Grand Prix Racing - The Science of Fast Pinewood Cars

Car Moment Of Inertia Worksheet

To compute your car's moment of inertia, you must find the moment of inertia of its parts, "transport" them to the car's center of mass, then add them all together. Below, we visit each part of the car in turn. For an example, see the model of car body rotation on which this method is based.

The body is probably not a rectangular block, so you may have to break it up into virtual pieces to get its moment of inertia. To do this, first draw a picture of the side of your car on graph paper.

Moments

Here is a list of shapes and formulas for determining the moment of inertia and center of mass for each three dimensional shape. Note that the width dimension is not drawn. Only the side having length and height is drawn. The center of mass location is along the length and height axies of the shape. For each shape, the axis of rotation passes through its center of mass and is at right angles to the side drawn. Note the difference between the cylinder and rod is that the cylinder turns end-over-end, but the rod spins about its center. The center of mass locations (cmh,cmx) are measured from the lower left of the shape drawing.

SHAPE NAME cmh cmx MOMENT OF INERTIA
Block
 Block h/2 l/2 m(h2+l2)/12
Cylinder
 Cylinder r l/2 m(3r2+l2)/12
Rod
 Rod r r mr2/2
Wedge
 Right Wedge h/3 2l/3 m(4h2+l2)/72

Matching up these shapes to your car's parts may not be easy, nor exact. The better it is done, the more acurate the result will be. But don't worry if it is not perfect. Fill in the chart below following these steps.

  1. Name each part and write it in the first column.
  2. To determine the mass of a part, weigh it in units of ounces and divide the result by 386.088 in/s2. Write the result for each part in the chart. You can use the weight of the standard AWANA or Boy Scout kit axles if you are using them. Don't forget to double the weight, since each axle is considered as a pair.
  3. h and l are the corresponding height and length of the shape matched to your car part. Measure them in units of inches on your part and record them in the chart.
  4. Compute cmh, cmx and the moment of inertia for each part using your measurements of m, h (or r) and l and the formulas for each part's shape from the chart above. Mark the point (cmh,cmx) on your drawing with an X. Note, cmh and cmx are measured from the lower left corner of the shape, not from the reference lines you drew.
Part SHAPE m ozs2/in h (or r) in l in cmh in cmx in MOMENT OF INERTIA ozins2
_Front_Axle_ _Rod____ ________ ________ ________ _r______ _r______ _mr2/2__
_Rear_Axle__ _Rod____ ________ ________ ________ _r______ _r______ _mr2/2__
____________ ________ ________ ________ ________ ________ ________ ________
____________ ________ ________ ________ ________ ________ ________ ________
____________ ________ ________ ________ ________ ________ ________ ________
____________ ________ ________ ________ ________ ________ ________ ________
____________ ________ ________ ________ ________ ________ ________ ________
____________ ________ ________ ________ ________ ________ ________ ________
____________ ________ ________ ________ ________ ________ ________ ________
____________ ________ ________ ________ ________ ________ ________ ________
____________ ________ ________ ________ ________ ________ ________ ________
____________ ________ ________ ________ ________ ________ ________ ________
____________ ________ ________ ________ ________ ________ ________ ________
____________ ________ ________ ________ ________ ________ ________ ________

Wheels can be measured for the following data or the values from the AWANA or Boy Scout kits can be used. Since the wheels are considered in pairs, double the mass and the moment of inertia and write them in the chart. This applies even if the wheels aren't touching the track. Mark the center of each pair of wheels with an X, as you did with the center of mass of the other parts.

Wheel Pair m ozs2/in MOMENT OF INERTIA ozins2
Front ________ ________
Rear ________ ________

Once you have each part's moment of inertia about its center, translate it to the center of mass of the car at (CMh,CMx). Do this by adding to the moment of inertia about the part's center the square of the distance from the block center to the car's center of mass times the part's mass. A measurement in height and length is needed from the center of mass of each part - its X at (cmh,cmx) - to the center of mass of the car at (CMh,CMx) measured from the rear axle (values of CMx increase away from the rear axle toward the front axle). These measurements are labeled Dh and Dx in the chart below.

Part m ozs2/in MOMENT OF INERTIA ozins2 Dh in Dx in I+m(Dh2+Dx2)
_Front_Wheels_ ________ ________ ________ ________ ________
_Rear_Wheels__ ________ ________ ________ ________ ________
_Front_Axles__ ________ ________ ________ ________ ________
_Rear_Axles___ ________ ________ ________ ________ ________
______________ ________ ________ ________ ________ ________
______________ ________ ________ ________ ________ ________
______________ ________ ________ ________ ________ ________
______________ ________ ________ ________ ________ ________
______________ ________ ________ ________ ________ ________
______________ ________ ________ ________ ________ ________
______________ ________ ________ ________ ________ ________
______________ ________ ________ ________ ________ ________
______________ ________ ________ ________ ________ ________
______________ ________ ________ ________ ________ ________
______________ ________ ________ ________ ________ ________
______________ ________ ________ ________ ________ ________
______________ ________ ________ ________ ________ ________
______________ ________ ________ ________ ________ ________
TOTAL ________ ________

Now add up the values in the "I+m(Dh2+Dx2)" column and you have your car's moment of inertia about its pitch axis. For a sanity check, you can add up the part masses and see if the total times 386.088 in/s2 matches the total weight of your car. If it doesn't, check your measurements and calculations.

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Grand Prix Racing - The Science of Fast Pinewood Cars
Copyright © 1997, 2004 by Michael Lastufka, All rights reserved worldwide.