Last time we introduced the free body diagram, applied it to a simple pulley, and discovered that in so doing lifting objects required 50% less effort. As an engineering expert, I’ve sometimes put this *improved* version of a *simple pulley* to work for me in designs. We’ll do *the math behind the* *improvement* today.
Here again is the *free body diagram* showing the *improved* *simple pulley* as introduced last week.
__The Math Behind the Improved Simple Pulley__
The illustration shows the three forces, *F*_{1}, *F*_{2}, and *W,* acting upon the *simple* *pulley *within the highlighted free body diagram. Forces *F*_{1} and *F*_{2} are exerted from above and act in opposition to the downward pull of gravity, represented by the weight of the urn, *W*. Forces *F*_{1} and *F*_{2 }are produced by that which holds onto either end of the rope that’s threaded through the *pulley.* In our case those forces are supplied by a man in a toga and a beam. By engineering convention, these upward forces, *F*_{1} and *F*_{2,} are considered positive, while the downward force, *W,* is negative.
In the arrangement shown in our illustration, the *pulley’s* rope ends equally support the urn’s weight, as demonstrated by the fact that the urn remains stationary in space, neither moving up nor down. In other words, forces *F*_{1} and *F*_{2} are equal.
Now, according to the basic rule of *free body diagrams,* the three forces *F*_{1}, *F*_{2}, and *W* must add up to zero in order for the *pulley* to remain stationary. Put another way, if the *pulley* isn’t moving up or down, the positive forces *F*_{1} and *F*_{2} are balancing the negative force presented by the urn’s weight, *W*. *Mathematically* this looks like,
*F*_{1} + F_{2} – W = 0
or, by rearranging terms,
*F*_{1} + F_{2} = W
We know that *F*_{1} equals *F*_{2}, so we can substitute* F*_{1} for *F*_{2} in the preceding equation to arrive at,
*F*_{1} + F_{1} = W
or,
*2 ×** F*_{1} = W
Using algebra to divide both sides of the equation by 2, we get:
*F*_{1} = *W* ÷ 2
Therefore,
*F*_{1} = *F*_{2} = *W* ÷ 2
If the sum of the forces in a free body diagram do not equal zero, then the suspended object will move in space. In our situation the urn moves up if our toga-clad friend pulls on his end of the rope, and it moves down if Mr. Toga reduces his grip and allows the rope to slide through his hand under the influence of gravity.
The net real world benefit to our Grecian friend is that the urn’s 20-pound weight is divided equally between him and the beam. He need only apply a force of 10 pounds to keep the urn suspended.
Next time we’ll see how the improved *simple pulley* we’ve discussed today led to the development of the compound pulley, which enabled heavier objects to be lifted.
Copyright 2016 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
____________________________________ |