Physical Pendulum, Torsion Pendulum

Transcription

1 [International Campus Lab] Physical Pendulum, Torsion Pendulum Objective Investigate the motions of physical pendulums and torsion pendulums. Theory Reference Young & Freedman, University Physics (14 th ed.), Pearson, Physical Pendulum (p.475~477) 9.4 Energy in Rotational Motion (p.307~312) 9.5 Parallel-Axis Theorem (p.312~313) 14.4 Application of SHM Angular SHM (p.471) When the body is released, it oscillates about its equilibrium position. The motion is not simple harmonic because the torque ττ zz is proportional to sin θθ rather than to θθ itself. However, if θθ is small, we can approximate sin θθ by θθ in radian. Then the motion is approximately simple harmonic. With this approximation, ττ zz = (mmggdd)θθ (2) 1. Physical Pendulum A physical pendulum is any real pendulum that uses an extended body, as contrasted to the idealized simple pendulum with all of its mass concentrated at a point. Figure 1 shows a body of irregular shape pivoted so that it can turn without friction about an axis through point OO. In equilibrium the center of gravity (cg) is directly below the pivot; in the position shown, the body is displaced from equilibrium by an angle θθ, which we use as a coordinate for the system. The distance from OO to the center of gravity is dd, the moment of inertia of the body about the axis of rotation through OO is II, and the total mass is mm. When the body is displaced as shown, the weight mmgg causes a restoring torque ττ zz = (mmgg)(dd sin θθ) (1) Fig. 1 The restoring torque on the body is proportional to sin θθ, not to θθ. However, for small θθ, sin θθ θθ, so the motion is approximately simple harmonic. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 1 / 12

2 Using the rotational analog of Newton s second law for a rigid body, ττ zz = IIαα zz, we find A body doesn t have just one moment of inertia. In fact, it has infinitely many, because there are infinitely many axes about which it might rotate. But there is a simple relationship, (mmggdd)θθ = IIαα zz = II dd2 θθ dddd 2 dd 2 θθ dddd 2 = mmggdd θθ (3) II called the parallel-axis theorem, between II cm (moment of inertia of a body about an axis through its center of mass) and II PP (moment of inertia about any other axis parallel to the origin axis) (Fig. 3): Comparing this with the equation for SHM, aa xx = (kk mm)xx, we see that the role of (kk mm) for the spring-mass system is played here by the quantity (mmggdd II). Thus II PP = II cm + MMdd 2 (6) where MM is the mass of body and dd is the distance between two parallel axes. ωω = mmggdd II TT = 2ππ II mmggdd (4) (5) From Eqs. (5), (6) (mm = MM), II cm = (1 12)MMLL 2, and Fig. 4(a), the period TT of a slender rod with length LL is Figure 2 gives moments of inertia for several familiar shapes in terms of their masses and dimensions. Fig. 2(b) shows that TT = 2ππ LL2 + 12dd 2 12ggdd (7) the moment of inertia of a rectangular plate through center of mass is II cm = (1 12)MM(aa 2 + bb 2 ), however, if aa bb(= LL) then it approximately becomes II cm = (1 12)MMLL 2 as Fig. 2(a). From Eqs. (5), (6) (mm = MM), II cm = (1 2)MMRR 2, and Fig. 4(b), the period TT of a solid cylinder with radius RR is TT = 2ππ RR2 + 2dd 2 2ggdd (8) Fig. 2 Moments of Inertia of Various Bodies Fig. 3 The parallel-axis theorem. Fig. 4 Various physical pendulums 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 2 / 12

3 The balance disk has a moment of inertia II about its axis. The twisted steel wire exerts a restoring torque ττ zz that is proportional to the angular displacement θθ from the equilibrium position. We write ττ zz = κκκκ, where κκ is a constant called the torsion constant. Using the rotational analog of Newton s second law for a rigid body, ττ zz = IIαα zz = IIdd 2 θθ/ddtt 2, we find κκκκ = IIII or dd 2 θθ dddd 2 = κκ II θθ (9) Fig. 5 A graph of the period TT as a function of distance dd from the center of mass for a 50cm-length slender rod pendulum. This equation is exactly the same as aa xx = (kk mm)xx for simple harmonic motion, with xx replaced by θθ and kk mm replaced by κκ II. So we are dealing with a form of angular simple harmonic motion. The angular frequency ωω and period TT are given by ωω = kk mm and TT = 2ππ mm kk, respectively, with the same replacement: ωω = κκ II (10) TT = 2ππ II κκ (11) Fig. 6 A graph of the period TT as a function of distance dd from the center of mass for a 10cm-radius solid cylinder pendulum 2. Torsion Pendulum A restoring force on a body undergoing periodic motion originates in difference ways in difference situations. Figure 7 shows a kind of torsion pendulum which consists of an elastic object such as a thin steel wire. When it is twisted, it exerts a restoring torque in the opposite direction. Fig. 7 The steel wire exerts a restoring torque that is proportional to the angular displacement θθ, so the motion is angular SHM. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 3 / 12

4 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 Rotary Motion Sensor (RMS) 1 Measures rotational or linear position, velocity and acceleration Slender Rod (or Long Rectangular Plate) 1 Length: 500mm Width: 20mm Pivot Point (Holes): 60, 100, 144, 190, 230mm from center of gravity Spherical Cylinder (or Disk) 1 Radius: 100mm Pivot Point (Holes): 30, 50, 70, 90mm from center of gravity Upper Wire Clamp Lower Wire Clamp 1 set Clamp wires. Wires (3 ea) (in the case) 1 set Exerts a restoring torque when twisted. Material: Steel Diameter: 0.8mm, 1.2mm, 1.6mm Balance Disk 1 Has a moment of inertia II = (1 2)MMRR 2 about its axis. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 4 / 12

5 Item(s) Qty. Description A-shaped Base Support Rod (600mm) 1 set Provide stable support for experiment set-ups. Ruler 1 Measures length. String / Scissors Shared Exerts a torque to twist a wire. Electronic Balance Shared Measures mass. 2. Details (1) Force Sensor Refer to the Circular Motion and Centripetal Force lab manual. (2) Rotary Motion Sensor It contains a small photogate sensor and an optical code wheel on which dark bands are printed in line. As the shaft of the sensor rotates, the bands block the infrared beam of the photogate, which provides very accurate signals for positioning or timing. The Rotary Motion Sensor is a bidirectional angle sensor designed to measure rotational or linear position, velocity and acceleration. It includes a removable 3-step pulley with 10mm, 29mm, and 48mm diameters. This allows you to convert a linear motion into a rotational motion. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 5 / 12

6 Procedure Experiment 1. Physical Pendulum (Slender Rod or Long Rectangular Plate) (1) Set up equipment. (3) Run Capstone software. 1 The interface automatically recognizes the RMS. Mount the RMS on the support rod so that the shaft of the sensor is horizontal (parallel to the table). 2 Adjust the sample rate of measurement. - [Rotary Motion Sensor]: Hz (2) Attach the slender rod to the RMS. Use the mounting thumbscrew to attach the slender rod to the shaft of the sensor through the end hole of the rod, so the pivot point is 230mm above the center of gravity. 3 Add a [Graph], and then select [Time(s)] for the xx-axis and [Angle(rad)] for the yy-axis. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 6 / 12

7 (4) Begin recording data. Click [Record] and then let the pendulum swing. 1 With the pendulum on the equilibrium position, click [Record] to begin recording data. 2 Gently start the pendulum swinging with a small amplitude (within 5 ). 3 After 5~6 oscillations, click [Stop] to end recording data. 3 Repeat measuring times for all oscillations and find the period of oscillation. Also, calculate and record the theoretical period of oscillation based on the length dd from the pivot point to the center of gravity TT average (s) TT theory (s) tt nn (s) TT = tt nn tt nn 1 (s) (5) Find the period TT of oscillation. 1 Choose any reference point of measurement (for example, peaks or zero up-crossings). TT = 2ππ LL2 + 12dd 2 12ggdd LL = 500mm dd = 230, 190, 144, 100, 60mm (7) 2 Use [Show coordinate ] to read off the time of the point. (6) Repeat measurement. Repeat steps (4) and (5) for the holes that are dd = 230mm, 190mm, 144mm, 100mm, and 60mm from the center hole. (7) Plot a TT-dd graph. Using your results in step (6), plot a TT-dd graph and compare it with Fig. 5 in Theory section. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 7 / 12

8 Experiment 3. Torsion Constant Q How does the period of this pendulum change when the pivot point moves towards the center of gravity? If it does not steadily increase or decrease, at what pivot point does the pendulum have minimum TT. Also, use Eq. (7) to calculate dd under the condition of minimum TT, and compare the theoretical value with your result. (1) Set up your equipment. A Q When the amplitude of this physical pendulum increases, should its period increase or decrease? Why? A 1 Slip the lower wire clamp onto the support rod. 2 Clamp the RMS at the top of the support rod so that the Experiment 2. Physical Pendulum (Solid Cylinder) Repeat the procedure of expt. 1 using a disk. shaft of the sensor is vertical. 3 Align the guide of the upper wire clamp with the slot of the shaft of the RMS. Slide the upper wire clamp onto the shaft and firmly tighten the thumbscrew. 4 Clamp each ends of the wire under the thumbscrew of the upper/lower wire clamp. Be sure that the elbow of the bend in the wire fits snugly against the axle of the thumbscrew. 5 Connect the sensors to the interface. TT = 2ππ RR2 + 2dd 2 2ggdd (8) RR = 100 mm dd = 90, 70, 50, 30 mm 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 8 / 12

9 6 Wind a string around the largest pulley. (2) Set up Capstone software. 1 Configure the Rotary Motion Sensor. - Click the RMS icon and then click the properties button ( ). - Select [Large Pulley (Groove)] for [Linear Accessory]. - [Change Sign] switches the sign of collected RMS data, which depends on the setup status or the rotational direction of the shaft. Check [Change Sign] if required. Caution When you slide the 3-step pulley onto the shaft of the RMS, be sure to align the guide of the pulley with the slot of the shaft. 2 Configure the Force Sensor. Caution If the retaining ring of the sensor shaft gets entangled in a string, SLOWLY and CAREFULLY remove the string. (NEVER apply a firm quick jerk to the string, which causes the retaining ring to warp, and as a result, the sensor to fail.) If it becomes warped, suspend your experiment immediately and visit lab office to replace the sensor. - Click the FS icon and then click the properties button ( ). - Check [Change Sign]. (The sign of FS data is initially negative for the pulling force.) 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 9 / 12

10 3 Configure calculator. (4) Begin recording data. Define the torque ττ as below, where [Force(N)] is measured data by the Force Sensor, and r is the radius of the 3 rd (largest) pulley (= 24mm). Hold the force sensor parallel to the table at the height of the largest pulley and slowly pull it straight out. If the angle shows negative, change the sign of RMS output (see step (2)-1). 4 Add a graph. Select [Rotary Motion Sensor Angle(rad)] for the xx-axis and [ττ(nm)] (defined in step3) for the yy-axis. (5) Analyze your graph. Find the torsion constant κκ. (3) Zero the Force Sensor. 1 Click [Select range(s) ] icon and then drag the data range of interest. NOTE To zero the sensor, press the [Zero] button on it WITH NO FORCE exerted on the sensor hook. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 10 / 12

11 2 Click [Select curve fits ] and select [Linear: mt+b] to find linear fit for selected data points. The torsion constant κκ is equal to the slope of the ττ-θθ graph. (2) Set up your equipment. Use the setup detailed in expt. 3. Remove the string and attach the balance disk to the 3-step pulley with the thumbscrew. (Be careful not to attach the disk directly on the shaft without the pulley.) (6) Repeat measurement for other wires. Repeat steps (4) to (5) using other wires. (3) Configure Capstone software. Follow the setup instruction of the experiment 1. Change [Sample Rate] to Hz or Hz. Experiment 4. Torsion Pendulum (Angular SHM) (4) Begin recording data. (1) Calculate the moment of inertia of the balance disk. Measure the radius and the mass of the balance disk and calculate the theoretical value of II. (Suppose the balance disk is a perfect solid cylinder and apply the relationship II = (1 2)MMRR 2.) Click [Record]. Twist the balance disk about 120~180 and release it. Keep recording data for about 5-6 oscillations and stop recording data. Determine the time for each period of oscillation and verify Eq. (11). TT = 2ππ II κκ (11) (5) Change the wire and repeat step (4). 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 11 / 12

12 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 from your lab computer. Turn off your lab Computer. Tighten all thumbscrews in position. Put the Wires in the storage case. Leave the Spools of String, Scissors in the basket on the lecture table. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 12 / 12