1: 2:
(ta initials) first name (print)
last name (print)
brock id (ab13cd)
(lab date)
Experiment 2
Angular Motion A friendly reminder: Is this your scheduled experiment? See the previous page.
In this Experiment you will learn • some basic principles of rotational motion around a central pivot point • to measure the specific rotational inertia Id of a solid disc by a multistep method • to manipulate equations and have them disclose information of interest to you • to use a computer-based fitting program to enhance the analysis of your data • to apply error analysis to experimental results and thus make your results relevant.
Prelab preparation Print a copy of this Experiment to bring to your scheduled lab session. The data, observations and notes entered on these pages will be needed when you write your lab report and as reference material during your final exam. Compile these printouts to create a lab book for the course. To perform this Experiment and the Webwork Prelab Test successfully you need to be familiar with the content of this document and that of the following FLAP modules (www.physics.brocku.ca/PPLATO). Begin by trying the fast-track quiz to gauge your understanding of the topic and if necessary review the module in depth, then try the exit test. Check off the box when a module is completed. FLAP PHYS 1-1: Introducing measurement FLAP PHYS 1-2: Errors and uncertainty FLAP MATH 1-4: Solving equations FLAP MATH 1-6: Trigonometric functions Webwork: the Prelab Angular Motion Test must be completed before the lab session Important! Bring a printout of your Webwork test results and your lab schedule for review by the !
TAs before the lab session begins. You will not be allowed to perform this Experiment unless the required Webwork module has been completed and you are scheduled to perform the lab on that day.
Important! Be sure to have every page of this printout signed by a TA before you leave at the end !
of the lab session. All your work needs to be kept for review by the instructor, if so requested.
CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT! 10
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Torque acting on a rotating body To understand what torque is, let us consider a rigid body with its centre of mass at a point r = 0, constrained to rotate about that point. The radius vector ~r of a point on the body has the origin at r = 0 and ends at the point, a distance r from the origin, as shown in Figure 2.1. ~ is applied at that point, in the plane If a force F of rotation of the disc, at a distance r from the centre, and at an angle φ to the radius vector ~r, the magnitude of the turning torque τ produced by this force is: τ = rF sin φ (2.1) The torque τ depends both on the distance from the centre of rotation and on the direction of the applied force F~ . Decomposing F~ into a radial com~r and a tangential component F ~t so that ponent F ~ ~ ~ F = Fr + Ft , it can be seen that only the tangential component of F~ causes torque and affects the magnitude of τ . A force applied through the centre of rotation has a zero tangential component (φ = 0, Ft = 0) and the radial component alone produces no torque and will not cause angular acFigure 2.1: Rotational coordinates celeration. As the angle of the force changes, so does the torque experienced by the body; for a given magnitude of the force, F , the maximum torque is ~ = F~t is perpendicular to ~r and τ = rFt = rF . produced when F Resisting the force F~ is the total mass P M of the rotating body. Suppose that this mass consists of many particles of mass mi so that M = i mi . In a translational motion, the force acts equally on all the P component particles of the body at once, according to Newton’s second Law F~ = i (mi~a) = M~a. In a rotation, the rotational effect of F~ on a particle is proportional to the distance r from the centre of rotation. While the entire rigid body experiences a single angular accelaration α, common to all its particles, each of them will experience a tangential acceleration ai that is proportional to the distance ri from the centre of rotation. The rotational moment of inertia I of a rigid body composed of many particles is simply the sum of the individual rotational moments of inertia of all particles: X I= mi ri2 (2.2) i
For a thin hoop, where all the particles are located at the same common radius away from the centre, ri = R, Eq. 2.2 reduces to I = M R2 ; for other shapes, the calculation may not be so simple. The rotational form of the Newton’s Second Law is τ = Iα: X X F = (mi a) = M a −→ τ= (mi ri2 α) = Iα. i
i
The torque τ plays the same role for rotational motion that the force F plays for translational motion.
Determining the rotational inertia of a disc Consider a homogeneous disc of radius R and mass M constrained to rotate without friction around the centre of mass. A massless string is wrapped around the outer edge of the disc and connected to a mass m that is subjected to the force of gravity Fg , as shown in Figure 2.2. The string experiences a tension T
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EXPERIMENT 2. ANGULAR MOTION
due to the weight of m; at the other end of the string the same tension T acts on the edge of the disc at a distance r from the centre of rotation. The displacement of the falling mass is given by Equation 2.3: y = y0 + v0 t + at2 /2
(2.3)
When a force is applied, the disc, initially at rest, begins to spin as m falls with linear acceleration a=g−
T . m
(2.4)
The rotational acceleration of the disk is α=
a τ Tr = = . r I I
(2.5)
By combining Eqs. 2.4 and 2.5, the rotational inertia I of the disc can be expressed in terms of a as follows: g I = mr 2 −1 . (2.6) a Measuring the acceleration of the falling mass, and comparing it to the acceleration of the free fall, yields an operational measurement of the moment of inertia of the disk.
Figure 2.2: A falling block causes the disc to rotate
Procedure The rotational inertia Id of a steel disc will be determined indirectly in three steps: 1. measure the rotational inertia Ic of a spinning cradle that will accept the disc; 2. measure the rotational inertia It = Ic + Id of the cradle and disc combination; 3. calculate Id from the difference of these two results. The cradle consists of a plastic disc bolted to a metal drum of four stacked pulleys of varying diameters. The cradle is centered on a ball-bearing post and can rotate nearly friction-free around this vertical axis as long as it is not rubbing against the side of the post. A string, wrapped around one of the drum pulleys and attached at the other end to a suspended mass m, allows a torque τ to be applied to the cradle at radius r of the pulley. The string leaving the drum must be perpendicular to the axis of rotation and to the radius vector, as shown in Figure 2.3.
? Suppose that the string was not perpendicu-
Figure 2.3: Experimental setup
lar. What adjustments, if any, would you need to make to your torque calculations?
13 The pulley A in Figure 2.3 is part of a photogate system. The C-shaped photogate has an infrared transmitter at one end and an infrared detector at the other end. As the pulley rotates, the spokes interrupt the beam, turning on and off the red light-emitting diode (LED). The LabPro device detects and counts these changes and Physicalab calculates the distance that the string has moved. The pulley has ten spokes and a circumference of 155 mm, so there are twenty pulses transmitted every rotation of the pulley. Each pulse represents a radial distance of 7.75 mm travelled by the string and hence the same distance y moved by the mass m. After the data acquisition is initiated, LabPro waits for the first transition and marks this event with the elapsed time and assigns it an initial distance y = 0. After all subsequent transitions, the new time and total distance travelled are sent. Note that the time t = 0 set by LabPro will not likely coincide with the start of the motion.
Data gathering and analysis using Physicalab 1. Wrap the string, trying not to overlap the strands, around the second smallest pulley (r = 0.02282 m) on the cradle and arrange the string path as shown in Figure 2.3. 2. Place a weight mw on the mass holder and rotate the cradle to raise these to the top of the assembly. Hold the cradle in place by putting a weight on the edge of the platform.
? Consider the geometry of the system, is the cradle likely to rub against the post? 3. Shift focus to the Physicalab software. Select Dig2, the channel that the photogate should be connected to, then choose to collect a nominal 20 points with 0.5 seconds between points. 4. Remove the weight from the platform. The cradle will begin to rotate. Press Get data to start gathering the photogate data. Note that the platform will not likely have a zero velocity v0 = 0, at time t = 0 when the data acquisition begins.
? Will this arbitrary starting point in your data collection have any effect on the quantity that you want to determine from the fit? 5. Stop the data acquisition and the platform rotation before the mass holder reaches the end of it’s trajectory; it will save you the trouble of having to re-spool the string on the pulleys. 6. At the end of the run, review your graphed data; it should approximate a smooth curve that represents the falling of a mass m under constant acceleration a. If this is not the case, repeat the trial.
? Could you delete a portion of your data set and still get the desired result from your fit? How might the result be affected by such a change? 7. Select fit to: y= and enter A+B*x+C*x**2/2 in the fitting equation box. Click Draw to perform a fit of your data. Click Send to: to email yourself and your partner a copy of the graph for later inclusion in your lab reports.
? How do you determine which of the fit parameters represents the acceleration a?
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EXPERIMENT 2. ANGULAR MOTION
mw (kg)
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
a1 (m/s2 ) a2 (m/s2 ) hai (m/s2 ) Table 2.1: Acceleration data for unloaded cradle assembly
Part 1: Determining the rotational inertia of the cradle • Fill in Table 2.1, then calculate an average acceleration hai for the various masses. You may now wish to evaluate the rotational inertia Ic for the unloaded cradle, using Equation 2.6 and the value of hai for a given mw in Table reftab:cradledata. This would not likely lead to the correct Ic value because m in Equation 2.6 represents the total mass m = mw + mh − mf that causes the tension T on the string. In your experiment, the effect of the applied mass mw and that of the mass holder mh is diminished by an unknown mass mf required to overcome the static and dynamic friction experienced by the platform and pulleys. You will have noted that the cradle sometimes begins to rotate with only the mass holder attached, when mw = 0. In this case, mh > mf and to prevent the system from rotating, mw would need to be negative (or the mass holder would need to be lighter). To get around the difficulty of not knowing mf , you could determine mh by weighing the mass holder, then use two sets of values from Table 2.1 and solve two simultaneous equations to eliminate the unknown variable mf . A better method makes use of the whole set of tabulated data values (hai, mw ) that are available while making unnecessary a knowledge of both mh and mf . Begin by rearranging Equation 2.6 so that mw is expressed as a function of a: g g Ic Ic = (mw + mh − mf )r 2 → mw = − 1 ≈ (mw + mh − mf )r 2 a + (mf − mh ). a a gr 2
(2.7)
The simplification is valid only if a ≪ g so that (g/a − 1) ≈ (g/a). Then the resulting equation is that of a straight line with slope Ic /(gr 2 ) and y-intercept (mf − mh ). • Enter into an empty data window your set of (hai, m) coordinates. The scatter plot of your data should show a linear behaviour. Select fit to: y= and enter A+B*x in the fitting equation box. Click Draw and record below the fit parameters A and B along with their appropriate units. A = ............... ± ...............
B = ............... ± ...............
• Calculate Ic using Equation 2.7 and use the appropriate error propagation rules to evaluate the corresponding uncertainty ∆Ic : Ic = ....................... = ....................... = .......................
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∆Ic = ....................... = ....................... = ....................... Ic = ............. ± .............
Part 2: Determining the rotational inertia of cradle and disc • Use the peg on the steel disc to center it on the cradle, then place a small piece of paper under the disc to prevent it from slipping. Repeat the previous steps to determine the rotational inertia It of cradle-plus-disc. mw (kg)
0.030
0.040
0.050
0.060
0.070
0.080
0.090
0.100
a1 (m/s2 ) a2 (m/s2 ) hai (m/s2 ) Table 2.2: Acceleration data for cradle assembly with disc
A = ............... ± ...............
B = ............... ± ...............
It = ....................... = ....................... = ....................... ∆It = ....................... = ....................... = ....................... It = ................. ± .................
Part 3: Determining the rotational inertia of the disc • Subtract Ic from It to obtain Id , the rotational inertia of the disc alone.
Id = ....................... = ....................... = ....................... ∆Id = ....................... = ....................... = ....................... Id = ................. ± ................. • Measure the mass M and radius R of the steel disc. Also weigh the mass mh of the mass holder. M = ......... ± .........
R = ......... ± .........
mh = ......... ± .........
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EXPERIMENT 2. ANGULAR MOTION
• Use the appropriate equation to calculate the theoretical rotational inertia for the disc that you used, as well an estimate of the error.
Id(th) = ...................... = ...................... = ...................... ∆Id(th) = ...................... = ...................... = ...................... Id(th) = ................. ± ................. • Perform a trial using the smallest pulley with pull mass m = 30 g on the unloaded cradle. Compare this a with the previous result for a obtained using the second smallest pulley and the same pull mass. The pulley radii from smallest to largest are: 1.539 cm, 2.282 cm, 3.350 cm and 4.343 cm. a(r = 1.539) = .................,
a(r = 2.282) = .................
? Do these two a values obtained by varying the pulley radius r agree with the results predicted by the theory?
Lab report Go to your course homepage on Sakai (Resources, Lab templates) to access the online lab report worksheet for this experiment. The worksheet has to be completed as instructed and sent to Turnitin before the lab report submission deadline, at 11:00pm six days following your scheduled lab session. Turnitin will not accept submissions after the due date. Unsubmitted lab reports are assigned a grade of zero. Notes:............................................................................. ...................................................................................... ...................................................................................... ...................................................................................... ...................................................................................... ...................................................................................... ...................................................................................... ...................................................................................... ...................................................................................... ...................................................................................... ...................................................................................... ...................................................................................... ......................................................................................
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