Grand Prix Racing - The Science of Fast Pinewood Cars

# Measure Rolling Friction Coefficient

It is a relatively simple matter to determine a wheel's tread friction coefficient. Tread friction is rolling friction. Rolling friction depends on the condition of both the tread and the track. The rougher either is, the more tread friction there will be. In order to get the best race-time estimate, you will need to find some flat boards that are finished in a way similar to the track, or better yet, use sections of the track itself.

## What you need

Some of the items listed below may not be on hand. You may have to borrow track sections or boards from your club or a handy neighbor. A few drafting tools will also help the measurements go faster, a large triangle or "T" square.

• Ramp: flat board or track section at least 2 feet long, one foot wide
• Flat: flat board or track section about 8 feet long, one foot wide
• Car wheel prepared so that it rolls straight
• A thick book: about 1.5 inches thick
• Measuring tape
• A sharp pencil
• A worksheet to record your measurements on
• A scientific calculator: one that has trigonometric functions

## Procedure

Find a hard floor area where the ramp and flat can be set up. Lay the "flat" on the floor. Lay the ramp next to it propped up by the thick book. In order to make the join smoother, run masking tape across it to insure that the ramp surface matches up with the flat surface as it will tend to fall just below it.

Draw a starting line across the ramp 18 inches from the join. The symbol for this ramp length of 18 inches is Lr. If you want the ramp length to be different, use your value for Lr. Now measure the height of the starting line above the floor. Measure it from the top of the borad to the floor. Subtract from that height the thickness of the flat. This is the distance the wheel will "fall" down the ramp. Its symbol is H.

Now that the track is set up and measured, practice rolling the wheel down the ramp. Try to position the wheel the same way each time balanced with its center over the edge of the starting line. If it rolls off the track, the wheel may need to be sanded on the outside edge of the roll to level it or the ramp may need lowering. If the ramp is lowered, measure H again.

You are ready to begin when you can roll the wheel the same way each time. The wheel should smoothly roll down the ramp and come to a stop on the flat.

1. Perform these steps 10 or more times.
1. Roll the wheel down the ramp and watch it come to a stop.
2. If the wheel jumps or swings unusually, do the run over again.
3. Mark the position of the wheel center on the flat with a pencil.
2. Place the tape measure beside the flat with "0" at the join.
3. Perform these steps for each mark on the flat.
1. Measure the distance to the mark to the nearest 1/16 of an inch. It will help to use a drafting triangle or "T" square to line up the mark with the measuring tape so you don't have to move the tape every time.
2. Record the measurement on your worksheet.
4. Follow the worksheet instructions to obtain the tread friction coefficient of your wheel on that particular type of surface.

Use this worksheet to record your measurements. Only Lr and H are measured once. O and u are calculated below. ya is an average of the measurements for yc that are recorded in the table below this one.

## Variables

Symbol Value Description
Lr __________ Length of ramp surface in inches
H __________ Distance the wheel falls in inches
ya __________ Average coasting distance on the flat measured from the join in inches
O __________ 'Theta', angle the ramp forms with horizon in radians
u __________ Rolling drag coefficient

## Coasting Distance Measurements

Record the distance the wheel rolled on the flat in each trial in this table.

 Trial yc (inches) 1 2 3 4 5 6 7 8 9 10 _______ _______ _______ _______ _______ _______ _______ _______ _______ _______

When this table is filled in, you can get the average coasting distance, ya, to record in the Variable Table above. Add up all of the values for yc and divide that total by the number of trials you ran. This value for ya is needed to calculate the tread friction coefficient.

Next, you must find a value for the ramp angle, O. To do this, you must evaluate the following expression. A scientific calculator makes this easy. Make sure the calculator is in RADIANS mode!

O = _________ radians = arcsin(H/Lr)

Now that we know the geometry of the ramp and how far the wheel rolled, we can use an equation from an analysis of the rolling friciton model to find the tread friction of the wheel on this surface. The equation is:

u = -tanO/(ya/Lrcos3O+1)

Let's do this a piece at a time to make sure it's done right.

Beginning on the bottom on the inside:

Lrcos3O = _________ = Lr times cosO cubed (only cosine theta is cubed)

ya/Lrcos3O = _________ = ya divided by the result of the preceeding line

ya/Lrcos3O+1 = _________ = 1 added to the result of the preceeding line

u = __________ = -tanO/(ya/Lrcos3O+1) = -tanO divided by the result of the preceeding line

This value can be recorded in the Variable Table and below in the Results Table. An easy way to see how much variation in the tread friction value there is with the measurement of coasting distance is to use the least distance and the greatest distance in the calculation for u instead of ya. Record the least measurement for yc in the result table, the average value for yc (that is ya) and the greatest value you measured for yc. Then calculate the friction coefficients using the two extreme values for yc and enter them in the Results Table.

## Results

yc u Description
__________ in __________ Shortest coasting distance
__________ in __________ Average coasting distance
__________ in __________ Longest coasting distance

How do the tread friction coefficients compare?

How many digits should you keep on the right of the decimal point for u?

If this experiment is repeated for different ramp angles, O, does the tread friction coefficient change significantly?

You can view an analysis of the sensitivity of results to measurements in this experiment and some results using an AWANA kit wheel.

 [Pit Area] [Title Page]