Circular Motion and Centripetal Force

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[For International Campus Lab ONLY] Objective Measure the centripetal force with the radius, mass, and speed of a particle in uniform circular motion.

1 [For International Campus Lab ONLY] Objective Measure the centripetal force with the radius, mass, and speed of a particle in uniform circular motion. Theory Reference Young & Freedman, University Physics (14 th ed.), Pearson, Motion in a Circle (p.106~109) 4.3 Newton s Second Law (p.132~136) 5.4 Dynamics of Circular Motion (p.174~175) The magnitude of the average acceleration during Δ is Δ Δ (2) Δ The magnitude of the instantaneous acceleration at point is the limit of this expression as we take point When a particle moves in a circle with constant speed, the motion is called uniform circular motion. There is no component of acceleration parallel to the path. The acceleration vector is perpendicular to the path and hence directed inward toward the center of the circular path. This causes the direction of the velocity to change without changing the speed. closer and closer to point lim Δ Δ lim Δ Δ (3) Fig. 1(a) shows a particle moving with constant speed in a circular path of radius with center at. The particle moves from to in a time Δ. The vector change in velocity during this time is shown in Fig. 1(b). The angles Δ in Fig. 1(a) and 1(b) are the same because is perpendicular to the line and is perpendicular to the line, so Δ or Δ (1) Fig. 1 Finding the velocity change, average accelerating, and instantaneous accelerating for a particle moving in a circle with constant speed 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 1 / 20

If the time interval Δ is short, Δ is the distance the particle moves along its curved path. So the limit of Δ Δ is the speed at point.

Then (4) Uniform circular motion, like all other motion of a particle, is governed by Newton s second law,.

The magnitude of the acceleration is constant, so the magnitude of the net force must also constant.

2 If the time interval Δ is short, Δ is the distance the particle moves along its curved path. So the limit of Δ Δ is the speed at point. Also, can be any point on the path, so we can drop the subscript and let represent the speed at any point. Then (4) Uniform circular motion, like all other motion of a particle, is governed by Newton s second law,. To make the particle accelerate toward the center of the circle, the net force on the particle must always be directed toward the center as in Fig. 3. The magnitude of the acceleration is constant, so the magnitude of the net force must also constant. If the inward net force stops acting, the particle flies off in a straight line tangent to the circle, as in Fig. 4. In uniform circular motion, the direction of the instantaneous acceleration is perpendicular to and inward along the radius, i.e. toward the center of the circle as in Fig. 1(c). Because of this reason, the acceleration in uniform circular motion is sometimes called centripetal acceleration. The word centripetal is derived from two Greek words meaning seeking the center. Fig. 2 shows the direction of the velocity and acceleration vector at several points for a particle moving uniform circular motion. The magnitude of the net force on a particle with mass in uniform circular motion is then (7) We can also express the magnitude of the acceleration in uniform circular motion in terms of the period (the time for one revolution) of the motion. In a time the particle travels a distance equal to the circumference 2π of the circle, so its speed is 2 (5) Fig. 3 Net force, acceleration and velocity in uniform circular motion Substituting Eq. (5) into Eq. (4) yields 4 (6) Fig. 2 Acceleration and velocity for a particle at uniform circular motion Fig. 4 What happens if the inward radial force suddenly ceases to act on a body in circular motion? 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 2 / 20

analyzes the data measured by various sensors.

sensors, including power supplies which provide up to 15 watts of power.

motion using electric motor. Swivel 1 Allows connected objects to rotate.

3 Equipment 1. List Item(s) Qty. Description PC / Software Data Analysis: Capstone 1 Records, displays and analyzes the data measured by various sensors. Interface 1 Data acquisition interface designed for use with various sensors, including power supplies which provide up to 15 watts of power. Force Sensor 1 Measures the magnitude of force. Range: 50N ~ 50N Resolution: 0.03N Photogate (Cable included) 1 Measures high-speed or short-duration events. Circular Motion Apparatus 1 Allows objects to undergo uniform circular motion using electric motor. Swivel 1 Allows connected objects to rotate. Wire 1 Connects the Mass to the Force Sensor Mass Set 1 set Includes 8 masses - Free mass: Hook embedded - Fixed mass: Perforated - Plastic, Aluminum, Al. with Brass, and Stainless Steel Fixed Mass Holder (Nut included) 1 Secures a fixed mass. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 3 / 20

Item(s) Qty. Description Bubble Level 1 Checks the level of a surface.

A-shaped Base Multi-clamp Support Rod 600mm Support Rod 300mm 1 1 1 1 Provide stable

Details (1) Photogate When the infrared beam is blocked, the output signal of the

When the beam is not blocked, the output signal becomes 1 and the LED goes off.

pendulum, the velocity of an object, etc.

4 Item(s) Qty. Description Bubble Level 1 Checks the level of a surface. Patch Cords (with banana plugs) 2 Carry electric current. A-shaped Base Multi-clamp Support Rod 600mm Support Rod 300mm Provide stable support for experiment set-ups. Electronic Balance Measure mass. 2. Details (1) Photogate When the infrared beam is blocked, the output signal of the photogate becomes 0 and the LED lamp on the photogate goes on. When the beam is not blocked, the output signal becomes 1 and the LED goes off. This transition of signal can be used to calculate quantities such as the period of a pendulum, the velocity of an object, etc. The Photogate sensor is an optical timing device used for very precise measurements of high-speed or short-duration events. It consists of a light source (infrared LED) and a light detector (photodiode). When an object moves through and blocks the infrared beam between the source and the detector, a signal is produced which can be detected by the interface. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 4 / 20

(2) Force Sensor (3) Mass Set The Force Sensor measures both pulling and pushing forces in the range of 50N to 50N.

The gauge consists of an insulating flexible backing which supports a metallic foil pattern.

The gauge is wired to form a full-bridge circuit that is driven by a constant voltage source. The voltage across the bridge circuit is proportional to the applied force.

5 (2) Force Sensor (3) Mass Set The Force Sensor measures both pulling and pushing forces in the range of 50N to 50N. Free Mass Fixed Mass Use Measuring Force Balancing with Hook with Center Hole Shape The sensor uses a strain gauge attached to an aluminum beam. The gauge consists of an insulating flexible backing which supports a metallic foil pattern. As the aluminum beam is deformed, the foil is deformed, causing its electrical resistance to change. The gauge is wired to form a full-bridge circuit that is driven by a constant voltage source. The voltage across the bridge circuit is proportional to the applied force. How Secure it to the rotating Hook it to the wire. to use arm using a mass holder. Mass Measure the mass using Do not have to measure an electronic balance. the mass. Material Color Mass range g Plastic White 11.50~12.50 Aluminum Light Gray 23.50~24.50 Al + Brass Gray / Yellow 43.00~44.00 Stainless Steel Dark Gray 66.50~ Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 5 / 20

Setup Setup 1. Equipment Setup CAUTION Do not tighten the swivel too hard.

NOTE 1 Thread the wire from the swivel hook through the pulley as shown below.

CAUTION To avoid possible injury from the rotating arm hitting the body, keep at least 1m distance from

To avoid damaging the apparatus, keep all cords away from the rotating arm.

6 Setup Setup 1. Equipment Setup CAUTION Do not tighten the swivel too hard. It could cause the sensor to fail. NOTE 1 Thread the wire from the swivel hook through the pulley as shown below. 2 Hook the free mass to the end of the wire. 3 Hold the fixed mass firmly using the mass holder. CAUTION To avoid possible injury from the rotating arm hitting the body, keep at least 1m distance from the rotating arm when running the motor. To avoid damaging the apparatus, keep all cords away from the rotating arm. NOTE To adjust the rotational radius of a free mass, lower or raise the height of the upper (horizontal) support rod. To avoid damaging any other equipment or facilities near the apparatus, especially your LCD monitor, keep the apparatus away from them. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 6 / 20

Setup 2. Software Setup 2 Select the timing device for the timer. (1) Run Capstone software.

Click the input port which you plugged the Photogate into and select [Photogate] from the list.

When the rod of the fixed mass holder passes the Photogate, the Photogate beam is blocked and a Blocked

The time interval between the signals equals the period of the motion, the time for one revolution.

) Click [ ] and then select [Blocked]. (3) Create and configure a timer.

7 Setup 2. Software Setup 2 Select the timing device for the timer. (1) Run Capstone software. Check [Photogate, Ch1] and then click [Next]. (2) Add a Photogate. Click the input port which you plugged the Photogate into and select [Photogate] from the list. 3 Arrange the sequence of timing events. When the rod of the fixed mass holder passes the Photogate, the Photogate beam is blocked and a Blocked signal is generated. The time interval between the signals equals the period of the motion, the time for one revolution. (You will use this measurement to calculate the speed of the mass in step (5).) Click [ ] and then select [Blocked]. (3) Create and configure a timer. To measure the period of the circular motion, click [Timer Setup] in the [Tools] palette and follow the steps below. Click [ ] again and select [Blocked] one more time. Now you have two [Blocked] events as below. Click [Next] for the next step. 1 Create a timer. Choose [Build your own timer]. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 7 / 20

4 Specify the timer s name. To define the equation 2/, type v=2*pi*r/ and [.

Make sure that you enter a left-bracket [ instead of T because you will insert measured data for.

through the process defined in step (4). Now you define and using [Calculator]. Select [period(s)], which you named in step (4).

8 4 Specify the timer s name. To define the equation 2/, type v=2*pi*r/ and [. Enter the name of the timer ( period is recommended) and click [Finish]. Make sure that you enter a left-bracket [ instead of T because you will insert measured data for. (Unlike numerals or math functions, measured data are given in brackets [ and ].) When you type [, a pop-up list appears as below. (5) Configure calculator. In a period, the mass travels a distance equal to the circumference 2π of the circle, so its speed is 2 (5) The period will be measured through the process defined in step (4). Now you define and using [Calculator]. Select [period(s)], which you named in step (4). When you select it, the equation is completed as shown below. When you press the enter key, two cells are automatically typed. One is for the unit of the parameter of the first equation, and the other is for the variable R which you do not define yet. Define the speed of the mass. Enter the equation of in the first row of the [Calculator]. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 8 / 20

In the [Properties] window, check [Change Sign].

force. In this experiment, you will measure the pulling force so you have to change the sign of the outputs.

9 Enter the unit of the speed. Click the Force Sensor icon in the [Hardware Setup] panel and then click the properties button ( ) in the lower right corner. Enter the value and unit of R. You have to enter a correct value for R whenever you vary the radius of the circular path of the mass. In the [Properties] window, check [Change Sign]. The sign of the outputs of the Force sensor is initially positive for the pressing force and negative for the pulling force. In this experiment, you will measure the pulling force so you have to change the sign of the outputs. (6) Configure the Force Sensor. The interface automatically recognizes the Force Sensor. If the sensor is not in the panel, click the input port which you plugged the sensor into. A drop down menu of sensors will appear. Select [Force Sensor, High Resolution] from the list and the sensor s icon will be added to the panel. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 9 / 20

(7) Configure the signal generator. Do not add any sensor for [OUTPUTS 1]. Set the [DC Voltage] to 4V.

Click [Signal Generator] in the [Tools] palette, and select [850 Output 1].

[On] : The signal generator always output the signal. [Off] : The signal generator is not activated.

10 (7) Configure the signal generator. Do not add any sensor for [OUTPUTS 1]. Set the [DC Voltage] to 4V. Click [Auto] button to automatically activate the generator. Click [Signal Generator] in the [Tools] palette, and select [850 Output 1]. NOTE Output options of [Signal Generator]: Change the [Waveform] of the generator to [DC]. [On] : The signal generator always output the signal. [Off] : The signal generator is not activated. [Auto] : The signal generator automatically outputs the signal when you start recording, and automatically stops when you stop recording. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 10 / 20

(8) Create displays. You need 4 displays.

[Displays] palette into the workbook page. You can see the table as below. 1 Configure [Table] display.

11 (8) Create displays. You need 4 displays. Data Table -- Graph 1 - Graph 2 - Digital Meter Click and drag [Table], [Graph], or [Digits] icon from the [Displays] palette into the workbook page. You can see the table as below. 1 Configure [Table] display. [Table] display initially has two columns. To insert one more column, click [Insert empty column] icon. 2 Configure [Graph] displays. Click <Select Measurement> of each column and select Time(s), Force(N), and v(m/s). Time(s) is a fundamental measurement which shows the elapsed time from when the measurement starts. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 11 / 20

Click <Select Measurement> to define the variables of each axis. Move and resize the individual displays for easy view. Make a - graph. Select Time(s) for the -axis and Force(N) for the -axis.

In the same way, make a - graph. Select v(m/s) for the -axis and Force(N) for the -axis Click the [Record] button at the left end of the [Controls] palette to begin recording data.

12 Click <Select Measurement> to define the variables of each axis. Move and resize the individual displays for easy view. Make a - graph. Select Time(s) for the -axis and Force(N) for the -axis. (9) Adjust the sample rate of the measurement. Select [10.00 Hz ] (collecting data every 0.1 second) for the sensor in the [Controls] palette. (10) Start/Stop recording data. In the same way, make a - graph. Select v(m/s) for the -axis and Force(N) for the -axis Click the [Record] button at the left end of the [Controls] palette to begin recording data. It will toggle to [Stop]. Click [Stop] to stop data collection. 3 Configure [Digits] display. Drag [Digits] from the [Displays] palette into the workbook page, and select v(m/s). You can delete any data run using [Delete Last Run] or drop-down menu in the [Controls] palette. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 12 / 20

Procedure NOTE Mass set Information Free Masses Fixed Masses Use Measuring Force Balancing with Hook with Center Hole Shape Experiment 1.

The radius and mass are held contant as the speed is increased. (1) Follow the equipment setup instruction. [Setup 1. Equipment Setup] on page 6 How to use Mass Hook it to the wire.

Software Setup] on pages 7-12 (3) Set the radius and the mass. Material Color Mass range g Plastic White 11.50~12.50 Aluminum Light Gray 23.50~24.50 Al + Brass Gray / Yellow 43.00~44.

13 Procedure NOTE Mass set Information Free Masses Fixed Masses Use Measuring Force Balancing with Hook with Center Hole Shape Experiment 1. In this experiment, you will vary the speed by changing the voltage to the electric motor as the centripetal force is continuously measured by the Force Sensor. The radius and mass are held contant as the speed is increased. (1) Follow the equipment setup instruction. [Setup 1. Equipment Setup] on page 6 How to use Mass Hook it to the wire. Measure the mass using an electronic balance. Secure it to the rotating arm using a mass holder. Do not have to measure the mass. (2) Follow the software setup instruction. [Setup 2. Software Setup] on pages 7-12 (3) Set the radius and the mass. Material Color Mass range g Plastic White 11.50~12.50 Aluminum Light Gray 23.50~24.50 Al + Brass Gray / Yellow 43.00~44.00 Stainless Steel Dark Gray 66.50~ Mount a set of masses (heavier set recommended). Measure the mass of free masses using the electronic balance on the lecture table at the front of the lab. Free Masses Plastic Aluminum Aluminum + Brass Stainless Steel Mass g 2 Adjust the radius (9cm is recommended) of the free mass. To vary the radius, adjust the height of the Force Sensor, keeping the wire tight. 3 Place the fixed mass at the same radius as the free mass to ensure balancing the arm as it rotates. NOTE Prior to recording each data run, press the [Zero] button on the Force Sensor to zero the sensor. 4 Be sure to enter the correct value for in the calculator. If the radius of the circular path of the mass is 9cm, enter 0.09m. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 13 / 20

(4) Set the speed. (7) Stop data collection. Speed is varied by changing the voltage supplied to the electric motor. This will be displayed on [Digits] display.

14 (4) Set the speed. (7) Stop data collection. Speed is varied by changing the voltage supplied to the electric motor. This will be displayed on [Digits] display. Apply 4, 5, 6, 7, and 8V DC to the motor. Click the [Stop] button to stop data collection. Set [DC Voltage] of [Signal Generator] at 4V. NOTE Collected data are stored in memory and appear in all displays. The data runs are listed in the legend for each display. (You can delete any run by clicking [Delete Last Run] or drop-down menu in the [Controls] panel.) (8) Analyze the data. (5) Measure the force. Click [Show coordinates ] to read off data points. Click the [Record] button at the left end of the [Controls] palette to begin recording data. As the rotating arm rotates, the displays show the speed of the mass and the force acting on the sensor. Wait a second for stable rotation. (6) Repeat measurement. Repeat measurement for 4, 5, 6, 7, and 8V DC. NOTE You can use [Pin] tool for easy view of the displays. (9) Repeat experiments. Repeat measurement more than 3 times for each voltage and calculate the average value to reduce the possibility of experimental error. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 14 / 20

(10) Record your results. NOTE Mass Radius Speed Force (Theo.) Force (Result) If - graph vibrates as below, kg m m/s kg m N (11) Plot - graph. 1 Make sure all the rotating parts are aligned.

15 (10) Record your results. NOTE Mass Radius Speed Force (Theo.) Force (Result) If - graph vibrates as below, kg m m/s kg m N (11) Plot - graph. 1 Make sure all the rotating parts are aligned. Plot - graph and verify the centripetal force is proportional to the square of the speed. Use the method of least squares if required. (Refer to the appendix of [Free Fall] experimental manual.) 2 It could occur due to mechanical problems of the swivel ball bearing. In this case, search the data for the time interval of interest and calculate the average value of the force to find the experimental value. CAUTION It is strongly recommended that you save the data file with a different file name if you need to change the value in the calculator. Changing makes all pre-measured speed recalculated with the new value. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 15 / 20

16 Experiment 2. (4) Set the mass. In this experiment, the radius and the speed are held contant as the mass is varied. By replacing the mass set, the mass of the system becomes increased. (1) Follow the equipment setup instruction. [Setup 1. Equipment Setup] on page 6 Use the lightest mass set. (5) Measure the centripetal force. Click the [Record] button at the left end of the [Controls] palette to begin recording data. (2) Follow the software setup instruction. [Setup 2. Software Setup] on pages 7-12 (3) Set the radius and the speed. 1 Set [DC Voltage] of [Signal Generator] at 7V. As the rotating arm rotates, the displays show the speed of the mass and the force acting on the sensor. Wait a second for stable rotation. Record the speed. This value will be used to fine-tune the voltage in the step (7). Repeat measurement more than 3 times for each mass and calculate the average value to reduce the possibility of experimental error. (6) Stop data collection. Click the [Stop] button to stop data collection. 2 Adjust the radius (9cm is recommended) of the free mass. 3 Place the fixed mass at the same radius as the free mass. 4 Enter 0.09m for the radius in the calculator. (7) Replace the mass set. For all mass sets, repeat the steps (5) and (6). Make sure the speed is equal to that of step (5). The speed could be slightly different for each mass set even if the voltage remains constant, i.e. you need to fine-tune the voltage for the each mass set. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 16 / 20

17 (8) Analyze the result. (9) Record your results. Click the icon below to select the data run of interest. Mass Radius Speed Force (Theo.) Force (Result) kg m m/s kg m N NOTE Recorded data run has a default name Run#%1, where %1 is an automatically generated run number. You can change the name of each data run if required. (10) Plot - graph. 1 Click [Data Summary] in the [Tools] palette 2 Select [Show Sensor Data] tap. 3 Right-click on the run name of interest. 4 Select [Rename] from the pop-up list. Plot - graph and verify the centripetal force is proportional to the mass. Use the method of least squares if required. (Refer to the appendix of [Free Fall] experimental manual.) CAUTION It is strongly recommended that you save the data file with a different file name if you need to change the value in the calculator. Changing makes all pre-measured Click [Show coordinates ] to read off data points. speed recalculated with the new value. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 17 / 20

18 Experiment 3. (6) Find the appropriate voltage output. In this experiment, the mass and the speed are held constant as the radius is varied. By lowering the Force Sensor, the radius increases. As the radius increases, the fixed mass must be moved to a matching radius to balance the rotating arm. Click the [Record] button to monitor the speed of the mass. You can monitor the speed of the mass in [Digits] display. (1) Follow the equipment setup instruction. [Setup 1. Equipment Setup] on page 6 (2) Follow the software setup instruction. [Setup 2. Software Setup] on pages 7-12 Fine-tune the [DC voltage] near 4V so you can find any voltage at which the speed shows easy value. This speed is the target speed of this experiment. (3) Set the mass. Choose any mass set. (4) Set the speed. (You will finally set the speed in step (6).) The (linear) speed is dependent to the radius of the circular path of the mass, i.e. whenever you change the radius, you have to adjust the voltage so the speed remains constant. (5) Set the radius. In the later steps, you have to adjust the voltage to rotate the mass under the same linear speed. (As the radius decreases, the voltage must be increased.) 1 Adjust the radius of the free mass to 9cm. 2 Place the fixed mass at the same radius as the free mass 3 Enter 0.09m for the radius in the calculator. In the pre-experiment, the linear speed was 2.0m/s in the conditions of 9cm and 4.0V output. And then 7.2V was supplied to rotate the mass with 2.0m/s in the condition of 5cm. (Required voltages depend on your motor status.) You should choose the voltage between 3V and 9V. The motor may not work in a lower voltage than 3V, and could make a noisy vibration in a greater voltage than 9V. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 18 / 20

19 (7) Measure the centripetal force. (10) Repeat measurement. If you have any useless data during monitoring the speed in the previous step, delete it using [Delete Last Run] or dropdown menu in the [Controls] palette. In the condition of 8cm, you must adjust the output voltage so the speed shows the predetermined value in the step (6). Repeat measurement for R 9, 8, 7, 6, and 5cm. In the condition of the voltage as you set in the step (6), begin recording data. (11) Record your results. (8) Stop data collection. Mass Radius Speed Force (Theo.) Force (Result) Click the [Stop] button to stop data collection. kg m m/s kg m N CAUTION You MUST save the data file with a different file name for each data run in this experiment. Changing makes all pre-measured speed in the prior step recalculated with the new value. (12) Plot - 1 graph. (9) Change the radius. Plot - 1 graph and verify the centripetal force is proportional to the inverse of the radius. Use the method of least squares if required. (Refer to the appendix of [Free Fall] experimental manual.) Adjust the radii of the masses to 8 cm. Change in the [Calculator] to 0.08m. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 19 / 20

Result & Discussion Your TA will inform you of the guidelines for writing the laboratory report during the lecture. End of LAB Checklist Please put your equipment in order as shown below.

20 Result & Discussion Your TA will inform you of the guidelines for writing the laboratory report during the lecture. End of LAB Checklist Please put your equipment in order as shown below. Delete your data files and empty the trash can from the lab computer. Turn off the Computer. Do NOT disassemble the equipment. Keep the Mass Set, Mass Holder, Wire, and Bubble Level in the compartment box. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 20 / 20

Free Fall and Projectile Motion

[International Campus] Free Fall and Projectile Motion Objective Investigate the motions of a freely falling body and a projectile under the influence of gravity. Find the acceleration due to gravity.