circular_motion-1 - Uniform Circular Motion Whenever a body...

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Unformatted text preview: Uniform Circular Motion 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. This force is called centripetal force. The reaction, which is equal and opposite, is the pull of the body on the restrain- THEORY ing medium and is called centrifugal force. The pur- pose of this experiment is to study uniform circular motion and to compare the observed value of the cen— tripetal force with the calculated value. 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 and that its magnitude is given by ”2 r a: where o 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, and this force is called centripetal force because it is always directed toward the center of rotation. By Newton’s second law of motion, the magnitude of this force is given by the relation F=m— r F=ma or where F is the force in newtons, m is the mass 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 because of the inertia of the body. This reac- tion is called centrifugal force. The centripetal force can also be expressed in terms of the angular speed, since w=27rf where v is the linear speed and r the radius of the path as before, to is the angular speed in radians per second, and f is the number of revolutions per sec- ond. Thus 0 = rm,» and F = mrw2 or F = 4772f2rm where F is the centripetal force in newtons. In the apparatus used for this experiment (see Fig. 6.1) a stretched spring provides the centripetal force necessary to keep a mass rotating ina circle of a particular measurable radius. The rotator is adjusted to a rotational speed such that the back of the mass just touches the sensing probe of a sensitive indica- tor. The indicator’s pointer then rises to stand oppo- Rotating mass figa 57'- Eye for 1 er hanging . L: ; apparatus on hook L—«av Indicator Index Figure 6.1 Centripetal force apparatus '41 42 UNIFORM CIRCULAR MOTION site an index ridge at the end of the shaft on which the apparatus tums. As this index and the pointer are near the center of rotation, they are easily visible re- gardless of the rotational speed. This speed is kept constant at the value just described and is measured APPARATUS 1. Centripetal force apparatus 2. Motor with coupler and means for counting turns 3. Stopwatch or stop clock PROCEDURE 1. When you come into the laboratory, examine the rotating assembly carefully in order to understand how it works. Do not change the spring tension ad- justment; it has already been set by your instructor. Note and record the value of the rotating mass. This value is either marked on the mass itself or will be given to you. 2. Set up the bench clamp, a vertical rod, right- angle clamp, horizontal rod, and hooked collar so as to suspend the apparatus vertically. Hook the weight hanger to the eye on the rotating mass and add weights carefully until this mass is pulled down just enough to touch the sensitive indicator and bring the I pointer to the index. Record the mass required to do this. Don’t forget to include the mass of the weight hanger. . 3. With the weights still on the hanger so that the rotating mass is in the position established in Pro- cedure 2, use the vemier caliper to measure the mass’s radius of rotation. The measurement is made from the point on the shaft above the index (the axis of rotation) to the line engraved around the mass, which marks the location of its center of gravity in that direction. 4. Unhook the apparatus from the weight hanger and supporting rods and mount it on the motor shaft by means of the coupling provided. Mount the motor unit securely on the bench using the bench clamp and check that all mounting clamps and screws are tight. 5. Make sure the speed control is at its lowest position and turn on the motor. As you gradually in- crease the speed, note that until the rotating mass has moved out to the radius at which it engages the indi- cator, the apparatus is highly unbalanced and will DATA Value of the rotating mass Mass hung on the spring Total mass necessary to stretch the spring by timing some convenient number of turns. The force required to stretch the spring by the same amount is subsequently measured by hanging weights from it. ~ '- 4. Supporting rods, hook, right—angle clamp, and bench clamp 5. Weight hanger and weights 6. Vernier caliper shake quite badly. Increase the speed until you see the pointer rise, but be careful not to increase it too much. CAUTION: It is dangerous to let the apparatus rotate at excessive speeds. Best results will be ob- tained if you get the pointer to rise just above the in- dex and then slow the rotation very carefully, making very small speed changes and watching the results. Your aim is to keep the motor speed at the value for which the pointer stands exactly opposite the index, but this adjustment is very critical, and you may want to practice a while before attempting to take data. Holding a sheet of white paper behind the apparatus as you watch the pointer and index will help you see them more clearly. 6. When you have the speed set at the correct value so that the pointer is opposite the index, mea— sure this speed by timing a counted number of turns. If you are using a revolution counter, note its reading and then engage it and start the stop clock simulta- neously. Disengage it and simultaneously stop the clock after about a minute and record the number of turns and the elapsed time. The setup shown in Fig. 6.1 uses a variable speed motor with an 18 : 1 reduc- tion gear on one end and a pointer mounted on the slowly turning shaft thus provided. Revolutions of this shaft are easily counted visually. In performing this experiment it is a good plan to have one observer pay strict attention to the proper adjustment of the speed while the other counts turns and operates the stop clock. 7. Repeat Procedures 5 and 6 three more times so as to have a total of four measurements of the rota- tion speed. Record the number of revolutions counted and the elapsed time in each case. Radius of rotation Centripetal force calculated from the theory Name __—__.__..__—____ Section __ Date __ EXPERIMENT 6 43 Measured value of the centripetal Percent discrepancy force Difference between calculated and measured value CALCULATIONS 1. From the data of Procedures 5—7, calculate the speed of rotation for each set of readings. 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 277 to get your result in radians per second. 2. Compute the average of your four speed measurements, the deviation of each from the average, and the average deviation (a.d.). 44 UNIFORM CIRCULAR MOTION _ 3. Calculate the centripetal force from the theory using the average speed of rotation obtained in Calculation 2. 4. Find the directly measured valueof the centripetal force by adding the value of the rotating mass given you in Procedure 1 to the mass found in Procedure 2 and multiplying by the acceleration of gravity to get the force stretching the spring in newtons. Record this result in the space provided in the data sheet. - ' 5. Compare the calculated value of the centripetal force with the value measured directly by sub- tracting one from the other and computing the percent discrepancy. Name —_______—__—___ Section ___ Date __ EXPERIMENT 6 45 6. Using the ad. found in Calculation 2 as the error in your rotational speed measurement, find the resulting error in your calculated value of the centripetal ferce and see whether the directly measured value falls within the limits of error so established. ' QUESTIONS ' 1. State what the experiment checks. 2. (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? 46 UNIFORM CIRCULAR MOTION ' 3. Distinguish between centripetal force and centrifugal force. Explain in what direction each force is acting and on what it is acting. 4. Calculate at what speed the earth would have to rotate in order that objects at the equator would have no weight. Assume the radius of the earth to be 6400 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 con- ditions? - 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 1). Taking 6 ft as a typical diameter for a propeller of a light airplane and 1100 ft/s as the speed of sound, find the upper limit on the rpm (revolutions per minute) of the propeller shaft. ' , ...
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