This is very exciting: a former student of mine, Anthony Attia, has submitted a solution to the Tarzan rope problem I posted some weeks ago. Anthony was in my Mechanics class at Vanier College in 2016. He is now pursuing undergraduate studies in mechanical engineering and simultaneously doing a stage at my former employer, MDA Space.

As is the case with some students, Anthony and I have stayed in touch since he graduated from college. This post, however, is the *first* one in more than ten years of this blog's existence that someone other than me has written; it is about time. Watch as Anthony analyzes a uniform rope, pinned at the top and vertically suspended, subjected to a horizontal uniform wind.

The following text appears here with Anthony Attia's consent:

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When
faced with a complex physical phenomenon, it is quite common to simplify the
problem to a point where an analytical solution can be formulated. The
simplification is done by stating assumptions throughout the approach. The more
assumptions we take, the more likely our approximated answer will diverge from
the true value. As students of science, it is our duty to ensure that we are
equipped with enough knowledge to apply the proper assumptions.

Tarzan’s rope problem can be as complex as we
want it to be. We can treat the rope as either flexible or rigid, we can treat
the wind force as a function of time or a constant, we can consider the effects
of cold temperature on the characteristic properties of air or we can neglect
them. For the sake of maintaining my sanity and that of the reader’s, we shall
treat the rope as a pinned rigid body who is subjected to a constant drag force
that is acting in the horizontal direction. An important fact about assumptions
is that there cannot be an incorrect one per say, however, every single one of
them must be justified.

In my preliminary analysis, I will assume the rope to be rigid, effectively assuming that the profile of the rope will be linear when displaced. Generally, this assumption would not be valid with a rope, but I will make it anyway and check the extent to which it was good later.

With that in mind, we can begin
trying to find the velocity of the wind, by relating the drag force *FD* and the weight *W*. Consider the model below, which depicts the scenario:

Given that the net
drag force is acting on the center of gravity in the horizontal direction and
the weight is acting in the vertical direction, the ratio of these forces, *FD/W*, ends up being equal to tan(*θ).* We can take the sum of all torques about the
pin and put them equal to zero. Then, using the following

definitions, we may express the wind speed as a function of the other parameters.

Surface gravity: *g*

Air density: *p*

Wind speed: V

Rope angle: *θ*

Rope mass: *m*

Rope diameter: *d*

Rope length: *L*

Rope shape (cylinder) coefficient of drag: *Cd*

The wind speed is then given by:

Knowing this, we may begin computations to determine the wind speed that causes a specific rope deflection. Assuming some reasonable 'Tarzan rope' values, it takes a 15 m/s wind to rotate the rope by 30°. This seems reasonable. But, we can only feel so much confidence in this result, as it is based on an assumption that may not be justifiable.

Say, however, that
we now want to treat the rope as a flexible body; how would we proceed? Before
answering that question, we must properly understand the behavior of weight and
drag. In the previous figure, the drag force was lumped into a single vector whose
line of action passes through the center of mass of the rope. Let us do a quick thought experiment: if we
were walking headwind, would our entire body feel pushed by the drag force or
just a single point? The answer is the former. So, why did we draw a single
vector? That vector is actually the resultant or net drag force acting on the
rope. If we were to properly illustrate the aerodynamic force that the body is
subjected to, we would have to draw many smaller vectors that are acting on the
entire exposed surface. These types of forces are called distributed load: though they act on every point of the body, we may sometimes use a single vector to
represent the resulting effect (note that gravity is similarly distributed and then a resultant is used). Every segment of the rope has a mass equal to *dm* and the sum of all segment masses will yield
the total mass *m*. Now, to solve the
flexible body problem, we must assess a differential segment *dm* that is exposed to a differential drag of *dFd* by drawing its free body diagram.

Newton’s second law in x and in y
yields:

These equations simplify to:

Equalizing the two equation we get:

It is evident that the equation obtained
for the flexible body problem is the same as the rigid body problem, however,
it is in a differential form. To remove the differentials, we must apply an
integrating operator to the equation. If we do so, the same expression linking the angle to the wind speed is obtained.

We conclude that both approaches lead to
the same answer, but one requires an understanding of calculus, whilst the
other requires only an understanding of mechanics. As one of my professors used
to say, the simplest solution is often the best solution!

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It gives me much pride to see a former student of mine express himself as he does here. I get the same result on my end as that which he found. The reason that the rigid body assumption works is because, due to the symmetry of the scenario, the uniformity of the fields and rope, the rope's profile must be linear.

In this problem, we have the weight force and the drag force. They act vertically, and horizontally, respectively, onto each element. While weight acts on dm elements, and drag acts on dA elements, both are uniform: we may think of each as a uniform field. Effectively, they combine to form a uniform net field, and the rope simply aligns itself with it. Though I initially thought the rope would have some curvature, it does not. I am almost disappointed that the result is so simple. I will try to pose a problem that has a stranger result in my next post.

Still, what I really want to emphasize here, is something greater than the problem itself. I am thrilled that *The Engineer's Pulse* just had its first guest writer; he happens to be a fine engineer in the making.