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Analysis: Rope around a bollard

This analysis produces the most delightful equation.
We'll state it without proof, but if you know calculus, look it up and revel in its elegance.
Here’s the experiment with the string over the circular rod. It’s like a rope wrapped around a bollard to make it easier for someone to secure a boat. To analyse this, we’ll need to use the exponential function. But don’t worry, we’ll explain it all. Here’s a diagram of a rope around a cylindrical rod. We need an expression for the ratio of tensions. Proving the analysis requires calculus, so we’ll just quote the equation. It’s a lovely one. The equation is T2 over T1 equals exp mu theta, where theta is the angle of wrap in radians. Don’t worry about that exponential function, exp mu theta. Just press the exp button on your scientific calculator.
Radian measure for angles is common in engineering mechanics. You can convert radian measure into degrees and vice versa by using the identity pi radians is 180 degrees. The figure shows 90 degrees, which is pi by 2 radians.
It should all be clearer when we use it on the experiment. Let’s see if we can find the coefficient of friction from our measurements with the rod.
In the first experiment, the angle of wrap was half a turn. That’s 180 degrees, or pi radians. We started with T1 at 4 washers. As T2 equals 10 washers, and this was when the string started to move around the rod. So we get exp mu pi equals 10 divided by 4 equals 2.5. But we’ve got a problem. We want to find mu, which is inside the exponential function. We need another trick. How do we get inside the exponential function? Just take it for granted that taking the natural logarithm undoes an exponential. So if we take the natural logarithm of both sides, we get mu pi equals the natural logarithm of 2.5. ‘ln’ is the abbreviation for natural logarithm.
Don’t understand natural logarithm? No problem. Just press the ‘ln’ button on your calculator. You should get 0.916. Finally, divide by pi to get mu equals 0.29. Now for the second experiment with its second turn. In this case, T1 was 4 washers and T2 was 44 washers. And we had 1 and 1/2 turns. Pause the video and work out the angle of wrap in radians. Remember, pi radians are 180 degrees.
Here’s the answer. The angle of wrap is 1 and 1/2 times, or 540 degrees, or 3 pi radians. T1 was 4 washers, T2 was 44 washers. You should now be able to calculate the coefficient of friction. Pause the video and try it.
Here’s the solution. We got coefficient of friction mu equals 0.25. Let’s compare the results for these two experiments, one with half a turn, one with 1 and 1/2 turns. Experiment 2 gave mu equals 0.25. And earlier, experiment 1 gave mu equals 0.29. In the first experiment, the weights almost moved with 9 washers. Repeating the calculations with this gives coefficient of friction equals 0.26. What do you think of the variation between the two experiments? Post your thoughts on the discussion. Excitement over for now. There are plenty of other engineering components we could have analysed - wedges, screws, clutches, brakes. But we have to move on to your design segment.

This analysis produces the most delightful equation.

We’ll state it without proof, but if you know calculus, look it up and revel in its elegance.

You’ll be using it for your design task.

Data from the experiment are given in the Downloads section below.

Talking points

  • What do you think of the variation in the estimate of coefficient of friction between the experiments with two angles of wrap?

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Through Engineers' Eyes - Expanding the Vision: Engineering Mechanics by Experiment, Analysis and Design

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