The amount of this may be computed with mathematical exactness. Since the tractive resistances are computed separately, we merely have to compute the tendency of the wheels to roll down the grade, or the resistances to pulling them up the grade, which are exactly equal when the frictional resistance is zero or when it is otherwise provided for. Assume that a ball or cylinder (Fig. 23) is being drawn up an inclined plane. If we represent its weight by W, as measured graphically by the line W in the figure, then N will measure the normal pressure against the plane, and G will measure the force required to draw it up the plane with a uniform velocity. It also measures the tendency of the weight to roll down the plane. From similar triangles we may write the proportion G:W::h:d or G =wh/d .....(1)

In the diagram d is very much larger than c, but, as will be shown, c is so nearly equal to d on all practicable railroad grades that there is no appreciable error in substituting c for d, and write the equation G = wh/c. But h/c equals the rate of grade. Therefore we have the very simple and mathematically exact relation that the grade resistance G = W times the rate of grade. In order to appreciate exactly the extent of the approximation in assuming that the slope distance equals its horizontal projection, the percentage of the slope distance to the horizontal projection is given in the tabular form on page 186. Incidentally the tabular form shows the amount of error involved when we measure with the tape lying on the ground instead of holding it horizontally. Since almost all railroad grades are less than 2% (where the error is but .02 of 1%) and anything in excess of 4% is unheard-of for normal construction, the error in the approximation is generally too small for practical consideration.

Grade resistance.

Fig. 23. Grade resistance.

Grade in per cent.

1

2

3

4

5

Slope dist. / hor. dist x 100...

100.005

100.020

100.045

100.08C

100.125

Grade in per cent.

6

7

8

9

10

Slope dist. /hor. dist. x 100...........

100.18C

100.245

100.319

100.404

100.499

If the rate of grade is 1:100, G equals W x 1/100, i.e., G = 20 pounds per ton; therefore, for any per cent of grade, G = (20 X per cent of grade) pounds per ton. When moving up and down grade this force G must be overcome in addition to all the other resistances. When moving down a grade the force G assists the motion, and the net force tending to move the car or train down the grade equals G minus the resisting forces. If the resisting forces are less than G, then the train will keep moving down the grade, and its velocity will increase until the added resistance to increased velocity just equals G. The train will then move at this uniform velocity as long as such conditions remain constant. If the resistance of a train averaged 6 pounds per ton for a velocity of 20 miles per hour and the train were started on a down grade of 0.3% at this velocity, then it would move indefinitely at this speed down such a grade. If a train were started down a 1% grade at a velocity of say 10 miles per hour, the grade force will equal 20 pounds per ton on the 1% grade. Under such conditions the velocity of the train would increase until the velocity resistances would equal 20 pounds per ton. The precise speed at which this will occur depends on whether cars are loaded or empty and on various other conditions which affect train resistance, but the velocity would probably be very high, perhaps 60 or 70 miles per hour. Since this would be too great a speed for safety with freight-cars, a 1% grade of indefinite length can never be operated without the use of brakes. As developed later in the chapters on Grade, the necessary use of brakes on a down grade is one of the objections to grade, in addition to the resistance to moving up the grade.