Exp. #1-1 : Measurement of the Characteristics of the Centripetal Force by Using Springs and a Computer Interface

Transcription

1 PAGE 1/13 Exp. #1-1 : Measurement of the Characteristics of the Centripetal Force by Using Springs and a Computer Interface Student ID Major Name Team No. Experiment Lecturer Student's Mentioned Items Experiment Class Date Submission Time Submission Place Introductory Physics Office Report Box # Students should write down Student s Mentioned Items at the cover page of Experiment Reports, and then complete Experiment Reports by adding contents to the attached papers (if needed) in terms of the following sections. Contents of the reports should be written by hand, not by a word processor. Instead, it is allowed that figures and tables are copied and attached to papers. Completed Experiment Reports should be submitted to the place due to the time specified by Experiment Lecturers. The Experiment Report score per each Experiment Class is evaluated by max. 50 points (basically 15 points). Solutions of Problems in Experiment Reports are not announced to the public according to the General Physics Laboratory - Administration Rule. If a student permits other students to pirate one s Experiment Reports or a student pirates Experiment Reports of other students regardless of permission of original creators, the corresponding Experiment Report score and Active Participation score will be zero in case of exposure of such situation. Unless Experiment Reports are submitted to the place due to the time specified by Experiment Lecturers, the corresponding Experiment Report score will be zero. If the submission rate of Experiment Reports is less than or equal to two thirds, the grade of General Physics Laboratory will be F level. In order to decide grades of General Physics Laboratory at the end of current semester, the detailed scores of General Physics Laboratory will be announced at Introductory Physics Office homepage. Based on the announcement, students can raise opposition of score error. Since the public evidence is needed for the confirmation of opposition, students should keep one s Experiment Reports completed evaluation by Experiment Lecturers until the Experiment Report score decision If a student is absent from the Experiment Class because of proper causes, the corresponding student should submit documents related to absence causes to Introductory Physics Office regardless of cause occurrence time until the grade decision of General Physics Laboratory. If a student moves the Experiment Class arbitrarily without permission of Introductory Physics Office, it is noted that the total Experiment Scores will be zero. Lecturer's Mentioned Items Submission Time/Place Check Experiment Report Score Evaluation Completion Sign 50

2 PAGE 2/13 1. Objective Student ID Name The centripetal force can be measured from the mass, the rotation radius, and the rotation period of an object under the uniform circular motion. By using the properties of the elastic force acting as the centripetal force, the validity of the formula describing the centripetal force will be investigated. 2. Theory (1) Centripetal force In addition, there are relations between the magnitude of the velocity, the angular speed, and the rotation period as follows: The relations between the unit vectors and in the rectangular coordinates and the unit vectors and in the cylindrical coordinates (or the polar coordinates) as shown in Fig. 1 are as follows:, (Eq. 6) (Eq. 7) Therefore, the magnitude of the acceleration of the object under the uniform circular motion is given as follows: (Eq. 8) From (Eq. 5), the direction of the acceleration of the object under the uniform circular motion is, that is, toward the center of the circular orbit. For this reason, the acceleration of the object under the uniform circular motion is called the centripetal acceleration, and the force applied to the object under the uniform circular motion is called the centripetal force. If the mass of the object is, the magnitude of the centripetal force is given as follows: (Eq. 9) Answer the following questions. cossin, (Eq. 1) sincos (Eq. 2) 1. Investigate other procedures deriving the formula about the centripetal force. Fig. 1. The relations between the unit vectors in the rectangular coordinates and the unit vectors in the cylindrical coordinates. (2) Elastic force By differentiating the unit vectors and in the cylindrical coordinates with respect to time, the following relations can be obtained. sincos, (Eq. 3) cossin (Eq. 4) Here, is called the angular speed of the object. Assume that an object is under the uniform circular motion on the plane with the rotation radius and the -axis as the rotation axis. Since the velocity of the object is ( is constant), the acceleration of the object is given as follows: (Eq. 5) When the spring is no longer under a load, the end position of the spring can be treated as the equilibrium point. When the external force applied to the spring causes the spring to change from its original length to new length, the magnitude of the force restoring the spring to its original length is equal to that of the external force applied to the spring. The force restoring the spring to its original length is called the elastic force or the restoring force. If the magnitude of the external force or the displacement of the spring is sufficiently small, the magnitude of the elastic force is proportional to the displacement of the spring ( ). This is referred to as Hooke's law, which can be expressed by introducing the proportionality constant as follows:

3 PAGE 3/13 Fig. 2. Elastic force in the spring. Fig. 4. The limit of Hooke's law and the elastic limit in the actual spring. (Eq. 10) Fig. 3. The graph showing the relation between the magnitude of the external force and the displacement of the spring. Here, is called the elastic constant of the spring or the spring constant briefly with the SI unit Nm. The sign in (Eq. 10) expresses the fact that the stretched spring ( ) tends to decrease its length ( ) while the compressed spring ( ) tends to increase its length ( ) due to the elastic force of the spring. That is, the direction of the elastic force is toward the equilibrium point of the spring. Note that the slope in Fig. 3 corresponds to the spring constant. A large spring constant means that much force is needed to stretch or compress the spring. The reciprocal of the spring constant is called the compliance. A large compliance means that the length of the spring changes easily. If the original length of the spring is long, the length of the spring is easily changed so that the spring constant is inversely proportional to the original length as follows: (Eq. 11) If the magnitude of the external force continues to increase, the magnitude of the elastic force is no longer proportional to the displacement of the actual spring as shown in Fig. 4. In the case of a large external force, the actual spring does not restore to its original length when the external force is no longer applied. That is, a large external force will exceed the elastic limit of the actual spring. In this case, the actual spring will experience permanent deformation and break eventually.

4 PAGE 4/13 3. Experimental Instruments Items Quantity Usage Clean up method Computer 1 set It is used to acquire and analyze data. Computer interface 1 set It is used to convert analog signals to digital signals. It should be placed at the center of the experiment table. It should be placed inside the basket of the experiment table. Computer interface -to-power adaptor & connection cable Photogate -to-computer interface connection cable 1 ea. It is used to connect the computer interface to the wall power. 1 ea. It is used to connect the photogate to the computer interface. It should be placed inside the basket of the experiment table. It should be placed inside the basket of the experiment table. USB cable 1 ea. It is used to connect the computer interface to the computer. It should be placed inside the basket of the experiment table. Spring balance 1 set It is used to measure the spring constant from the gravitational force. It should be placed inside the basket of the experiment table. Additional masses for spring balance 2 ea. They are used to measure the spring constant from the gravitational force. They should be mounted on the spring balance. Power supply -to-power connection cable Centripetal force measurement instrument-to-power supply connection cable 1 ea. It is used to connect the power supply to the wall power. 2 ea. They are used to connect the centripetal force measurement instrument to the power supply. It should be placed inside the basket of the experiment table. They should be placed inside the basket of the experiment table. Power supply 1 set It supplies the power to the centripetal force measurement instrument. It should be placed at the center of the experiment table.

5 PAGE 5/13 Items Quantity Usage Clean up method Centripetal force measurement instrument 1 set It is used to rotate two objects mounted on the bar which is attached in the centripetal force measurement instrument. It should be placed at the center of the experiment table. Measuring scale 1 ea. It is used to measure the rotation radius of two rotating objects and the displacement of the spring. It should be attached to the centripetal force measurement instrument. Rotating object 2 ea. As two objects rotate, two springs are compressed until the elastic force acts as the centripetal force. Spring 2 ea. They provide the elastic force to two rotating objects. They are used to fix two rotating object and two springs mounted Spring fix screw 2 ea. on the bar which is attached to the centripetal force measurement instrument. Photogate 1 ea. It is used to measure the rotation period of two rotating objects. They should be mounted on the bar which is attached to the centripetal force measurement instrument. They should be mounted on the bar which is attached to the centripetal force measurement instrument. They should be mounted on the bar which is attached to the centripetal force measurement instrument. It should be attached to the centripetal force measurement instrument. Electric balance 1 set It is used to measure the mass of two rotating objects and additional masses for the spring balance. It should be placed at the center of the common experiment table.

6 PAGE 6/13 < How to Use the Power Supply > [7] In order to prevent an electricity accident, note that one's body must not be in contact with connection parts of the experimental instrument-to-power supply connection cables during the measurement. If the electric circuit becomes open during the measurement, the current will suddenly become zero and the current control indication lamp will turn off. In this case, after rotating all voltage and current adjust knobs to the minimum, turn off the power supply and correct the electric circuit in the same manner of [3]. [1] After confirming that the power supply is off, use the power supply-to-power connection cable to connect the power supply to the wall power and keep the power supply off. Use the experimental instrument-to-power supply connection cables to connect the experimental instrument to and terminals of the power supply. Without special condition, do not connect the ground terminal (GND or COM) of the power supply. [8] After the experiment is finished, turn off the power supply in the way opposite to turning on the power supply. That is, after rotating the current adjust knob to the minimum, rotate the voltage adjust knob to the minimum and turn off the power supply. Note that you must turn off the power supply when all voltage and current adjust knobs are at the minimum, or else a severe electricity accident may be happened. After turning off the power supply, clean up the experimental instruments according to the suggested method. [2] After confirming that all voltage and current adjust knobs of the power supply are at the minimum, turn on the power supply. In some models of the power supply, the power lamp or the voltage control indication lamp (CV) will turn on when turning the power supply on. [3] Rotate the voltage adjust knob (VOLTAGE - COARSE) slowly to increase the voltage. When approaching to a sufficiently high voltage for a normal electric circuit, the current control indication lamp (CC) will turn on. But for an open circuit, no matter how high voltage is, the current will remain zero and the current control indication lamp will be off. In this case, after rotating all voltage and current adjust knobs to the minimum, turn off the power supply and correct the electric circuit. The inspection of the electric circuit must be done only after the power supply is off. [4] In specific experiments that need a current supply, rotate the current adjust knob (CURRENT) slowly to set the desired current while checking if the current control indication lamp is on. [5] Even in properly functional electric circuits, when rotating the current adjust knob to increase the current, it can be observed that the current control indication lamp will suddenly turn off and the nonzero current does not increase any longer. This is caused by an insufficient voltage, which can be solved by rotating the voltage adjust knob more to increase the voltage. After the current control indication lamp turns back on, the current can be increased by rotating the current adjust knob. [6] Note that all voltage and current adjust knobs should be rotated slowly, or else the abrupt change of the voltage or the current may cause an electricity accident. Do not rotate the voltage fine adjust knob (VOLTAGE FINE) unless it is absolutely needed, and keep it at the minimum.

7 PAGE 7/13 < How to Use the Computer Interface) > [8] Save the data in the computer by selecting File Export Data ******.txt in the menu of the Capstone program and copy the text files to a USB memory prepared beforehand. [9] After the experiment is finished, close the Capstone program and turn off the computer interface and the computer. Clean up the experimental instruments according to the suggested method. [1] After confirming that the computer interface is off, use the computer interface-to-power adaptor & connection cable to connect the computer interface to the wall power and keep the computer interface off. [2] Connect the sensors to the proper analog channels or digital channels and use the USB cable to connect the computer interface to the computer. Note that the connection among all experimental instruments should be completed before the computer interface is turned on. Note that the computer interface must not be damaged by connecting the computer interface to the computer while the computer interface is on. [3] After turning on the computer and the computer interface, open the Capstone program. If the Capstone window appears, select Tools Hardware Setup Choose Interface PASPORT or ScienceWorkshop 750. [4] Select the sensors at the positions in the screen that are equivalent to the positions of the computer interface in the actual experiment by clicking. [5] After selecting Displays Graph or Table by double-clicking, show graphs or tables on the screen. [6] Click the Record button of the Capstone program to start the measurement and check if the acquired data is displayed in the screen. After checking if the zero status is displayed in the data, click the Stop button of the Capstone program to stop the measurement. [7] After completing the check of the zero status, click the Record button of the Capstone program again to start the measurement. If the data is acquired, click the Stop button of the Capstone program to stop the measurement. Repeat this procedure to acquire the correct data.

8 PAGE 8/13 4. Experimental Procedures (1) Measurement of the spring constant from the centripetal force 1) After confirming that the computer interface is off, use the computer interface-to-power adaptor & connection cable to connect the computer interface to the wall power and keep the computer interface off. 2) Use the photogate-to-computer interface connection cable to connect the photogate to the digital channel 1 of the computer interface. Use the USB cable to connect the computer interface to the computer. 3) After turning on the computer and the computer interface, open the Capstone program. If the Capstone window appears, select Tools Hardware Setup Choose Interface PASPORT or ScienceWorkshop ) Select Pendulum Timer at the position in the screen that is equivalent to the position connected to the computer interface in the actual experiment by clicking. 5) After selecting Displays Digits by double-clicking, show the Period digits on the screen. 6) After measuring masses and of two rotating objects by using an electric balance, mount two rotating objects on the bar which is attached to the centripetal force measurement instrument. Since undesirable vibration of the centripetal force measurement instrument can be caused by two rotating objects with a large mass difference, two rotating objects with similar masses should be used. 7) After mounting two springs and two spring fix screws in the bar which is attached to the centripetal force measurement instrument, tighten two spring fix screws firmly to fix two springs. Since undesirable vibration of the centripetal force measurement instrument can be caused by two springs with a large spring constant difference, two springs with similar spring constants should be used. 8) After confirming that the power supply is off, use the power supply-to-power connection cable to connect the power supply to the wall power and keep the power supply off. Use two centripetal force measurement instrument-to-power supply connection cables to connect the centripetal force measurement instrument to the power supply. 9) After confirming that all voltage and current adjust knobs of the power supply are at the minimum, turn on the power supply. After rotating the current adjust knob slightly, rotate the voltage adjust knob slowly to increase the voltage and make the objects rotate under the uniform circular motion. (Note that the usage of the power supply in this experiment is somewhat different from the general usage of the power supply.) Do not use extremely large voltage making the objects rotate too fast. Otherwise, it may cause an accident where the spring fix screws become loose and the rotating objects fly out. 10) As the voltage applied by the power supply increases, the rotation radii of the rotating objects increase until the elastic force acts as the centripetal force. By observing the centripetal force measurement instrument from above, the continuous rings formed by the motion of the rotating objects can be observed and the rotation radii of the rotating objects can be measured from the radii of the rings by using a measuring scale which is attached to the centripetal force measurement instrument. In the case of extremely fast rotation, the springs will be in their maximum compression state and the positions of the rotating objects will not be changed. Therefore, control the voltage applied by the power supply to set the rotation radius as cm. Note that the sufficiently long stabilization time is needed after the change of the rotation radius and the lack of the stabilization time may cause a measurement error so that the rotation period increases despite the increase of rotation radius. 11) Click the Record button of the Capstone program to start the measurement. Measure the rotation period and calculate the magnitude of the centripetal force. Use the average mass of the two rotating objects as the value of. In the case of similar masses and similar spring constants, measure the radius of one ring formed by one of the rotating objects and regard it as the rotation radius. In the other case, measure the radii and of the two rings formed by the rotating objects and regard the average of two values as the rotation radius. Click the Stop button of the Capstone program to stop the measurement.

9 PAGE 9/13 12) Calculate the displacement of the spring from the rotation radius and the original position of the rotating objects. Draw the magnitude of the centripetal force vs. the displacement of the spring graph using a proper program and evaluate the spring constant from the slope of the graph. 4) Measure the value of additional masses by using an electric balance and calculate the gravitational force. Here, is given by the sum of additional masses excluding the reference mass. It is possible to set the original length of the spring without the reference mass. In this case, is given by the sum of all masses including the reference mass. 13) After the measurement is finished, rotate all voltage and current adjust knobs to the minimum and turn off the power supply. Answer the following questions. (2) Measurement of the spring constant from the gravitational force 1) Release two spring fix screws mounted on the bar which is attached to the centripetal force measurement instrument and take out two springs and two rotating objects from the bar. 2) Place the spring #1 removed from the centripetal force measurement instrument on the pole of the spring balance. Place the reference mass on the pole of the spring balance and measure the original length of the spring from the scale which is pointed by the pin attached to the reference mass. Note that the pin attached to the reference mass should be handled carefully in order to prevent poking one's body. 3) When additional masses are added on the reference mass, the length of the spring decreases until the gravitational force and the elastic force of the spring are at the equilibrium. Change the value of additional masses and measure the changed length of the spring. 2. Explain why the value of additional masses is given by the sum of all masses including the reference mass in the case of the original length of the spring set without the reference mass 5) Calculate the displacement of the spring from the original length and the changed length of the spring. Draw the magnitude of the gravitational force vs. the displacement of the spring graph by using a proper program and evaluate the spring constant from the slope of the graph. 6) Repeat the experimental procedures for the spring #2. If the spring constants of two springs are and respectively, regard the average of two values as the spring constant measured from the gravitational force. 7) If the measurement is finished, turn off the computer interface and the computer. Clean up the experimental instruments according to the suggested method. (3) Comparison of the spring constant between two measurements Regard the spring constant measured from the gravitational force as the reference value and compare it with the spring constant measured from the centripetal force. Investigate the validity of the formula describing the centripetal force by comparing the spring constant between the two measurements.

10 PAGE 10/13 5. Experimental Values (1) Measurement of the spring constant from the centripetal force Masses of rotating objects (g) Original position of rotating objects (cm) # of measurement Rotation radii of rotating objects (cm) Rotation period (s) Displacement of the spring (m ) Centripetal force (N ) #1 ( cm) #2 ( cm) #3 ( cm) #4 ( cm) #5 ( cm) Slope Nm -intercept -intercept m N

11 PAGE 11/13 (2) Measurement of the spring constant from the gravitational force 1) Spring #1 2) Spring #2 Original length of the spring (cm) Original length of the spring (cm) Gravitational acceleration (ms ) Gravitational acceleration (ms ) Mass of the additional masses (g) Gravitational force (N ) Changed length of the spring (cm) Displacement of the spring (m ) Mass of the additional masses (g) Gravitational force (N ) Changed length of the spring (cm) Displacement of the spring (m ) Slope Nm Slope Nm -intercept m -intercept m -intercept N -intercept N (3) Comparison of the spring constant between two measurements Spring constant Spring constant measured from the centripetal force (Nm ) Spring constant measured from the gravitational force (Nm ) [Reference value] Error (%)

12 PAGE 12/13 6. Results and Discussions (This page should be used as the first page of the corresponding section. If the contents exceed this page, additional contents should be written by attaching papers. Contents should be written by hand, and not by a word processor. Attaching copied figures and tables to the report is allowed.) Write down contents in terms of the following key points. 1. Discuss the validity of the formula describing the centripetal force by using the error between the spring constant measured from the centripetal force and the spring constant measured from the gravitational force. 2. Discuss the case that the masses of two rotating objects are equal and the spring constants of two springs are different. 3. Discuss the case that the masses of two rotating objects are different and the spring constants of two springs are equal.

13 PAGE 13/13 7. Solution of Problems (This page should be used as the first page of the corresponding section. If the contents exceed this page, additional contents should be written by attaching papers. Contents should be written by hand, and not by a word processor. Attaching copied figures and tables to the report is allowed.) 8. Reference