When a train moves on a rising gradient, it requires extra effort in order to move against gravity as shown in Fig. 25.2.
Fig. 25.2 Resistance due to gradient
Assuming that a wheel of weight W is moving on a rising gradient OA, the following forces act on the wheel.
(a) Weight of the wheel (W), which acts downward
(b) Normal pressure N on the rail, which acts perpendicular to OA
(c) Resistance due to rising gradient (R4), which acts parallel to OA
These three forces meet at a common point Q and the triangle QCD can be taken as a triangle of forces. It can also be geometrically proved that the two triangles QCD and AOB are similar. From A QCD,
It may be noted here that when a train ascends a slope, extra effort is required to overcome the resistance offered by the gradient. The position is, however, reversed when the train descends a slope and the resistance offered by the gradient helps in the movement of the train.