Here are a couple of examples of interesting results that come out of a trigonometric analysis when either the sine or cosine function ends up in the denominator. Remember that sin(θ) or cos(θ) can equal zero, so the graph of any function that has either in the denominator will include vertical asymptotes. Physically, this means that a force represented by a vector in a function can be infinitely large in the right circumstances.
Take a look at the rock climbing example below. When setting up a rock-climbing anchor, one wants always to have some redundancy — more than one anchor, and to share the load (e.g. of a falling person) among two or more
anchors (we'll just use two in this analysis). The angle θ in the left figure is the angle between the anchors. Each anchor shares some fraction of the downward load. But that share is only one-half of the load if θ = 0˚. Otherwise, the tension in each of the anchor "arms" gets larger as θ grows.
Study the force diagram on the right side of the figure. The forces in each of the two anchor arms (we'll assume they're the same for simplicity) is the sum of vector forces in the upward and left—right directions. The left-right forces cancel at equilibrium, so the upward forces (F1y and F2y) must equal the normal force, FN.
Because of the left-right symmetry of this problem, we only need to find the tension in one arm of the ancor system. The other is the same (we'll use the right side).
The sum of the upward forces must equal the normal force (the negative of the load or gravitational force).
We can use the cosine function to relate F2y to F2, as shown, then we rearrange to get F2, which we call the tension in that anchor. Because the tension contains the cosine of an angle in the denominator, and we know that cos(90˚) = 0, we see that when θ = 180˚, the tension becomes infinitely large.
Note: Force in a rope or wire is usually called tension.
This analysis shows that it will pay to keep the angle between your climbing anchors as close to zero as possible in order to minimize the actual force applied to each anchor point. Those forces can multiply dramatically during the downward acceleration of a falling climber.
This graph shows that as the angle θ approaches 180˚ any function proportional to [cos(θ)]-1 asymptotically approaches infinity.
Our rock-climbing example has a lot of other implications. For example, think about overhead wires like the power and communicaion wires likely strung through your neighborhood. Notice that all of them sag just a bit.
If you think of one span of these wires forming a climbing anchor, with each pole as an anchor point and an invisible load in the middle of the wire (representing the weight of the wire itself), you can see that it would take an infinite amount
of tension in the wire in order not to have a bit of sag. It's impossible to get rid of that sag. Even the tightest cable strung between the most solid anchors will sag just a little. Even a guitar string sags just a little under the force of gravity, it's just very light and doesn't sag a noticable amount.
The shape of a sagging cable, rope or chain, which looks a bit like a parabola, is actually a catenary curve. I will describe catenary curves in more detail if I ever find the time!
Image: Associated Press
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