My recent post ontightropes reminded me of a great “hack.” Here’s the situation. Your car is stuck in the mud, so you grab a rope and tie it to the front of the car and then the other end to a very sturdy tree. Now for the trick—grab the rope in the middle of the length and pull perpendicular to the rope. Here is a diagram.
It’s a cool and useful trick, but how does it work? In short, it’s the same as standing on a tightrope. The forces at the point of contact have to add up to the zero vector if it’s in equilibrium.
Let’s take a closer look at the point that you would pull on the rope. At this point, there are essentially three forces.
With the contact point in equilibrium, these forces have to add to zero. The only component of force that is interesting is that perpendicular to the rope. Assuming the magnitude of the two tensions is the same, then I get the following expression.