Figure 9 is a graph of the engine output torque of a 1986 Honda Accord LX Sedan. One might have expected the results of the four gears to fall on the same line. After all, the engine torque should merely be a function of the throttle and the engine speed. Indeed engine torque is specified at a particular engine speed but with no reference to a particular gear. But the engine torque listed in manuals is measured on a test bench with the engine running at constant speed or slowly accelerating while working against a load. In our case, the output torque of the engine is not just accelerating the mass of the car (including itself) linearly. It must also use some of its torque to provide angular acceleration of its parts, chiefly the flywheel. In the lower gears, the engine rotational speed and acceleration are large compared to the car's speed and acceleration, so this noticeably reduces the net output torque of the engine. To account for this, we define the total engine torque as a sum of the engine output torque and the torque needed for this angular acceleration of the engine.

where I is the moment of inertia and _E the angular acceleration of the engine, which is given by:

Figure 9.

Figure 10 is a graph of the total engine torque of the same car. The engine moment of inertia I was determined graphically by finding a value that made the first-gear data agree with the higher gears. The value we found for this car was 0.20 kg m². This value is reasonable compared to a few examples taken from car racing information, but we have no other source of data on any cars we have measured. In any case the effect is mainly confined to first gear.

Figure 10.

A similar but less noticeable effect is the rotational inertia of the wheels.¹ The acceleration of the car also involves the angular acceleration of the wheels. Since the wheels always have the same ratio of rolling speed to linear speed, the net effect is simply to increase the effective mass of the car. In a rolling wheel, the mass at the surface of the wheel is effectively doubled while mass at the center of the wheel has no added effect. The radius of gyration of the wheel is the radius that gives the same moment of inertia if all the mass were concentrated there. The radius of gyration of a uniform disk is 0.707R. Clearly the wheels have mass distributed unevenly, but we estimate the radius of gyration of the wheel and tire combination to be about 0.8R. The tangential speed at this radius is just 0.8v. So the total kinetic energy including this term (but not the engine term that we dealt with in a different manner above) is:

We have found that the mass of wheels and tires on various size cars is very close to 5% of the total car's mass. So the adjustment we make to account for this effect is to increase the mass of the car by 5% × (0.8)² 3%.

Note that the torque is very constant over the range of engine speeds measured. This is true of most cars. The torque ultimately comes from the force that the hot combustion gases exert on the piston. The actual details are more complicated, but let us assume that the combustion is completed at the top of the stroke before the expansion begins. Then the force on the piston is not dependent on the speed of the piston but only on the pressure of the gases (the piston speed is always well below the speed of sound). So as long as each intake stroke supplies the same mix, the pressure and torque will be the same regardless of the speed of the receding piston. At high engine speeds, the cylinder is less completely filled during the intake stroke.² In addition, the combustion is not completed before the piston begins receding.³ Both of these cause the torque to drop off. An example of the summary page from a student report is given in Fig. 11. Other reports are available on our website.⁴

Figure 11.

Note the drag coefficient of the car has been determined to be 0.39. Actual drag coefficients of real cars vary from 0.5 down to about 0.25 for very well designed cars.⁵ Although our results generally fall in this range, we have not been able to find manufacturers' quoted values for the cars we have measured.

The total engine power is graphed on the lower right of this report. Since the torque is nearly constant, it is to be expected that the power would increase linearly with engine speed. The power quoted from the manual is specified at the highest engine speed that allows the manufacturer to list a more impressive number. In this example, as in most cases, the measured values of torque and power are somewhat lower than the quoted specifications. Some of this is due to losses in the drive train and the tires that we don't account for. Our measurement of drag does account for the effects of rolling friction of unpowered tires, but when the wheels are powered, there are additional losses. Total losses from engine to road are reported⁶ to be in the range of 15%, which is consistent with our results.