Experiment 0 Whenever a body moves in a circular path, a force directed toward the center of the circle must act on the body to keep it moving in this path. That force is called centripetal force. The reaction, which is equal and opposite, is the pull of the body on the restraining medium and is called centrifugal force. The purpose of this experiment is to study uniform circular motion and to compare the observed value of the centripetal force with the calculated value. THEORY If a body moves with constant speed in a circle, it is said to be moving with uniform circular motion. Even though the speed is constant, the velocity is continuously changing because the direction of the motion is continuously changing. Thus, such a body has an acceleration. It can be shown that the direction of the acceleration is always toward the center of the circle (because it is only the direction and not the magnitude of the velocity that is changing) and that its magnitude is given by v a = r where v is the speed of the body in meters per second and r is the radius of the path in meters. A force is necessary to produce this acceleration. Because it must be in the same direction as the acceleration-namely, toward the center of the circle as noted above-it is called centripetal force. Newton's second law now requires that the magnitude of this force be equal to the mass times the acceleration produced, so that in our present case v F = m r where F is the force in newtons, m is the mass of the rotating body in kilograms, and v and r are the same as before. But by Newton's third law of motion, an equal and opposite force is exerted by the body on the restraining medium. This reaction is called centrifugal force. The centripetal force can also be expressed in terms of the angular speed, since v = rω and ω = π f where v is the linear speed and r the radius of the path as before, ω is the angular speed in radians per second, and f is the number of revolutions per second. Thus, F = mrw or F = π f rm where F is the centripetal force in newtons as already described.
Figure () Centripetal force apparatus, manual form Apparatus for the experiment is shown in Fig. (). A stretched string provides the centrifugal force to keep the rotating mass moving in a circle, but the complete assembly is mounted on a horizontal bar (the rotating platform) mounted on a vertical shaft carried in very good bearings. A knurled portion of the shaft provides a convenient grip for the experimenter's fingers, and you'll find with a little practice that you can keep the assembly rotating at a constant speed quite accurately. Note that at the bottom of the spring there is a small pink disk, and because the spring and disk are at the center of rotation, the position of the latter is easily observed even with the apparatus in motion. The mass to be kept rotating at a particular radius is suspended by threads from a post mounted on the rotating platform at that radius. Weights hung on a thread passing over a pulley at the end of the platform are used to bring the rotating mass to the selected radius with the apparatus at rest, and the marker ring on the central post is moved to the position now occupied by the pink disk. The weights are then removed and the apparatus rotated to a speed that again brings the pink disk into the marker ring. The centripetal force exerted by the spring must then be the same as that exerted by the weight when the apparatus wasn't turning, and the rotating mass must be back at the selected radius. The apparatus parameters have been chosen so that no rotational speed will be high enough to make counting the turns difficult. Be careful, however, to count only one end of the platform as it comes around. APPARATUS. Centripetal force apparatus. Motor with coupler and means for counting turns (Procedure A only). Stopwatch or stop clock
. Supporting rods, hook, right-angle clamp, and bench clamp (Procedure A only) 5. Weight hanger and weights 6. Vernier caliper (Procedure A only) 7. Level (Procedure B only) 8. Triple-beam balance (Procedure B only) PROCEDURE. When you come into the laboratory, examine the rotating assembly carefully in order to understand how it works. Make sure that it is level by placing your level on the rotating platform and adjusting the leveling screws in the base until the platform's top surface remains exactly horizontal no matter which way it is turned. Observe that the rotating mass consists of three parts and is suspended by two threads from a supporting post that may be set at any point along the rotating platform. A centimeter scale ] on this platform and a vertical line on the post allow the distance of the post from the center of rotation to be measured quite accurately. Unhook the complete rotating mass from the threads, weigh it, and record your result. Then remove the two outer sections, weigh the central section alone, and record this result also. You will thus have two possible values for the rotating mass.. Replace the central piece of the complete rotating mass on the supporting threads and note that in addition to the hook by which you hang this mass, it has two other hooks, one facing toward the center of the apparatus and the other facing outward. A piece of thread about 0 cm long should run from the inner hook, under the pulley on the central post, and up to the bottom of the pink indicator disk. Check that this thread is in place (install it yourself if it is not) and that the run from the hook to the pulley is horizontal. If it is not, the height of the bracket holding the supporting threads may be adjusted until it is.. Run another thread from the outer hook on the rotating mass over the pulley at the end of the platform and down to a weight hanger. Place 50 g on the hanger and record the total mass (weights plus the hanger mass) so suspended. This will pull the rotating mass out and stretch the spring.. Carefully loosen the clamp fixing the supporting post to the platform and slide it along its track until the threads supporting the rotating mass are exactly vertical. You can do this by lining up the two threads with each other and with the vertical line scribed on the post. Be sure the post itself is vertical by sliding it so that its bottom edge bears on the track on the platform's lower edge. Clamp the post in position and read the radius at which you have set it from the scale on the platform. Record this value. Finally, move the indicator ring on the central post so that the pink indicator disk is centered in it and fix the ring in this position. 5. Now remove the weight hanger, weights, and thread from the outer hook on the rotating mass. The spring will pull this mass in toward the center of the apparatus, and your plan is to spin the assembly up to the speed at which the rotating mass returns to the position in which it was held by the weights with the apparatus at rest. Do this by carefully turning the knurled portion of the
supporting shaft with your fingers, gradually increasing the rotational velocity until the pink indicator disk is pulled back down into the indicator ring. Practice for a while until you can keep this velocity constant at the value for which the pink disk is centered in the ring. Then start the stop clock as one end of the platform comes around and stop it after ten turns. Measure the time for ten turns in this manner four times and record your results. Note that it is a good plan in making these measurements to have one observer pay strict attention to keeping the speed constant at the proper value while the other counts turns and operates the stop clock. 6. Repeat Procedures -5 with 00 g on the weight hanger and again using 50 g. 7. Reassemble the two outer portions of the rotating mass to the central piece and repeat Procedures -6 using this larger rotating mass. DATA Value of the rotating mass ---------- Measured value of the centripetal force ------ Difference between calculated and measured value --------- Centripetal force calculated from the theory ------ Percent discrepancy ---- Trial of turns interval speed Deviation
Value of the rotating mass ---------- value --------- ------ Percent discrepancy ----------- ------ Trial of turns interval speed Deviation Value of the rotating mass ---------- ------ Percent discrepancy ----------- ------ value --------- Trial ofturns Interval speed Deviation
Value of the rotating mass ---------- value --------- ------ Percent discrepancy ----------- ------ Trial of turns interval speed Deviation Value of the rotating mass ---------- value --------- ------ Percent discrepancy ----------- ------ Trial of turns Interval speed Deviation \
Value of the rotating mass ---------- Percent discrepancy ----------------- value --------- ------ Trial of turns interval speed Deviation CALCULATIONS. From the data of Procedures 5-7, calculate the speed of rotation for each reading you have taken. Note that dividing the number of turns by the elapsed time gives you revolutions per second, that if you used a reduction gear you must multiply by the gear ratio to get the rotational speed of the apparatus, and that you must then multiply by π to get your result in radians per second.. Compute the average of your four speed measurements, the deviation of each from the average.. and the average deviation (a.d.) for each set of readings. Note that you will have six sets if you used the manual apparatus of measurement. 7
. Calculate the centripetal force from the theory using the average speed of rotation obtained in Calculation. You will have to do this and the next three Calculations six times. The mass whose weight stretches the spring is just the mass hung on the weight hanger in Procedures, 6, and 7 plus the mass of the weight hanger. Multiply by the acceleration of gravity to get the force stretching the spring in newtons in each case and record your three results in the appropriate spaces. 5. Compare the caculated value of the centripetal force with the value measured directly by subtracting one from the other and computing the percent discrepancy. 6. Using the a.d. found in Calculation as the error in your rotational speed measurement, find the resulting error in your calculated value of the centripetal force and see whether the directly measured value falls within the limits of error so established. 8
QUESTIONS. State the main results of the experiment.. (a) How does the centripetal force vary with the speed of rotation for a constant radius of the path? (b) How does it vary with the radius of the path for a constant speed of rotation?. Distinguish between centripetal force and centrifugal force. Explain in what direction each force is acting and on what it is acting.. Calculate at what speed Earth would have to rotate in order that objects at the equator would have no weight. Assume the radius of Earth to be 600 km. What would be the linear speed of a point on the equator? What would be the length of a day (time from sunrise to sunset) under these conditions? 5. Engines for propeller-driven aircraft are limited in their maximum rotational speed by the fact that the tip speed of the propeller must not approach the speed of sound in air (Mach I). Taking 6 ft as a typical diameter for a propeller of a light airplane and 00 fils as the speed of sound, find the upper limit on the rpm (revolutions per minute) of the propeller shaft. 9