results consistent with the theoretical picture for uniform circular motion? Discuss possible sources of error in your measurements. C. Make 2 suggestions on how this experiment could be improved. TABLE 1. Data of Periods of Oscillations. RUN Spring R (m) M (kg) N average T std dev σ(t) error of avg σ(t) 1 weak 0.16 M B 2 strong 0.16 M B 3 strong 0.16 M B + 0.15 4 strong 0.20 M B + 0.15 4-7
TABLE 2. Data of Centripetal Force Experiment. Mass of the bob M B =, δm B = RUN Sprin g R (m) δr (m) M (kg) δm (kg) T (s) δt (s) m (kg) δm (kg) 1 weak 0.16 M B 2 strong 0.16 M B 3 strong 0.16 M B + 0.15 4 strong 0.20 M B + 0.15 TABLE 3. Data Analysis. Run F c δf c F s δf s F c - F s δ(f c - F s ) 1 2 3 4 4-8