EGR244 ROLLING CYLINDER LAB

EGR244 Lab 2: ROLLING CYLINDERTracy Lu

(Dated: 10 March 2017)

The purpose of this experiment was to explore the principles of physics, inertia, Newtons laws, and EulersLaws and their effects on four cylindrical rigid bodies. These bodies included solid cylinders and hollowcylinders composed of aluminum or acrylic, and were released from rest along a track that was elevated to5 different bank angles. Using the principles listed above, equations were derived to theoretically model themotion of and forces acting on the bodies as they traveled down the tracks. At the end of the experiment theexperimental data was compared to the theoretical model. In general, the standard deviations and deviationfrom expected values increased as bank angle increased, but overall the experiment yielded very similarresults to the predictions despite minor discrepancies due to assumptions and small, influencing factors inthe experiment environment. The ideas from this lab could have many applications to any type of rolling orrotating motion, especially wheels, transportation, turbines, and power generation.

I. INTRODUCTION

Using an ultrasound position sensor to track positionvs. time, the experiment of different cylindrical bod-ies released from rest along a track elevated at differentbank angles demonstrates the principles of physics, in-ertia, Newtons Laws, and Eulers Laws and their effectson rigid bodies. With the principles listed above, equa-tions were derived to theoretically model the motion ofand forces acting on the bodies as they traveled down thetracks.

The net force equation, constraints, Eulers SecondLaw, and inertia, ultimately allow the derivation of anequation of motion. Using Newtons Laws and a free bodydiagram, shown in Figure 1, a net force equation can bederived. Because the body is on an incline, the iner-tial reference frames axis is tilted at the same angle asthe bank angle. An essential force that is incorporatedinto the net force equation is frictional1 force, which isthe force created by two surfaces contacting and slid-ing against each other. Frictional force always opposesthe direction of motion and in this experiment occursbetween the cylindrical body and the track. The frictionalso creates an additional constraint equation used to de-rive the equation of motion. Another equation needed isEulers Second Law2, which requires that the sum of themoments of the external forces acting on the rigid bodymust equal the change in the angular momentum of therigid body. The moment of inertia of each body is alsoa significant factor influencing the motion of the rolling.Moment of inertia2 of a body is a body’s tendency to re-sist angular acceleration, and determined by its physicalcharacteristics.

The ideas from this lab could have many applica-tions to any type of rolling or rotating motion, especiallywheels and transportation or turbine design. For exam-ple, if you wanted an object to spin faster, it would bebetter to design the object with a smaller moment ofinertia. This could allow for more energy efficient trans-portation or generation.3

The rest of the report will go into detail about deriv-ing the theoretical models, the experimental setup, datacollection, and data analysis. Overall, despite slight de-viations, our experimental data was very close and exhib-ited the same general trends compared to the theoretical,expected calculations.

II. THEORY

Figure 1: Free Body Diagram labeling forces andexperimental design parameters for cylinder, track, andbank angle. These values are measured and referenced

throughout the experiment.

EGR244 ROLLING CYLINDER LAB 2

Parameter ValueR (Radius of Cylinder) about 4in

Gravity (g) 385.8267 in/ s −2

Mass (m) measured in Table 2Track length 84 in

n (number of bricks) brick height is 2.25inh (track height) 2.25(n+1) in T

O release gate locationx distance from Oθ bank angleG Center of Mass locationIg Moment of Inertia about GKg Radius of Gyration about GMg Moments about point GF Frictional ForceN Normal Force

Table 1: Parameter variables and corresponding valuesthat are referenced throughout experiment. Values areeither calculated experimentally through measurement,

or obtained from references1.

The free body diagram in Figure 1 shows the relation-ship between the forces acting on the cylinder. The netforce equation modelling this based on Newtons SecondLaw is

Fnet = ma = (mgsin(θ) − F )i+ (N −mgcos(θ))j (1)

Where m is the mass of the cylinder, mg is the gravita-tional force, mass times gravitational constant, θ is thebank angle, N is normal force, and frictional force is rep-resented by F. Note that aerodynamic drag has been leftout of the net force equation due to it being a negligiblevalue in comparison to the other forces.

However, more equations are needed to solve for theequation of motion. Because there is friction, the mo-tion is assumed to be rolling without slipping, and anadditional constraint can be applied, modeled by

x = −Rθ (2)

Where x is distance from the release gate and R is theradius of the cylindrical body.Eulers Second Law also requires that the sum of the mo-ments of the external forces acting on the rigid body mustequal the change in the angular momentum of the rigidbody. This gives another equation represented by

Mg = Ig θ = −Rj × (−F i+Nj) = −RFk (3)

Where θ is angular acceleration, k is the direction out ofthe page, and Ig is the moment of inertia

Ig = mk2g (4)

and k2g is the radius of gyration, the distance from anaxis at which the mass of a body may be assumed tobe concentrated.2 Radius of gyration squared was foundto be .5R2 for the solid bodies based on integrating the

formula for moment of inertia (r is a variable representingradius),

Is =

∫r2dm =

m

πR2

∫ R

0

∫ 2π

0

dθdr = mR2 (5)

For the hollow bodies, the moment of inertia and radiusof gyration was calculated by assuming an infinitesimallythin hoop, because the thickness of the hollow cylinderwas comparatively negligible. Radius of gyration squaredwas found to be R2 for the hollow bodies based on inte-grating the formula for moment of inertia (r is a variablerepresenting radius),

Ih =

∫r2ds =

m

2πR

∫ 2π

0

r3dθ = .5mR2 (6)

Where s is the arclength being integrated. Setting thek direction (out of the page) components equal to eachother

θ = −RF/Ig (7)

Substituting equations 2-5 into Eq(1), and then setting

the i components on each side of the equation equal toeach other (there is no acceleration in the j direction), anEquation of Motion is derived and algebraically simplifiedto

θ =−gRsin(θ)

K2g +R2

(8)

Then multiplying by R to convert from angular acceler-ation to linear acceleration,

x =−gR2sin(θ)

K2g +R2

(9)

where x is linear acceleration in the i direction.

III. EXPERIMENTAL METHODS

Type Weight R (in) Thickness Inner WallSolid Metal 526.4 g 3.9988 0.944 in n/aSolid Plastic 230.68.4 g 3.982 0.946in n/aHollow Metal 31.26 g 3.94 0.944 in 0.0614 inHollow Plastic 13.44 g 3.985 0.9311in .0614 in

Table 2: Measured variables and corresponding valuesthat are referenced throughout experiment for each typeof body used.

EGR244 ROLLING CYLINDER LAB 3

Figure 2: Picture of the cylindrical bodies. From top tobottom: Solid Metal, Solid Plastic, Hollow Plastic,

Hollow Metal

Figure 3: Closer photograph of top of track wherecylinder is released. The sensors, release gate, and

alignment mount are seen in more detail here.

The experiment used 4 different cylindrical bodies fortesting, solid metal, solid plastic, hollow plastic, hollowmetal, each with their own physical properties as shownin Table 1. The experiment was set up by using an alu-minum track with one end propped up against a varyingnumber of bricks that controlled the bank angle (Fig-ure 4). A release gate (Figure 3) held the cylinder inplace and at rest before it was pulled to release the bodyand trigger the start of data collection by the ultrasoundposition sensor which measures the time required for atransmitted signal to reflect off the body. The sensorsupplies an output voltage, which is proportional to thedistance from the falling object, thereby allowing the po-sition to be measured. Data from the sensor was then

collected every .02 seconds as a function of time and ex-ported through LabView. When the body reached theend of the track, it was caught by hand to end the pathand trajectory of the body. A total of 60 trials was con-ducted, 3 trials for each of the 4 cylinder types and eachof their 5 bank angles.

Figure 4: Closer photograph of the bricks that controlthe height and the ultrasonic position sensor.

Figure 5: Closer photograph of the end of the metaltrack where the cylindrical body is caught. Each brick is

2.25 inches

Bank angle was calculated by taking the inverse sin of hdivided by 84 in, which is the track length. Since eachbrick was 2.25 in, h = 2.25(n+ 1).

For creating the first graph, Figure 6, the displacementof one type of cylinder for each bank angle of the solidmetal cylinder disk was averaged to create a new array ofaverage displacement. This was then plotted along timeto show the trajectories of each bank angle for a cylinder

EGR244 ROLLING CYLINDER LAB 4

type. The second graph, Figure 7, shows the experi-mental and theoretical acceleration plotted against timefor each type of cylinder. The experimental accelerationwas calculated by quadratically fitting the displacementvs time trajectories of each type of cylinder for each angleand each of the three trials. This was done with Matlabspolyfit command for second order during the time pe-riod before the trajectory plateaus. The coefficient of thesquared term represented the experimental accelerationof the block divided by two, which is done because tak-ing derivatives of the initial displacement vs time func-tion would result in a .5 in the coefficient that must beaccounted for. The error bars were drawn by calculatingthe standard deviation between each of the 3 trials foreach cylinder type and bank angle. The theoretical ac-celeration was calculated for each cylinder type and bankangle by using the derived Equation 9 . When actuallyplotting the graphs, a vertical shift was applied to slightlyoffset the lines for clarity and comparison purposes. Thesame amount of shift was applied to the experimentaland corresponding theoretical data.

IV. RESULTS/DISCUSSION

Figure 6: Experimental position data for Solid MetalCylindrical Disk trajectory at each of the 5 angles. Path

plateauing represents the completion of the track.Dashed lines represent quadratic fit in the colorcorresponding to the experimental angle data.

The completion of the track is represented by the generalplateauing of the path. The plateau is not a smooth hor-izontal line, but rather a series of peaks and dips dueto the the cylinder being removed and the sensor re-flecting off of other environmental objects. As expected,the smaller the angle, the longer it takes to completethe track, and as per Eq.(9) ,the greater the angle, thegreater the curvature of the graph, representing acceler-ation. There is a small dip at around the middle of eachtrials trajectory, which is most likely due to a small bumpor crack in the track at that location. See Appendix forfull code.

Figure 7: Theoretical acceleration calculated by Eq.(9)(dashed lines) are plotted as a function of bank angle

next to the experimental points (circles of correspondingcolor). Each trial is offset for visual clarity. Error barsrepresent the standard deviation across the three trials

for each cylinder and bank angle.

The dashed lines representing the theoretical accelerationof each type of cylinder as a function of angle are veryclose to their corresponding experimental accelerationsobtained as a function of bank angle. Some interestingtrends are the fact that the solid cylinders have a fasteracceleration than the hollow cylinders, because referringto Equations 4 and 9, the solids have a smaller radiusof gyration. Also, the standard deviations, representedby error bars, increase as the angle increases for everytype of cylinder. Also, the deviation between the theo-retical dashed line and the line fitting the experimentalacceleration points as a function of angle increase as theangle increases. These trends probably exist because asthe angle increases, the cylinder rolls faster, as shown byFigure 6 and equation 9. Therefore it is more likely towobble, become unstable, or roll with slipping. Since ourtheoretical model is for rolling without slipping, the slip-ping would cause more deviation. And, since our bankangles are fairly small, the sine values of them are ap-proximately linear as they change; however, as the angleincreases the Taylor approximations are not as accurate.

V. CONCLUSION

In this experiment of releasing different cylindricalbodies along a track elevated at different bank angles,data collected was compared to their theoreticallyderived counterpart models of the motion and forcesacting on the bodies by using the principles of physics,inertia, Newtons laws, and Eulers Laws and their effects.

Overall our data matched our theoretical modelspretty closely and exhibited similar general trends interms of slope and relative values between the differentbodies. There were slight differences, which are mostlikely due to some assumptions and experimental inac-

EGR244 ROLLING CYLINDER LAB 5

curacies. For example, we assumed that there was noaerodynamic drag, the radius of gyration was calculatedfor a hollow cylinder of infinitesimally thin thickness,and that the motion was purely rolling without slipping.It is important to note that friction may cause themeasured accelerations to be smaller than that predictedby theory. An estimate of the coefficients of frictioncan be estimated with the following equations based onFigure 1.Solid: µs = 3

2g (atheory − aexperimental)1

cosθ

Hollow: µh = 12g (atheory − aexperimental)

1cosθ

Based on the data values collected and fitted and thecalculated theoretical values, µs = .0212 and µh = .0252

Also, when we triggered the release gate, it shook thetrack a bit, which would affect the path of the body.It also appeared that there was a slight bump or dentin the middle of our track based on a consistent smalldip in the graphed trajectories of each body and angle.To increase the accuracy of the experiment and reportwe could conduct more than three trials for each of theangles, modify the release gate to move smoother andwith less impact, tape down the stand holding the trackto prevent shaking, and consider doing calculations forand factoring in aerodynamic drag, thickness, or rolling

with slipping for larger bank angles.The theories and results from this experiment provide

insight on rolling or spinning motion which has manyreal world applications such as wheels, transportation,motors, energy generation, etc.3 Our analysis shows usthat the physical qualities influencing moment of inertiaand friction would affect the slipping and rolling motionor spinning, which would be an essential fact to applywhen designing better mechanical systems.

VI. ACKNOWLEDGEMENTS

Lab Partners for data collection: Hailey Prevett,Laura Perez.TA’s: Roozbeh and Sarah Land.

1 Physics Classroom. ”Types of Forces.” The PhysicsClassroom. N.p., n.d. Web. 10 Mar. 2017.

2 Real World Physics. ”Radius Of Gyration.” Real WorldPhysics Problems. N.p., 2008. Web. 10 Mar. 2017.

3 ”Chapter 12.” 12. ROLLING, TORQUE AND ANGU-LAR MOMENTUM. N.p., n.d. Web. 10 Mar. 2017.