## 3. Triangle of forces

Any force has both magnitude and direction. Imagine someone pulling along a loaded trolley, attached to them by a rope and harness. If the rope makes a very shallow angle with the direction of travel, most of the force generated by the person pulling is applied to moving the trolley. Conversely, if the rope is at a steep angle, some of the force is trying to lift the trolley rather than pulling it along. Look at these two diagrams, where the force along the rope, F, has been divided into a horizontal component h and a vertical component v:

In both cases, the magnitude and direction of the pulling force are represented by the hypotenuse of a right-angled triangle. We can use trigonometry to determine the proportion of the force that acts horizontally, pulling the trolley along. If we assume that the force is the same in each case, this is shown as the length of the hypotenuse, F. The horizontal force, h, is the side of the triangle that is adjacent to the angle between the rope and the horizontal, which we can again represent as α. (For simplicity, the third side of the triangle, which is equal to v, has been omitted from the diagram.) The trigonometric function that relates the hypotenuse and adjacent is the cosine, so that:

cos α = h/F

h = F . cos α

If the rope were horizontal, the angle α would be zero and its cosine would have the value 1, so that the entire towing force acts to pull the trolley along. As the rope is inclined a a greater angle to the horizontal, the value of the cosine decreases. When the rope is vertical, cos α = 0 and there is no horizontal force – all the person's effort is going into trying to lift the trolley off the ground! At an angle of 60°, half of the force along the rope is acting horizontally.

 Angle of rope above horizontal 0° 15° 30° 45° 60° 75° Proportion of force along rope acting to tow the trolley 100% 97% 87% 71% 50% 26%