Objective: CIRCULAR MOTION The purpose of this experiment is to test the validity of Newton's Z* Law as it applies to th

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Objective: CIRCULAR MOTION The purpose of this experiment is to test the validity of Newton's Z* Law as it applies to the case of an object moving with constant speed in a circular path and therefore, being acted on by a centripetal force. Apparatus: Self-contained Uniform Circular Motion (UCM) unit, mass suspended from strings, spring attached to multihole ladder, weights, timer clock. Theory: Uniform Círcutar Motion describes the motion of an object that moves along a circular path with velocity v whose magnitude is constant. The direction of the velocity changes of course, as the object moves along the circular path and being always tangent to the path, it is always perpendicular to the radius of the circle. When this occurs, we can show that the object has an acceleration that is constant in magnitude and always pointing along the radius toward the center of the circular path. For this reason, the acceleration in UCM is called "centripetal acceleration"aco Its (constant) magnitude is given by acp = v?/R (1) where R is the radius of the circle. In this case, recalling Newton's 2nd Law of Motion F=ma, the net external force Facting on the object must also point toward the center of the circle, as its direction is always parallel to the acceleration. For this reason the net external force in UCM is called "centripetal force" F. Therefore, we rewrite Newton's 2nd Law for UCM as: Fo= macp=m v/R (2) Em m R For the purposes of this experiment it is useful to further rewrite (2), by noting that the speed v can be written in terms of the radius R and frequency f of the circular motion The frequency fis the number of revolutions per second, which is the inverse of the period Tof the motion, the time required for the object to go around once), Recall v = distance/time = circumference/period = 2R/T = 2nRf. Therefore, Egn. (2) can be rewritten as: Fcp = 4n’mRF (3) Note: If the values of m and R are fixed, Egn. (3) is the equation of a straight line where Fcp is the dependent variable, fis the independent variable and 4n MR is the slope (recall y = constant x). This equation gives the theoretical prediction for the centripetal force and is the equation that actually will be tested in this experiment.
When the shaft is spinning and the bob is moving on its circular path, the bob is subjected to three forces its weight mg directed downward, the tension in the two sections of the string. "generally upward, and the spring force F, along the spring and pointing toward the center of the cirde (vertical shaft). There are three possibilities for the direction of T and For as shown below, depending on the magnitude of the rotational speed : Frie MR MR ma In order to easily isolate the force that supplies the centripetal force to be measured in this experiment, it is important for the plane of the strings to be vertical while the experiment is performed, so that is vertical and for is horizontal (and in the plane of rotation). Under these conditions T and mg will cancel each other out and For being now the only force in the plane of rotation and pointing toward the center of the circular path, is identified as the centripetal force. For mg In order for this to happen, the radius of the circular path of the bob at which the strings will be vertical must be established in advance of spinning. This is done by disconnecting the bob from the spring, letting it hang freely and adjusting the position of the horizontal shaft so that the bob hangs over the "cog" of the desired radius. When the bob is reconnected to the spring and the shaft is placed in rotation, the rotational speed (and, therefore, the frequency of rotation) will be adjusted so that the spring extends enough for the bob to be exactly over the chosen cog. The specific steps for preparing the setup are: Disconnect the ladder from the bob. → Remove the bob including the nut and screw on top of it, and measure and record the combined mass. This will be the mass m in Egn. (3). Suspend the bob from the string again, orienting it so that the horizontal hole on its side is toward the vertical shaft. Let it hang freely. Adjust the position of the horizontal arm (by loosening the screw fixing it to the vertical shaft and sliding it) so that the bob hangs down freely, directly over the cog at the desired radius, e.g. R = 18 cm. This establishes the radius of the circle at which the bob must be maintained during rotation. (There is nothing special about R = 18 cm, you may choose another cog for your radius if you prefer). Connect one end of the spring to the horizontal threaded bolt located near the bottom of the vertical shaft, and the other end to the outermost position on the 5-hole ladder.
The First Data-taking Run: 1 Attach the hook fixed to the ladder to the horizontal hole on the side of the bob. Confirm that in this configuration the spring exerts some pull on the bob, bringing closer to the vertical shaft from the R = 18 cm cor it doesn't, unhook the spring from the threaded bolt and unscrew the bolt (moving it away from the bob) enough so that the spring when reconnected to it, will be pulling the bob toward the shaft 2. Spin the shaft by hand faster and faster until the bob is swinging directly over the R=18 cm cog. Be sure to sight the motion at right angles to the cog bar, le tangentially to the circular motion at the position of the cog. To make the motion as circular as possible it is better to twist the shaft gently several times per revolution rather than harder and less frequently. When this is done correctly the bob will during each revolution, pass precisely over the correct cog This step is very important if you are to obtain good results! When you can maintain the right radius reasonably steadily measure the time t for N revolutions, eg, 30-40 revolutions. Remember to start your timer when you count N as "o", not "1". (This is a two student task: One student should get the bob spinning at the right speed at the right radius, starta countdown of the revolutions Le -4,-3,-2-1,0,1,2,and a second student should start a clock on the count of O. The student spinning the shaft should maintain the bob at the correct radius all the while counting the turns aloud. The second student should stop the clock at the chosen count of N revolutions) 3. Stop the run. Record N and the total time t. The period is given by the ratio t/N. The First Independent Measurement of F 4. Keep the spring and ladder connected to the bob as in the previous section Hook the provided string-with-paper-clip to the vertical (outer) hole of the bob, run the string over the pulley of the UCM unit, and attach a weight holder at the other end. Add to the holder enough additional weight to stretch the spring until the tip of the bob is again directly over the R = 18 cm cog. This duplicates the conditions when m was rotating at R = 18 cm, therefore the total hanging weight Mag becomes equal to the spring force during rotation. That is, Magis equal to the centripetal force acting during rotation and this determination of Mag is the independent measurement of Fonoted in the M "Theory" section. Record the total hanging mass Me remembering to include the mass of the weight holder. 5. Prepare a table with columns for Hole #, the number of counted revolutions, the total time t for these revolutions, the calculated values for the period T, frequency f, f, total hanging mass M, and the corresponding weight Mag Record all the values for this first data point in this table. 6. Remove all the weights and horizontal string from the bob. m
The Remaining Data Runs 7. Connect the spring to the remaining a positions (holes) of the ladder, one at a time, and repeat steps 2-6 above for each position. At the end of this step you should have 5 rows of data in the table you created in step 5 . Analysis: 1 Graph Mg vertically vs. f horizontally and draw the best fit straight line through your 5 data points. If a data point is way off the line that the other points would fall on, try to figure out what went wrong and/or retake the data for that spring position 2. The straight line now represents the results of your experiment, ie what you have discovered about the relationship between F. (M.2) and f for uniform circular motion. Determine the slope of this line. This is your experimental value of the slope 3. According to theory (Egn. 3) the slope should be 4nm. Compare your experimental slope to this theoretical prediction by calculating the difference between the two values. 4. Again, according to theory, the "best fit" line should cross Mag=0 when f = 0. Extrapolate (extend) your line to f = 0. What is your experimental value for Mag at F.0? Is the difference from within what you would consider a reasonable uncertainty in the measurement? Explain. Suggested Layout of Data in Logbook Mass m of bob: (kg) Radius Rof circular motion: (m) % difference: Measured slope: Theoretical slope:
Data: M 9 (N) Mass of bob: 365.3 g, Radius of circular motion: 19.0+0.2cm Hole # N t(s) T($) f (Hz) f?(Hz) 1 20 21.26 1.063 .9407 2 20 17.27 .8635 1.158 3 20 14.85 .7425 1.347 4 20 13.39 .6695 1.494 5 20 12.35 .6175 1.619 M.(9) 305.8 431.0 540.7 650.5 770.2