To Infinity and Even Farther--A Question of Levers

Here's a question concerning the FOSS Levers and Pulleys Module (grades 5-6) from teachers Jil Brown-Curry and Linda Miller in Cobb County, Georgia.

What is the actual explanation for Graph A being curved vs. Graph B being a straight line? (Activity 1, Levers)

Here's the answer from Larry Malone of the FOSS staff at the Lawrence Hall of Science.

Ok, here goes...

The force (F) (either the load or the effort) multiplied by the length of the lever arm (d) (distance from the fulcrum to the point at which the force acts) is what tends to rotate a lever on its fulcrum. This is called torque.

If two equal forces are applied on opposite sides of the fulcrum, each the same distance from the fulcrum, the torques will be the same and no motion will occur. Similarly, if one of the forces is moved to a new position half as far from the fulcrum, a force twice as large must be applied at the new position to achieve balance. Force X distance on one side = force X distance on the other side. The trick to achieving balance (no motion of the lever arm) is to get the force times the distance on one side of the fulcrum to exactly equal the force times the distance on the other side.

If you place a load (FL) at a fixed distance from the fulcrum and apply an effort (FE) at the same distance on the other side, the beam will balance when the pulling force of the load is exactly equal to the force of the applied effort. But if you move the position at which the effort is applied farther from the fulcrum, and apply the same amount of force, force times distance on the effort side produces more torque than force times distance on the load side. The system is unbalanced; the force applied farther from the fulcrum creates more torque and the system rotates. In order to achieve balance with the effort applied at its new position, a smaller force must be applied.

This is what happens in Experiment A. The load is placed at 10 cm from the fulcrum on one side, where it stays, and the effort is applied at various distances from the fulcrum on the other side. Using a spring scale, we quantify the force (effort) needed to achieve balance at different distances. When the distances are equal (both at 10 cm from the fulcrum), the force of the load (2.4 N) equals the force of the effort (2.4 N). In both cases the torque is 24 N cm. But as the distance at which the effort is applied gets smaller (gets closer to the fulcrum), the amount of force has to increase in order to achieve balance. At a distance of 5 cm, a force of 4.8 N will be needed to produce 24 N cm of torque to balance the load which has not changed. Now you can see that as the distance at which the effort is applied approaches zero, what must happen to the force to achieve balance? It approaches infinity! That's why the graph heads off to infinity when we plot lever-arm distance vs. effort.

On the flip side, as the distance at which the effort is applied goes beyond 10 cm, less and less effort is required to achieve balance. So, as the effort distance approaches infinity, the force required to achieve balance approaches zero. Great! An infinitely small force is required to lift the load on the other end! This is why Experiment A gives a graph that has an interesting curve with both ends heading off to infinity in opposite directions. It is also this realization that moved Archimedes to say, "Give me a lever of sufficient length and a place to stand and I shall lift the Earth into heaven."

Now consider Experiment B. If we apply the effort at a fixed position, say 10 cm from the fulcrum, and move the location of the load around on the opposite side, what happens to the magnitude of the effort required to achieve balance? We already know what happens when the load is placed 10 cm from the fulcrum. 10 cm X 2.4 N = 24 N cm. What happens to the effort needed to lift the load as it is moved toward the fulcrum? As the distance at which the load is positioned gets smaller and smaller, less and less effort is needed on the other side to balance it. (Remember, force X distance = torque.) When the load is moved to zero, the force required to "lift" it is also zero. Our graph of distance of load from the fulcrum (independent variable) versus force required to balance the lever (dependent variable) has its origin at (0,0). As the load moves farther and farther from the fulcrum, the force required to balance the load goes up. This is a linear relationship--as one variable increases, so does the other. The result is a straight line starting at the origin and heading out to the great beyond. As the distance from the fulcrum at which the load acts approaches infinity, so does the effort required to lift it. The relationship in Experiment A is an inverse relationship--as the length of the lever arm you are using to apply the effort increases, the effort required to do the job decreases. The relationship in Experiment B is linear. As the distance at which the load is placed increases, so does the effort I need to apply to get the job done.

Now, as to WHY it happens, I don't have an answer. That's just the way our local universe is set up. We can understand the laws and principles that govern the behaviors of levers, describing them in words and mathematics, but the reasons for these behaviors lapse into deep philosophical ruminations. The graphs are simply ways we have devised to describe the relationship for all to ponder.

That's all I know...hope it helps.

Larry Malone