Moment of Inertia

The moment of inertia has units of mass times the square of distance and is a measure of how hard it is to get something to start (or stop) rolling. In that way, it acts sort of like a mass in that if the energy input is equal, if the moment of rotational inertia goes up, the wheel will turn more slowly.

The moment of inertia is equal to the sum of all the masses that make up an object times the square of their distance from the center of mass. So, the more mass is moved out away from the center of mass, the more energy it takes to turn the wheel once because the radius of the mass is getting bigger.

The model for the chapter, shown in Figure 5-1, looks like a hockey puck with holes punched in it. We call it the weighted wheel. The one we used in this chapter, printed in PLA with 10% infill, has a mass without any pennies in it of 46 grams. There are penny-sized holes into which we can place coins to add mass at different radii from the center.

Figure 5-1. The weighted wheel model

Use a bit of tape (blue tape of the type used on 3D printer platforms works nicely—see Figure 5-2) to hold pennies in place by wrapping a set of pennies in blue tape. We found that the pennies fell out if we just put blue tape over the holes, and if they rattled a bit it affected the results. We used slugs of 13 pennies per hole for the experiments in this chapter.

Figure 5-2. The weighted wheel model with pennies in the “outer ring” held with blue tape

Predicted Moments of Inertia for This Model

For the model in Figures 5-1 and 5-2, we will figure out the moment of rotational inertia as follows:

• Moment of rotational inertia = Rotational inertia of the wheel + Sum of the rotational inertias of the pennies

Wikipedia ( https://en.wikipedia.org/wiki/Moment_of_inertia ) and other physics references tell us that the moment of inertia of a solid cylinder is (1/2) * M * R ² . Our model is not a solid cylinder—it is a 3D print with an outer shell, some holes in it, and 10% infill—but for getting general trends, it should be close enough. In this case, the radius R is 50 mm, or 5 cm. With a mass of 46 grams, we get the moment of inertia of the empty wheel without coins as 575 g-cm².

Predicting Velocity of the Rolling Wheel

Once we have these numbers, we can predict how fast the wheel should roll downhill. First, we need to compute a formula for the velocity the wheel will have after it rolls down the slope of our experiment, when it will have converted all its potential energy (at the top of the slope) to kinetic and rotational energy. As mentioned in the last section, we know:

• Potential energy = Translational energy + Rotational energy

Combining the formulas for each term from the last section gives us:

• M * g * h = (1 / 2) * M * v ² + (1 / 2) * I * v ² / R ²

Which we can clean up a bit to this:

• 2 * g * h = v ² * (1 + I / (M * R ² ))

Or solving for velocity,

• v ² = 2 * g * h / (1 + I / (M * R ² ))

where g = 980 cm/s², R = 5 cm, h = distance * sin(incline angle) , and I = moment of inertia. If we want to get the ratio of the velocities for our current weighted wheel cases, we get the following equation, after dividing out some of the things that we can hold constant from one test to another (like the geometry of the slope, the distance travelled, and so on, assuming here the radius R of the overall body is the same):

• v ₁ / v ₂ = sqrt ((1 + I ₂ / ( M ₂ * R ² )) / (1 + I ₁ / (M ₁ * R ² )))

The results of plugging in some pairs of values for comparisons are shown in Table 5-2. The moments of inertia vary, but the masses do too. Because this is a dynamic system, the results can be a little surprising. The higher the moment of inertia, the slower the wheel will accelerate, and longer it will take to reach the bottom of the slope. However, the higher moment of inertia in some cases was offset by differences in mass.

Results

We tried rolling the wheel on a smooth table with one end raised up a bit, and also outdoors on some sloping concrete. We took our weighted wheel out, rolled it in the configurations in Table 5-1, and measured the distance rolled and the time.

Rolling it outdoors encountered a variety of problems. The concrete was not very smooth, and had decorative inlaid bricks. The empty wheel stopped very soon after starting and in some cases even rolled backward (in the nominally uphill direction) a bit! However, qualitatively it was interesting to see how much faster the wheel rolled with the pennies on the inside versus the outside.

We would expect the velocity calculated by v ² = 2 * g * h / (1 + I / (M * R ² )) to be twice the average velocity over the whole time rolling down the table, because gravity is making it accelerate. We got the average velocity by just dividing the distance the wheel rolled by the time it took to travel it.

The first setup we used was a smooth table 150 cm long, raised 3.14 cm at one end. We let the wheel go on one end and recorded at least three times how long it took each configuration to cross the far end (Figure 5-3). A strategic pillow on a chair and another on the floor at the end is good for catching it so it does not break or dump pennies everywhere when it goes off the edge.

Figure 5-3. The empty wheel heading downhill (to the upper right in the photo) to its pillows

The second setup was outdoors on some gently sloping concrete. We tried to note when the wheel got to 12 feet (366 cm) centered where we were releasing it, trying to repeat the initial release angle as much as possible. We marked our starting spot with a bit of blue tape (Figure 5-4) and you can take advantage, as we did, of decorative markings as start or stop points. It is not a perfect solution because the slope will vary, but the wheels did not roll all that repeatably and other things we tried (see the “Learning Like a Maker” section of this chapter) had other issues. We found an average slope of 3 degrees on the concrete and used that to calculate a height difference of 19.1 cm in 366 cm.

Figure 5-4. Marking the start of a test run on concrete for the wheel with the pennies on the outer ring. Downhill is to the right.

Table 5-3 summarizes these results. We predicted the velocity based on the moment of inertia calculated and the measured heights of the ramps, and then divided by 2 to compare it to the average velocity (just the distance over the time). The empty ring and pennies in center models were only tested indoors. The empty wheel was too easily perturbed by random bumps in the concrete to get any repeatability.

As the physics predicts, higher moment of inertia wheels go more slowly. The measured ratio of the speeds of the wheel with pennies on the inner ring to the outer is 1.14 : 1 outdoors and 1.09 : 1 indoors, bracketing our theoretical ratio of 1.12 : 1 in Table 5-2. There is, of course, a spread in the results because of the many uncertainties, which we note in the following Caution.

The Model

The 3D-printable model for the wheel, shown in Listing 5-1, is very simple. It starts with a cylinder and subtracts other cylinders from it. The parameter coin is the diameter of the coins you want to use (plus a bit of margin—we used 20 mm for 19.5 mm diameter pennies, which left room for tape around them).

You can vary the diameter, d, to make a bigger or smaller wheel too. If you change the diameter, holes will remain near the edge, but the ring nearer the center will move outward to be halfway between the center hole and the outer ones. Obviously, you cannot make this a lot smaller than it is now. The radius R we use in previous sections is one-half the diameter d.

The model is pretty straightforward to print. This one is 20 mm high (2 cm), which accommodates 13 pennies. We tried making one with quarter-sized holes and making it thinner, but it was too unstable.

Listing 5-1. The Weighted Wheel

// A model of a weighted wheel
// To demonstrate conservation of angular momentum
// file weightedWheel.scad
// Rich Cameron, March 2017

d = 100; // diameter of disk, in mm
h = 20; //height of disk, in mm
t = 1; // minimum wall thicknesses
coin = 20; // diameter of coin in use, mm

$fs = .2;
$fa = 2;

difference() {
   cylinder(r = d / 2, h = h);
   for(i = [0:6]) rotate(120 * i + 60 * ceil(i / 3))
      translate([ceil(i / 3) * (d - coin - t * 2) / 4, 0, t])
         cylinder(r = coin / 2, h = h);
} // end model

3D printed “cylinders” are made up of small flat surfaces. If you want the wheel to be made up of smaller increments (and thus be rounder and smoother) change the $fs and $fa parameters. In OpenSCAD, the number of faces in the regular polygons used to approximate circles are specified using the special $fs, $fa, and $fn special variables. $fs specifies the minimum size of the facets in mm, with a default of 2 mm. $fa is the minimum angle between facets in degrees, defaulted to 12 degrees. Depending on the size of the object you are printing, one or the other will be important, as follows:

• For small, circular 3D printed objects, $fs will keep the number of facets high enough to make the circle look round. A small-enough value for $fs will prevent holes from ending up too much smaller than the specified diameter. This is because the radius of a regular polygon is measured from the center to the vertices, and the apothem (the distance from the center of the polygon to the center of a side) is smaller.

• For larger circular objects, a larger $fa value will prevent the number of facets from becoming unnecessarily high, which increases rendering time. $fa needs to be a number that divides evenly into 360 (if it is not, OpenSCAD will round to a number that is).

• $fn overrides both of the other special variables and allows the user to specify a specific number of facets. It is usually a bad idea to set $fn globally. Any of these variables can also be specified for each individual object, which provides an easy way to create a regular polygon of $fn sides.