Quote of the Day
Once you get to earth orbit, you’re halfway to anywhere in the solar system.
I have been tutoring math and physics at the local library for the last few months. As part of this tutoring, I have been looking for good graphics that illustrate basic science concepts. One common high-school physics problem involves computing the tension in ropes tied to an anchor by a pulley. Figure 1 is a graphic that nicely illustrates the tension between two ropes connected to an anchor point by a carabiner.
In this post, I will show how to derive a couple of formulas for the various angles and forces shown in Figure 1. The derivation assumes that there is no friction associated with the carabiner, which is not true. This idealization would be better if the carabiner was replaced with a pulley.
I find infographics like this useful because I frequently use ropes, pulleys, and winches at my cabin to perform tasks like removing tree stumps, pulling docks out of the water, and helping folks whose cars are stuck in the mud.
Figure 2 shows a free body diagram of the rigging and my derivation of the angles and forces involved.
I created an Excel Workbook that computes the values shown in Figure 3. It also computes the angle of the gray rope in Figure 1, which is not shown in that diagram.
In Figure 2, I derived . This equation is not defined when φ = 0. You can circumvent this discontinuity using one of three techniques:
- Assume that φ is very small rather than 0. The classic approach for an engineer and the one I typically use.
- Derive an alternate expression that removes the discontinuity, which is shown in Figure 4.
- Just eliminate φ from the equation for F as shown in Figure 5.