Gyroscope. Objective. Theory. 1. Torque. 2. Angular Momentum. Observe the motions of gyroscope.

Transcription

1 [International Campus Lab] Objective Observe the motions of gyroscope. Theory Reference Young & Freedman, University Physics (14 th ed.), Pearson, Torque (p.327~330) 10.5 Angular Momentum (p.341~343) 10.7 s and Precession (p.346~349) Fig. 1 shows a force FF applied at a point PP described by a position vector rr with respect to the chosen point OO. Then the magnitude of torque depends on FF, rr, and the angle φφ between FF and rr. ττ = FFFF = rrrr sin φφ = FF tan rr (1) or ττ = rr FF (2) 1. Torque Torque (or moment) is the tendency of a force to rotate a body about an axis. We use the Greek letter ττ for torque. 2. Angular Momentum The angular momentum LL in the rotational motion is the analogue of (linear) momentum pp = mmvv in the translational motion. Its relationship to momentum pp is exactly the same as the relationship of torque to force, ττ = rr FF. We define angular momentum LL as LL = rr pp = rr mmvv (3) When a net force FF acts on a particle, its velocity and momentum change, so its angular momentum may also change. To show this, we take the time derivative of Eq. (3): Fig. 1 The torque ττ of the force FF about the point O is ττ = rr FF. In this figure, rr and FF are in the plane of the page and the torque vector ττ points out of the page toward you. ddll dddd = ddrr ddvv mmvv + rr mm dddd dddd = (vv mmvv ) + (rr mmaa ) (4) 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 1 / 19

2 The first term is zero because it contains the vector product of the vector vv = ddrr dddd with itself. In the second term we replace mmaa with the net force FF, obtaining ddll dddd = rr FF = ττ (5) We can also find the total angular momentum of a rigid body rotating about the zz-axis with angular speed ωω. Consider a thin slice of the body lying in the xxxx-plain (Fig. 2). Each particle in the slice moves in a circle centered at the origin, and at each instant its velocity vv ıı is perpendicular to its position vector, rr ıı as shown. A particle with mass mm ii at a distance rr ii from OO has a speed vv ii = rr ii ωω. Then the magnitude LL ii of its angular momentum is LL ii = mm ii vv ii rr ii = mm ii (rr ii ωω)rr ii = mm ii rr ii 2 ωω (6) The total angular momentum of the slice of the body lying the xxxx-plain is the sum LL ii LL = LL ii = mm ii rr ii 2 ωω = IIII (7) And for a rigid body rotating around an axis of symmetry, we have the vector relationship 3. When the axis of rotation changes direction, some quite unexpected physical phenomena can occur. For example, consider a gyroscope that is supported at one end (Fig. 3). If we hold it with flywheel axis horizontal and let go, the free end of the axis simply drops owing to gravity if the flywheel isn t spinning. But if the flywheel is spinning, what happens is quite different. One possible motion is a steady circular motion of the axis in a horizontal plane, combined with the spin motion of the flywheel about the axis. This motion of the axis is called precession. To study this strange phenomenon of precession, we need the relationship given by Eq. (5), ττ = ddll /dddd. We take the origin OO at the pivot and assume that the flywheel is symmetrical, with mass MM and moment of inertia II about the flywheel axis. The flywheel axis is initially along the xx-axis. The only external forces on the gyroscope are the normal force nn acting at the pivot and the weight ww of the flywheel that acts its center of mass, a distance rr from the pivot. nn has zero torque with respect to the pivot, and ww has a torque ττ = rr ww in the yy-direction, as shown in Fig. 4(a). LL = IIωω (8) Fig. 2 In a rigid body rotating about zz-axis at angular speed ωω, the angular momentum LL ii of a particle of mass mm ii is perpendicular to the plane of motion and has magnitude LL ii = mm ii vv ii rr ii = mm ii (rr ii ωω)rr ii = mm ii rr ii 2 ωω. Fig. 3 A gyroscope supported at one end. When the flywheel spins, it and its axis float in the air while moving in a circle about the pivot. The horizontal circular motion of the flywheel and axis is called precession. The angular speed of precession is Ω. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 2 / 19

3 Initially, there is no rotation, and the initial angular momentum LL ıı is zero. From Eq. (5) the change ddll in angular momentum in a short time interval dddd following this is ddll = ττ dddd (9) This change is in the yy-direction because ττ is. As each additional time interval dddd elapses, the angular momentum changes by additional increments ddll in the yy-direction because the direction of the torque is constant (Fig. 3(b)). The steadily increasing horizontal angular momentum means that the gyroscope rotates downward faster and faster around the yy-axis until it hits the table. If the flywheel is spinning initially, the initial angular momentum LL ıı is not zero (Fig. 5(a)). Since the flywheel rotates around its symmetry axis, LL ıı lies along the axis. But each change in angular momentum ddll is perpendicular to the axis because the torque ττ = rr ww is perpendicular to the axis (Fig. 5(b)). This causes the direction of LL to change, but not its magnitude, the change ddll are always in the horizontal xxxx-plane, so the angular momentum vector and the flywheel axis with which it moves are always horizontal. In other words, the axis doesn t fall, it just precesses. 4. Precession Angular Speed Fig. 4 (a) If the flywheel in Fig 3 is initially not spinning, its initial angular momentum is zero (b) In each successive time interval dddd, the torque produces a change ddll = ττ ddtt in the angular momentum. The flywheel acquires an angular momentum LL in the same direction as ττ, and the flywheel axis falls. At the instant shown in Fig. 5(a), the gyroscope has angular momentum LL. a short time interval dddd later, the angular momentum is LL + ddll ; the infinitesimal change in angular momentum is ddll = ττ dddd, which is perpendicular to LL. As Fig. 6 shows, this means that the flywheel axis of the gyroscope has turned through a small angle dddd = ddll / LL. The rate at which the axis moves, dddd dddd, is called the precession angular speed; denoting this quantity by Ω, we find Ω = dddd dddd = ddll LL ddtt = ττ zz = wwww LL zz IIII (10) Thus the precession angular speed Ω is inversely proportional to the angular speed ωω of spin about the axis. A rapidly spinning gyroscope precesses slowly. If friction in its bearings causes the flywheel to slow down, the precession angular speed increases. Fig. 5 (a) The flywheel is spinning initially with angular momentum LL. ı The forces (not shown) are the same as those in Fig 4(a). (b) Because the initial angular momentum is not zero, each change ddll = ττ ddtt in angular momentum is perpendicular to LL. As the result, the magnitude of LL remains the same but its direction changes continuously. Fig. 6 Detailed view of past of Fig 5(b). In a time dddd, the angular momentum vector and the flywheel axis precess together through an angle ddφφ. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 3 / 19

4 We can use Eq, (10) when only the weight ww of the flywheel acts force on the gyroscope. In this experiment, you will use the gyroscope equipped with several sensors and components on the axis of the flywheel. They all have torques in the yy-direction. Because we do not know their weights and distances from the pivot, we cannot calculate net torque. If the axis is exactly balanced in the horizontal position by adjusting the position of weights on the counterweight arm, we don t have to consider the torque of any component (including the flywheel) on the axis. Instead, to produce the turning torque, you will attach a small add-on weigh to the front of the flywheel. If the add-on weight has mass MM and distance RR from the pivot, Eq. (10) becomes 5. Nutation As explained above, if a gyroscope is set at a tilt on a horizontal surface and spun rapidly, its rotational axis starts precessing about the vertical. After a short interval, the gyroscope settles into a motion in which each point on its rotation axis follows a circular path. Initially, however, there is no precession, and the gyroscope falls downward. This gives rise to an imbalance in torques that starts the precession. In falling, the gyroscope overshoots the level at which it would precess steadily and then oscillates about this level, as shown in Fig. 7. This oscillation is called nutation. Ω = ττ LL = (MMgg)RR IIII (11) 6. Moment of Inertia II : Moment of inertia of the flywheel ωω : Angular speed of the flywheel MM : Mass of the add-on weight gg : Acceleration due to gravity RR : Distance from the pivot to the add-on weight We can find the moment of inertia of the flywheel experimentally. In the figure 8, the weight with mass mm hangs from the thread which is wrapped around the pulley attached on the flywheel. The weight speeds up with linear acceleration aa. Applying Newton s second law gives FF = mmmm = mmgg TT and solving for the tension in the thread gives TT = mm(gg aa). Thus the torque ττ caused by the weight hanging from the thread is ττ = rrrr = rrrr(gg aa), where rr is radius of the pulley. Fig. 7 Nutation (a) released from rest (b) released with forward speed (c) released with backward speed The relationship of the linear acceleration aa to the angular acceleration αα of the pulley (or the flywheel) is aa = rrrr. Since ττ = IIII, the moment of inertia of the flywheel is II = ττ αα rrrr(gg aa) = (aa rr) = mmrr 2 gg 1 (12) aa Fig. 8 Rotating flywheel and free-body diagram mm : Mass of the weight rr : Radius of the pulley aa : Acceleration of the weight 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 4 / 19

5 Equipment 1. List Item(s) Qty. Description PC / Software Data Analysis: Capstone : SensorLAB 1 set 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. Apparatus 1 Allows studies of the motion of a gyroscope. USB to RS-232 Cable DC Adapter (12V) 1 1 Connects the gyroscope apparatus to PC. Supplies DC power to the gyroscope apparatus. Photogate (Pulley, Rod, and Cable included) 1 set Measures high-speed or short-duration events. A-shaped Base (Small) 1 Provide stable support for experiment set-ups. Support Rods (600mm) 2 Provide stable support for experiment set-ups. Multi-clamps 2 Provide stable support for experiment set-ups. Hanging Weight Add-on Weight 1 1 Mass: 50g for the moment of inertia experiment Mass: 100g for acting downward force 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 5 / 19

6 Item(s) Qty. Description String 1 Spins the gyroscope flywheel. Thread Scissors Hang a weight. Flywheel Part Gives its mass and dimension so that you can compute the moment of inertia of the gyroscope flywheel. Vernier Caliper 1 Measures external, internal diameter or depth of an object with a precision to 0.05mm. Electronic Balance Measures mass with a precision to 0.01g. 2. Details (1) Apparatus (2) Vernier Caliper 1 22 mm is to the immediate left of 0 on the vernier scale. 2 Aligned (13 th ) line corresponds to 0.65 mm (= ) = mm 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 6 / 19

7 Procedure Caution Caution The gyroscope apparatus could easily malfunction or produce incorrect results. It should be treated with extra care. 3. Never touch the sensor modules. 1. Keep tight hold of the gyroscope axle while you pull the thread to rotate the flywheel, or while you stop the flywheel. Note You will use two kinds of thread/string in this experiment. Use Thread For suspending a weight from the flywheel pulley (Experiment 1) String For spinning up the flywheel of the gyroscope (Experiment 2 ~ 5) Where Lecture Table Each Lab Table Wound on a spool Thicker one with loops Shape 2. Subjecting the apparatus to strong impact could cause the sensors to fail, or the axles to be bent. Note Color: Black or White Do NOT cut the string. Do NOT untie endknots. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 7 / 19

8 Experiment 1. Moment of Inertia of Flywheel 1 Clamp the gyroscope axle. Using a support rod and a multi-clamp, clamp the gyroscope axle in the horizontal position. 2 Install the photogate assembly. Assemble the photogate assembly (the raised slot on the photogate housing provides a seat for attaching a pulley using a pulley mounting post). And then mount it on the support rod using a multi-clamp. 3 Arrange a weight to accelerate the flywheel. Tie a length of thread to the hole of the pulley of the flywheel. Tie a mass of 50g at the other end of the thread. And then pass the thread over the photogate pulley. 4 Adjust the position of the photogate pulley. (1) Set up equipment. Adjust the position of the photogate pulley so the thread runs parallel to the table and the weight can fall out of the table. 5 Adjust the length of the thread, if required. The weight must stop falling before it reaches the ground. 6 Connect the Photogate to the interface. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 8 / 19

9 (2) Configure the Capston software. 1 Add [Photogate with Pulley]. (Make sure you do not select [Photogate].) Make sure that [Linear Speed] is checked, the spoke arc length is 0.015m, and the spoke angle is Create a graph display. xx-axis: [Time(s)], yy-axis: [Linear Speed(m/s)] 2 Create a timer. (3) Wind the thread around the pulley. By rotating the flywheel, wind the thread around of the pulley, until the weight is located near the photogate pulley. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 9 / 19

10 (4) Click [Record]. (8) Repeat the measurement. Repeat the steps (3) to (7) more than three times. (5) Release the flywheel. The torque of a tension force rotates the flywheel. Click [Stop] before the weight reaches the ground. 1 st 2 nd 3 rd average mm (kg) rr (m) aa II (6) Find the acceleration aa of the system. (9) Compare the result with the theoretical value. The flywheel part is prepared on lecture table. Measure the mass, the inner radius, and the outer radius of the flywheel, and calculate the theoretical value II flywheel of the flywheel. (7) Calculate II of the flywheel. II = mmrr 2 gg 1 (12) aa RR 1 = (m) RR 2 = (m) MM = (kg) Because the moment of inertia of the pulley is relatively smaller than that of the flywheel, you can ignore it. If required, use the pre-calculated value II pulley = kg m 2. mm kg : Mass of the weight rr m : Radius of the pulley 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 10 / 19

11 Experiment 2. Motion of, Part 1 (3) Observe the motion of the gyroscope without rotating. (1) Set up equipment. 1 Remove the photogate assembly. 2 Remove the support rod and the multi-clamp clamping the gyroscope axle. 3 Remove the thread connected to the weight. Do not spin the flywheel. Tap the end of the gyroscope axle in the horizontal direction, and observe the motion of the gyroscope. (4) Observe the motion of the gyroscope with rotating. Spin the flywheel. And then tap the end of the gyroscope axle in the horizontal direction. Observe the motion of the gyroscope. (2) Balance the gyroscope axle in the horizontal position. Adjust the position of the counterweights until the gyroscope is balanced without an add-on mass. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 11 / 19

12 (5) Compare the results. Experiment 3. Motion of, Part 2 Q Describe the differences of two cases and explain why. (1) Spin the flywheel. A (2) Apply a steady force on the gyroscope axle. With the flywheel rotating, pull the gyroscope axle downward using a string (or push it downward using your finger). Observe the motion of the gyroscope. Caution When you stop the spinning flywheel, follow the instructions below. Q Can you simply swing the gyroscope axle as you pull it? If not, which direction is the axle going in? Explain why. A Q If you change the direction of the force, which direction is the axle going in? Explain why. A 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 12 / 19

13 Experiment 4. Precession Angular Speed 1 Connect or disconnect communication. Caution Before you run SensorLAB, you should shut down PAS- CO Capstone. If not, they suddenly crashes and you may lose your experimental data. 2 Begin or stop recording. 3 Displays the sensor readings. 4 Plots graphs. 5 Select graph type. (1) Set up the equipment. 6 Span graphs. Connect the gyroscope apparatus to the computer using Serial USB cable. Provide power to the apparatus using 12V DC adapter. (Different type of DC adapter can be provided.) 7 Converts the value of rotating angle to absolute value. 8 Change the sign of sensor values. 9 Adjust the ranges. 10 Shows data table. (2-1) Click [Connect] to receive the sensors values. (2) Run SensorLAB software. [Received Data] displays the real-time values of the sensors. Rotation : Rotating angle of the rotating axle ( ) (by the rotary motion sensor) Tilt : Tile of the gyroscope axle ( ) (by the tilt sensor) DiskRPM : RPM of the flywheel (revolutions per minute) (by the RPM sensor) 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 13 / 19

14 (2-2) Set [Absolute Value]. (2-4) Select a graph type. You can change graph types to display on the graph area. Select any type of graph in [Graph Option]. The data collection will automatically end when the graph point reaches the right end of the graph display. By [Absolute Value] icon, [Rotation] value will [Continually increase regardless of the rotating direction] [increase or decrease according to the rotating direction] Watch the [Rotation] values of [Received Data] and rotate the rotating axle of the gyroscope clockwise or counterclockwise. It is not recommended to select the [Rotation-Tilt] graph because the data collection finishes in a relatively short time. (This graph type is for the 5 th experiment, nutation.) Select a flow mode graph type by depressing [Flow mode] icon. If [Rotation] value continually increases regardless of the rotating direction, depress [Absolute Value] icon. Then it will increase or decrease according to the rotating direction. (2-3) Set [Flip Horizontal] and [Flip Vertical]. [Rotation] value initially increases when the rotating axle of the gyroscope rotates counterclockwise. If you depress [Flip Horizontal] button, [Rotation] increases when the axle rotates clockwise. (2-5) Set the range of each axis. Set the range of xx-axis (time) at maximum (60 seconds). The data collection will automatically end after 60 seconds. [Tilt] value is initially positive when the arm with the flywheel of the gyroscope axle is higher. If you want to change the sign of [Tilt], depress [Flip Vertical]. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 14 / 19

15 (3) Level the gyroscope. Follow the steps below. Do not move the apparatus after finishing leveling it. Note In this experiment, you will use the gyroscope equipped with several sensors and components on the gyroscope axle. They all act downward force in the yy-direction. Because we do not know their weights and distances from the pivot, we cannot calculate net torque. If the axle is exactly balanced in the horizontal position by adjusting the position of counterweights, we don t have to consider the torque of any component (including the flywheel) on the axle. Instead, to produce the turning torque, you will attach an add-on weight to the front of the flywheel. Then you can easily calculate the torque by measuring the mass and the position of the add-on weight. (5) Spin the flywheel. Spin up the flywheel to high speed using the string. (4) Balance the gyroscope axle. The counterweight arm has three weights. Adjust the positions of these weights so that the gyroscope axle is exactly balanced in the horizontal position. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 15 / 19

16 (6) Hang an add-on weight. To produce a tipping torque, hang the 100g weight on the RR = 0.15 m hanger. (The hangers are 0.15, 0.17, 0.19, and 0.21 m away from the pivot.) If you touch the end of the axle, you can feel the speed of the gyroscope. Help the gyroscope start precessing with the gyroscope axle horizontal by moving it along in the correct direction at the correct speed, and then release gently. If you push it faster or slower than the correct speed, it will move upward or downward. When it precesses without nutation, begin data collection. (7) Let the gyroscope precess. (8) Begin data collection. If you spin the flywheel in the direction as shown above, the gyroscope will precess counterclockwise. In the general case, i.e. for arbitrary initial conditions, the motion of the gyroscope is a superposition of precession and nutation. Nutation is caused by a possible small deviation of the vector of own angular momentum from the axis of symmetry. This deviation is absent only for carefully chosen specific initial conditions. It means you have to help the gyroscope start precessing with the gyroscope axle horizontal by moving it along in the correct direction at the correct speed, and then release gently. Click [Start] to begin data collection. The display will start plotting graph. The data collection will automatically end after 60 seconds (as you set in the step (2-5)). 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 16 / 19

17 (9) Extract collected data. Click [Datasheet] icon to extract data table Click [Save] to save the data as text format file (*.txt). (10) Open the data file. You can use Microsoft Excel to import data from a text into a worksheet. To start the Text Import Wizard, on the Data ( 데이터 ) tab, in the Get External Data ( 외부데이터가져오기 ) group, click From Text ( 텍스트 ). Then, in the Import Text File ( 텍스트파일가져오기 ) dialog box, double-click the text file that you want to import. In the step 1 of 3 of the Text Import Wizard, select Delimited ( 구분기호로분리됨 ) for Original data type ( 원본데이터형식 ). In the step 2 of 3, check Comma ( 쉼표 ) and Space ( 공백 ) for Delimiters ( 구분기호 ). (11) Analyze the data. Cells in the column B contain the values of the rotating angle of the rotating axle. The table below shows that the difference between B9 cell and B1061 cell is 360 degrees (1 complete turn). Column A (time column) shows that it has taken (= ) seconds for one complete turn. The average RPM (rev/min) of the flywheel during this time interval, is given by Average(D9:D1061). 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 17 / 19

18 (12) Calculate the precession angular speed. Verify equation (11). Spin up the wheel to high speed using the string. With the add-on weight producing a tipping torque, hold the axle horizontally, and then Ω = dddd dddd = ττ LL = MMggRR IIII (11) Experimental value of the precession angular speed is Ω = 2ππ/TT, where the period TT of the gyroscope is experimentally obtained in the step (11). 1 Gently tap it in the counter-precession direction 2 Suddenly release it from rest 3 Gently tap it in the precession direction Theoretical value of the precession angular speed is Ω = MMggRR IIII, where MM is the mass of the add-on weight, RR is the distance from the pivot to the add-on weight, II is the moment of inertia of the flywheel which is obtained in the experiment 1, and ωω is the flywheel angular speed which is obtained in the step (11). (1 rpm = 1 rev/min = 1/60 rev/s = 2ππ 60 rad/s) (13) Repeat experiments. Vary conditions and repeat experiment. MM RR II ωω Ω = MMggRR IIII Ω = 2ππ TT 1 st 2 nd 3 rd The gyroscope undergoes nutation, as in patterns below. Experiment 5. Nutation Change [Graph Option] to [Rotation-Tilt]. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 18 / 19

19 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 and the Interface. Keep the Apparatus away from strong impact. Never touch the sensor units of it. Unplug the DC Adapter. Do not unplug the Serial-USB Cable from the computer. Be careful not to knot or tangle the String. Keep the Flywheel Parts, Spools of Thread, Scissors in the basket on the lecture table at the font of the laboratory. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 19 / 19