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. Related topics Rotation, angular velocity, torque, angular acceleration, angular moment, moment of inertia, rotational energy. Tasks 1. Measure 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 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 these two energies. Fig 1: Experimental set-up with turntable P2131315 PHYWE Systeme GmbH & Co. KG All rights reserved 1
Moment of inertia and angular acceleration with Cobra3 Equipment 1 Tripod -PASS- 02002-55 1 Precision bearing 02419-00 1 Inertia rod 02417-03 2 Turntable with angle scale 02417-02 1 Aperture plate for turntable 02417-05 1 Connecting cord, l = 150 cm, red 07364-01 1 Connecting cord, l = 150 cm, blue 07364-04 1 Light barrier with Counter 11207-30 1 Adapter, BNC plug/4 mm socket 07542-26 1 Foil capacitor in casings G1, 0.1 μf 39105-18 1 Power supply 5 V DC/2.4 A 11076-99 1 Precision pulley 11201-02 2 Right angle clamp PASS- 02040-55 1 Support rods PASS- l = 400 mm 02026-55 2 Bench clamp -PASS- 02010-00 1 Silk thread, l = 200 m 02412-00 1 Circular level 02122-00 1 Weight holder, 10 g 02204-00 1 Weight holder, 1 g 02407-00 20 Slotted weight, 1 g, 03916-00 10 Slotted weight, 10 g, black 02205-01 2 Slotted weight, 50 g, black 02206-01 1 Holding device with cable release 02417-04 1 Measuring tape, l = 2 m 09936-00 Set-up and procedure 1. Set the experiment up as seen in Fig. 1. Connect the light barrier with counter according to Fig. 2 to the cable release. 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. 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 precision pulley 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 sector mask and does not disturb the movement after release. Start 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. 2 PHYWE Systeme GmbH & Co. KG All rights reserved P231315
Fig 2: Connection of the compact light barrier to the Cobra3 Basic Unit P2131315 PHYWE Systeme GmbH & Co. KG All rights reserved 3
Moment of inertia and angular acceleration with Cobra3 Fig. 3: Cobra3 Translation / Rotation settings Start the measurement with the "Continue" button and release the turntable. Data recording should start automatically. Stop 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 ma such that the torque T = r a 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 mn m 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. Start the turntable by a shove with the hand. Nearly unaccelerated movement should be observable. 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 4 PHYWE Systeme GmbH & Co. KG All rights reserved P231315
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). Theory and evaluation The angular momentum J of a single particle r at place with velocity v, mass m and momentum p = m v is defined as J = r p and the torque T from the force F is defined as t = r F, With torque and angular momentum depending on the origin of the reference frame. The change of J in time is d J dt and with = d d r p ( r p) = p+ r d d t d t d t d r p = v m v = 0 d t and Newton's law F = d p d t the equation of motion becomes T = d J d t (1) For a system of N particles with center of mass R c.m. and total linear momentum P = m i v i is N J = i=1 N m i ( r i R c.m. ) v i + m i R c. m. v i = J c. m. + R c.m. P i=1 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 1 r j fixed. The velocity of particle i may be written as v i = ω r i with vector of rotation ω = d φ d t Constant throughout the body. Then (2) J = m i r i v i = m i r i ( ω r i ) with a ( b c) = b( a c) c( a b) is P2131315 PHYWE Systeme GmbH & Co. KG All rights reserved 5
Moment of inertia and angular acceleration with Cobra3 J = m i ( ω r i 2 r i ( r i ω)) with r i ω = x i ω x + y i ω y + z i ω z and J z = ω z m i (r i 2 z i 2 ) ω z m i z i x i ω y m i z i y i. The inertial coefficients or moments of inertia are defined as I x, x = m i (r i 2 x i 2 ) I x, y = m i x i y i I x, z = m i x i y i (3) and with the matrix ^I={I k,l } it is J = ^I ω and for the rotational acceleration α = d ω dt is then (4) T = d J = ^I d ω = ^I α dt dt The rotational energy is E = 1 2 m i v i 2 = 1 2 m i ( ω r i ) 2 = 1 2 I k,l ω k ω l Sum convention: sum up over same indices using ( a b )( c d ) = ( a c)( b d) ( a d )( b c) The coordinate axes can always 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 ω ^e z ω z ^e z ω with the unit vector ^e Z. The energy is then E = 1 2 I Z,Z ω2 The torque T = ma (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. So with (1), (2) and (3) d 2 φ T = I Z, Z = m d t 2 a gr a (6) ω(t ) = ω(t=0)+ m g r a a t = ω(t=0)+ T t (7) I Z,Z I Z,Z φ (t ) = φ (t=0)+ 1 2 m a g r a I Z, Z t 2 = φ (t=0)+ 1 2 T I Z, Z t 2 (8). 6 PHYWE Systeme GmbH & Co. KG All rights reserved P231315
The potential energy of the accelerating weight is with (8) and (7) E = m a gh(t ) = m a gφ (t) r a = 1 2 2 m a g 2 2 r a t 2 = 1 I Z, Z 2 I Z, Z ω2, thus verifying (4). If a weight m i is mounted to a rod in a distance r i to the fixed axis z, which is the axis of rotation perpendicular to the rod, with the rod lying along the y-axis, then are the coordinates of the weight (0, r i,0). According to (3) the moment of inertia about the z-axis 2 2 is I Z,Z = m i r i, about the x-axis 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. Select 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 "Scale 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: Fig. 4: Angle vs. square of time for one turntable P2131315 PHYWE Systeme GmbH & Co. KG All rights reserved 7
Moment of inertia and angular acceleration with Cobra3 Fig. 5: Angle vs. square of time for two turntables In Fig. 6 the angular acceleration values of Fig. 5 and Fig. 6 were plotted vs. the used torque T = m a gr 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 mn m s 2 = 268 kg cm 2 for two discs. Fig. 6: Angular acceleration vs. accelerating torque. 8 PHYWE Systeme GmbH & Co. KG All rights reserved P231315
Fig. 7 shows a plot of moment of inertia I z,z = T/a = m i r i2 + I Rod;z,z vs. the weight m i. The regression values yield r i2 = 0.047 m 2 r i = 21.7 cm and I Rod;z,z = 83 kg cm 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. 7: Moment of inertia vs. weight at r = 200 mm with constant torque of 3.09 mnm Fig.8 shows a plot I z,z = T/a = m i r i2 + I Rod;z,z vs. square of radius r i2. The regression values yield a mass m i = 240 g and I Rod;z,z = 82 kg cm 2 P2131315 PHYWE Systeme GmbH & Co. KG All rights reserved 9
Moment of inertia and angular acceleration with Cobra3 Fig. 8: Moment of inertia vs. square of radius with m = 2 100 g and constant torque of 3.09 mn m Fig 9: Bilogarithmic plot of angle vs. time 10 PHYWE Systeme GmbH & Co. KG All rights reserved P231315
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 φ 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 mn m. Since φ = 1 2 α t2, reads the plot 1 2 α = 0.103 1 s 2 α = 0.203 1 s 2 On the other hand T = 3.09 mn m α ^I ^I z = 150 kgcm 2 compared to theoretical ^I z = 2 80g (21cm) 2 + ^I rod, z = 70.6 kg cm 2 +72 kg cm 2 = 143 kg cm 2. P2131315 PHYWE Systeme GmbH & Co. KG All rights reserved 11