Centripetal Force

Purpose: In this lab we will study the relationship between acceleration of an object moving with uniform circular motion and the force required to produce that acceleration. Introduction: An object moving in a circle with constant tangential speed is said to be executing uniform circular motion. This results from having a net force directed toward the center of the circular path. This net force, the centripetal force, gives rise to a centripetal acceleration toward the center of the circular path, and this changes the direction of the motion of the object continuously to make the circular path possible.

The equation which governs rotational dynamics is Fc = mac, where Fc is the centripetal force, ac is the centripetal acceleration, and m is the mass of the object. This is nothing but Newton’s second law in the v2 radial direction! From rotational kinematics we can show that ac= r , where v is the tangential speed and mv2 r is the radius of the circular path. By combining the two equations, we get Fc = r . The period T of circular motion is the time it takes for the object to make one complete revolution. The tangential speed v of 2πr the object is v = T , which is the distance traveled divided by the time elapsed (i.e. circumference/period). Thus we finally get: 4π2mr F = c T 2

In this laboratory session we will keep the centripetal force Fc and the mass m constant and vary the radius r. We will investigate how the period T need be adjusted as the radius r is changed.

Laboratory Procedure: Part I - Taking Direct Measurements 1. Using the digital scale, measure the mass m of the brass cylinder with hooks, and the mass M of the brass hexagon. Record these masses in your lab notebook.

m 2. The mass M times the magnitude of the gravitational acceleration (g = 9.81 s2 ) will provide us with a force of tension with the same magnitude as the centripetal force Fc, which we will verify in Part II. Calculate this force for Part I (Fc = M ∗ g), and enter its value in your notebook. 3. Determine δm and δM from the precision of the balance.

4. Determine the fractional uncertainty (δm/m and δM/M) for these measurements and record this in your data table.

5. Determine δFc for part I based on the fractional uncertainty δM/M. See the example in Part III for help.

Part II - Taking Indirect Measurements 1. Hang the brass cylinder from the side post (see figure), and connect the string from the spring to the cylinder (you have a choice of four loops, choose any to start). The string must pass under the pulley on the center post as shown in the figure. Attach a string to the cylinder and let the string pass over the clamp-on pulley. 2. Hang the hexagon on the end of the string. Adjust the side post assembly so that the brass cylinder hangs vertically as in the figure.

3. Record the radial position, r of the side post. 4. Determine δr from the precision of the scale attached to the apparatus.

1 Complete Rotational System 012-05293F

side post center post assembly string assembly

clamp-on rotating pulley platform

hanging mass "A" base

Figure 3.1: Centripetal Force Apparatus 2. Attach the clamp-on pulley to the end of the track nearer to the hanging object. Attach a string to 5.the Determine hanging object the fractional and hang uncertaintya known mass (δr over/r) th fore clamp-on this measurement pulley. Record and this record mass this in Table in your data table. 3.1. This establishes the constant centripetal force. 3.6. Select Align a the radius indicator by aligning bracket the line (see on figure) the side on post the with center any post desired with po thesition pink on the indicator, measuring again making sure the brass cylinder is vertically aligned. tape. While pressing down on the side post to assure that it is vertical, tighten the thumb screw on 7.the Remove side post the to hanging secure its mass position. (M) Record and the this string radius which in Table goes 3.1. over the clamp-on pulley. 4. The object on the side bracket must hang vertically: On the center post, adjust the spring bracket 8.vertically Begin to until rotate the the string platform from whic andh the thereby object put hangs the on cylinder the side inpost circular is aligned motion. with the Increase vertical the speed until linethe on pink the indicator side post. is again centered in the indicator bracket. This indicates that the string supporting the hanging cylinder is again vertical and thus the cylinder is at the desired radius. 5. Align the indicator bracket on the center post with the orange indicator. 6.9. Remove Maintaining the mass this that speed, is hanging use a ov stopwatcher the pulley to measureand remove the the time, pulley.t, it takes for the cylinder to make ten 7. Rotaterevolutions the apparatus (N=10). by hand, increasing the speed until the orange indicator is centered in the indicator bracket on the center post. This indicates that the string supporting the hanging object is 10.once Record again the vertical time andand thus calculate the hanging the period, object Tis, at (T = t/N). 11.the Repeat desired steps radius. a total of three times. Table 3.1: Varying the Radius 8. Maintaining this speed, use a stopwatch to time 12.ten Calculate revolutions. and Divide record the the time average by ten period and (Tavg). Mass of the object = record the period in Table 3.1. Mass hanging over the pulley = 13. Determine δTavg by first determining the uncertainty in your reaction time. Find your reaction time 9. Moveby starting the side and post stopping to a new theradius stopwatch and repeat quickly, then divideSlope that from time graph by your = ten revolutions. the procedure. Do this for a total of five radii. reactionRadius time Period (T) T2 δT = Analysis avg N 1. The weight of the mass hanging over the pulley 14. Determine the fractional uncertainty (δT /T ) for this measurement and record this in your data is equal to the centripetal force applied by theavg avg table. spring. Calculate this force by multiplying the 15.mass Repeat hung steps over 1the - 13,pulley varying by “g” the and radius record ofthis the brass cylinder by choosing a new loop on the string and forcerecording at the thetop of period. Table 3.2. Do this for a total of four different radii. 2. Calculate the square of the period for each trial 16. Calculate and record the square of the average period, T 2 , for each one of the four radii. and record this in Table 3.1. avg 17.3. Plot Plot the the radius radius versus versus the square the square of the ofperiod. the averageThis will period.give a straight This line should since: be a linear plot, because the F equation obtained on the first page can be------written2 as r = ⎛⎞2 T ⎝⎠4π m Fc 4. Draw the best-fit line through the data points andr = measure ( the)T slope2 of the line. Record the slope (4π2m) avg in Table 3.1. r 18. Fit a trend line through the data points and determine the slope of the line. Note that the slope ( T 2 ) Fc equals ( 2 ). (4π m) 20 19. From your equation for the slope, calculate and record Fc for Part II.

2 Part III - Determining Uncertainties in Your Final Values In the results section of your notebook, state the results of both parts of your experiment in the form Fc±δFc. Note, δFc in part I should be equal to the fractional uncertainty from your value of mass (M) fractional uncertainty multiplied by your value of Fc from Part I. For Part II, δFc should be equal to the largest fractional uncertainty from your values of radius or average period or mass (m) fractional uncertainties multiplied by your value of Fc from Part II. Example for Part II;   δm δTavg δr δFc = Fc ∗ max , , m Tavg r You should also address the following question:

1. Do your results for Fc in the two parts agree within their uncertainties? Be sure to clearly state the quantitative values you are comparing. If there are any large discrepancies, quantitatively comment on their possible origin.

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