Moment of inertia and angular acceleration with Cobra3

Transcription

1 Moment of inertia and angular acceleration with Cobra3 LEP Related Topics Rotation, angular velocity, torque, angular acceleration, angular momentum, moment of inertia, rotational energy. Principle A known torque is applied to a body that can rotate about a fixed axis with minimal friction. Angle and angular velocity are measured over the time and the moment of inertia is determined. The torque is exerted by a string on a wheel of known radius with the force on the string resulting from the known force of a mass in the earth's gravitational field. The known energy gain of the lowering mass is converted to rotational energy of the body under observation. Tasks 1. Measure the angular velocity and angle of rotation vs. time for a disc with constant torque applied to it for different values of torque generated with various forces on three different radii. Calculate the moment of inertia of the disc. 2. Measure the angular velocity and angle of rotation vs. time and thus the moment of inertia for two discs and for a bar with masses mounted to it at different distances from the axis of rotation. 3. Calculate the rotational energy and the angular momentum of the disc over the time. Calculate the energy loss of the weight from the height loss over the time and compare. Equipment Tripod -PA Precision bearing Inertia rod Turntable with angle scale Aperture plate for turntable Cobra3 Basic Unit, UB Cobra3 power supply unit Translation/Rotation oftware Light barrier, compact Connecting cord, l =150 cm, red Connecting cord, l =150 cm, blue Connecting cord, l =150 cm, yellow Boss head upport rod, l = 250 mm Bench clamp -PA ilk thread, l = 200 m Circular level Weight holder, 10 g Weight holder, 1 g lotted weight, 1 g, natur. colour lotted weight, 10 g, black lotted weight, 50 g, black Holding device w.cable release Measuring tape, l = 2 m PC, WINDOW 95 or higher Fig.1: Experimental set up with turntable PHYWE series of publications Laboratory Experiments Physics PHYWE YTEME GMBH & Co. KG D Göttingen P

2 LEP Moment of inertia and angular acceleration with Cobra3 et-up and procedure 1. et the experiment up as seen in Fig.1. Connect the compact light barrier to the Cobra3 unit according to Fig. 2. Connect the COBRA3 Basic Unit to a UB port of the computer. Adjust the turntable to be horizontal it must not start to move with an imbalance without other torque applied. Fix the silk thread (with the weight holder on one end) with the screw of the precision bearing or a piece of adhesive tape to the wheels with the grooves on the axis of rotation and wind it several times around one of the wheels enough turns, that the weight may reach the floor. Be sure the thread and the wheel of the compact light barrier and the groove of the selected wheel are well aligned. Place the holding device with cable release in a way that it just holds the turntable on the "Aperture plate" and does not disturb the movement after release. tart the "measure" program and select "Gauge" > "Cobra3 Translation / Rotation", select the tab "Rotation" and set the parameters as seen in Fig.3. In the field "Axle diameter" you have to enter the diameter of the chosen wheel with groove on the axis of rotation. It may be best to start with the one with 15 mm radius and 30 mm diameter (the smallest), because with this one the highest number of turns of the disc is observable before the weight reaches the floor. Fig. 2. Connection of the compact light barrier to the Cobra3 Basic Unit Fig.3: Cobra3 Translation / Rotation settings tart the measurement with the "Continue" button and release the turntable. Data recording should start automatically. top data recording just before the weight reaches the floor by key press (in order not to have irrelevant data at the end of your recorded data). 1. Record measurements with different accelerating weights m a up to 100 g. For heavy weights you may set the "Get value" time to 200 ms and for low weights you may set this time to higher values. Record measurements with the thread running in different grooves adjust the position of the light barrier to align thread and groove and wheel of light barrier (thread has to be horizontal) and enter the correct axle diameter in the "Rotation" chart and choose especially the weight on the end of the thread m a such that the torque T ra F is constant e.g. m a = 60 g for r a = 15 mm and m a = 30 g for r a = 30 mm and m a = 20 g for r a = 45 mm, each time the torque being T = m a gr a = 8.83 mnm with the earth's gravitational acceleration g = 9.81 N/kg = 9.81 m/s 2. Also choose a weight with which you take a measurement for each groove radius. You may also take a measurement with the weight m a so light that it nearly does not affect the movement (like the weightholder of 1 g) but is enough to drive the light barrier's wheel. tart the turntable by a shove with the hand. Nearly unaccelerated movement should be observable. yellow blue red 2. Record measurements with two turntables mounted on the precision bearing with weight values used on the single turntable and with double the weight used on the single turntable for comparison. Remove the turntables and mount the inertia rod to the precision bearing and the two weight holders symmetrically to the rod with both the same distance to the axis r i. Take measurements with varied masses m i at constant r i and also with constant masses m i at varied r i (both masses of course still mounted symmetrically) accelerated with the same weight m a (or with the same series of weights for high precision). 2 P PHYWE series of publications Laboratory Experiments Physics PHYWE YTEME GMBH & Co. KG D Göttingen

3 Moment of inertia and angular acceleration with Cobra3 LEP Theory and evaluation The angular momentum J of a single particle at position r with velocity n, mass m and momentum p m n is defined as J r p The inertial coefficients or moments of inertia are defined as I x,x a m i 1r 2 i x 2 i 2 (3) I x,y a m i x i y i and the torque T from the force F is defined as T r F, I x,z a m i x i z i and with the matrix Î 5I k,l 6 it is with torque and angular momentum depending on the origin of the reference frame. The change of J in time is d J d 1 r p 2 d r p r d p J Î v and for the rotational acceleration dv a is then (4) and with d r and Newtons's law becomes dj T p n m n 0 dp F the equation of movement For a system of N particles with center of mass R c.m. and total linear momentum P a m i n i the angular momentum is (1). N J a m i 1 r i R c.m.2 N n i a m i R c.m. n i i 1 i 1 dj T The rotational energy is Î dv I ˆ a E 1 2 a m i n 2 i 1 2 a m i 1v r i I k,l v k v l um convention: sum up over same indices using 1 a b 21 c d 2 1 a c 21 b d 2 1 a d 21 b c 2. The coordinate axes can allways be set to the "principal axes of inertia" so that none but the diagonal elements of the matrix I k,k 0. In this experiment only a rotation about the z-axis can occur and v êz v z ê z v with the unit vector ê z. The energy is then J c.m. R c.m. P. E 1 2 I z,zv 2 (5) Now the movement of the center of mass is neglected, the origin set to the center of mass and a rigid body assumed with r i r j fixed. The velocity of particle i may be written as n i v r i with vector of rotation The torque T = m a (g-a)r a is nearly constant in time since the acceleration a = a r a of the mass m a used for accelerating the rotation is small compared to the gravitational acceleration g =9.81 m/s 2 and the thread is always tangential to the wheel with r a. o with (1), (2) and (3) v d w constant throughout the body. Then (2) T I z,z d2 w 2 m a gr a (6) J a m i r i n i a m i r i 1 v r i2. v1t2 v 1t 02 m a gr a I z,z t (7) With a 1 b c 2 b 1 a c 2 c 1 a b 2 is w1t2 w 1t m a gr a I z,z t 2 (8). J a m i 1 v r 2 i r i 1 r i v 22 with The potential energy of the accelerating weight is and r i v x i v x y i v y z i v z J z v z a m i 1r i 2 z i 2 2 v z a m iz i x i v y a m iz i y i. E = m a gh(t) = -m a gw(t) r a thus verifying (4) m a 2 g 2 2 r 172 a 1 t 2 I z,z 2 I z,z v 2, PHYWE series of publications Laboratory Experiments Physics PHYWE YTEME GMBH & Co. KG D Göttingen P

4 LEP Moment of inertia and angular acceleration with Cobra3 If a weight m i is mounted to a rod that can rotate about the fixed axis z perpendicular to it in a distance r i, maybe the rod lying along the y-axis, then are the coordinates of the weight (0,r i,0)and according to (3) is the moment of inertia about the 2 2 z-axis I z,z m i r i, about the x-axis it is I x,x m i r i and about the y-axis it is I y,y, = 0. Plot the angle of rotation vs. the square of time using "Analysis" > "Channel modification". The correct zero of time t 0 (the begin of movement without digital distortion) may be found with the "Regression" tool of measure used on the "omega" values or on the data table as the first value where the angle exceeds zero. elect time as source channel and enter the operation "(t-t 0 )^2" with your actual value t 0 in your channel modification window. Then use "Measurement" > "Channel manager " to select the square of time as x-axis and "angle" as y-axis. The slope of the obtained curve is half the angular acceleration and may be determined with the "Regression" tool. Fig. 4 and Fig. 5 show obtained data of several measurements which were put into one diagram with "Measurement" > "Assume channel " and scaled to the same value with the "cale curves" tool and the option "set to values". The linearity vs. the square of time can be seen well. Compare these slope values with the slope of the "omega" values of the raw data: In Fig. 6 the angular acceleration values of Fig. 5 and Fig. 6 were plotted vs. the used torque T = gm a r a with "Measurement" > "Enter data manually ". The inverse slope is then the moment of inertia and reads I = 13.3 mnms 2 = 133 kg cm 2 for one disc and I = 26.8 mnms 2 = 268 kgcm 2 for two discs. Fig. 6 : Angular acceleration vs. accelerating torque. Fig. 7 shows a plot of moment of inertia I z,z = T/a = m i r i 2 + I Rod;z,z vs. the weight m i. The regression values yield r i 2 = m 2 1 r i = 21.7cm and I Rod;z,z = 83 kgcm 2. The angular acceleration data were evaluated from the measurements with the "Regression" function from the "omega" values and entered with "Measurement" > "Enter data manually " into a "Manually created measurement". The moment of inertia values were calculated by "Analysis" > "Channel modification". Fig. 4: Angle vs. square of time for one turntable Fig. 5: Angle vs. square of time for two turntables Fig. 7: Moment of inertia vs. weight at r = 200 mm with constant torque of 3.09 mnm 4 P PHYWE series of publications Laboratory Experiments Physics PHYWE YTEME GMBH & Co. KG D Göttingen

5 Moment of inertia and angular acceleration with Cobra3 LEP Fig.8 shows a plot I z,z = T/a = m i r i 2 + I Rod;z,z vs. square of radius r i 2. The regression values yield a mass m i = 240 g and I Rod;z,z.= 82 kg cm 2 Else you may evaluate the data using a bilogarithmic plot but then it's crucial to correct the data for the right zero point of time and angle using "Analysis" > "Channel modification " subtracting t 0 and adding/subtracting a w 0 and making a plot of the changed channels with "Measurement" > "Channel manager ", erasing the values lower than zero in the data table and using the "Display options" tool to set the scaling of both axes to "logarithmic". Fig. 7 shows an example for the bar with 2 80 g mounted 21 cm from the axis and accelerated with a weight of 21 g at 15 mm i.e. a torque of 3.09 mnm. ince w 1 2 a t 2, reads the plot 1 2 a s 2 1 a s 2, on the other hand T = 3.09 mnm compared to theoretical a Iˆ 1 Îz 150 kgcm 2 Î z 2 80 g 121 cm2 2 Î rod,z 70.6 kgcm 2 72 kgcm 2 Fig. 8: Moment of inertia vs. square of radius with m = g and constant torque of 3.09 mnm = 143 kgcm 2. Fig. 9 Bilogarithmic plot of angle vs. time PHYWE series of publications Laboratory Experiments Physics PHYWE YTEME GMBH & Co. KG D Göttingen P

6 LEP Moment of inertia and angular acceleration with Cobra3 6 P PHYWE series of publications Laboratory Experiments Physics PHYWE YTEME GMBH & Co. KG D Göttingen