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A friend recently sent me this cute video from YouTube, and I thought it might be worth analyzing:
As you can see, this is a pretty simple system consiting of what looks like a book, a string, two pulleys, and some weights. The string seems to be passed along the inside of the book's spine with each end going up to a pulley and then down to a weight (actually 3 small weights on each side, but we can consider each set of 3 weights as a single weight).
What's interesting is that when the strings are all parallel, the weights are sufficient to balance the book's weight. But when the strings supporting the book are crossed, those same weights are NOT sufficient to balance the book's weight. So what's going on here?
Let's start by naming the total weight on each side as "W" pounds (each of the three small weights would be W/3 pounds). So the tension in the strings between the pulleys and the weights is a tension of "W" pounds.
In real life, the pulleys will introduce some small amount of friction, but for this analysis, we'll ignore that. So the tension between each of the pulleys and each end of the book will be the same tension as is between the pulleys and the weights. In our example, we've said that was a tension of "W" pounds.
If you think about it, you'll realize that the string carries the same tension of "W" pounds in either configuration. So why does that tension support the book in one case (where the strings are parallel) but not in the other (where the strings are crossed)?
The answer lies in the direction of the tension, and that brings up the topic of force vectors. A force vector is a specified force along a specified direction. In both configurations, the amount of force (or tension) along the strings is the same "W" pounds. But the directions are different.
In the parallel string case (where the book is suspended), the force vectors are vertical so 100% of the tension works toward supporting the book. Since the book is balanced in this case, we can infer that the book weighs 2xW pounds since it takes both of the "W" weights to balance it.
But in the crossed string case (where the book is NOT supported), the force vectors are NOT vertical. In the crossed string case, some of the tension in the strings is vertical and supporting the book, but some of the tension is horizontal which is not helping to support the book at all. In fact, the horizontal components of the two strings are actually squeezing the spine of the book between them.
So with the crossed strings configuration, some of the tension in the strings is "wasted" as the strings "work against" one another squeezing the spine of the book. That leaves less tension in the vertical direction to support the book, and it falls.
Thanks for that fun diversion!!
