Heavy-Duty Forces

Here's a little more advanced discussion of curve shape, which you can skip if it's too much. The way to analyze the curve shapes on the first page, the parabola and catenary, from the forces that form them is to consider the slope of the curve at any given point. You've probably studied the arc of a projectile, like the water jet, and the slope of that curve (a parabola, if we can neglect air resistance) is the angle of the velocity vector at each point, since the curve is actually formed by the motion of the projectile along that vector. The horizontal component of the velocity is constant, and the vertical component of velocity increases uniformly with time, by the constant acceleration of gravity. (Yes, I'm assuming you already know this.) Thus the slope (the ratio of vertical to horizontal velocity) increases uniformly with time...the curve points more and more downward. But we want to know slope as a function of position not of time! The trick to getting one from the other is that constant horizontal velocity: horizontal distance increases uniformly with time, so the slope increases uniformly with ("is proportional to") horizontal distance from the peak of the parabola. If you are twice as far in x-coordinate from the peak, twice as much time has passed from when you were at the peak, and so the slope is twice as great. In the diagram, the slope at x is s, and the slope at 2x is 2s. (Note that this s value on the vector is its slope, not its length like usual.) This is a property of the parabola that comes up in calculus, where slopes are a major topic. Why do we care about the slope here? Read on!

The slope of the the suspension bridge cable at any point can be found by thinking along generally similar lines ;) In this case, the slope is still the ratio of vertical to horizontal components of a physical vector, but now the vector is force not velocity, since nothing is moving. Remember, the cable shape is defined by the force being along the cable. We emphasize the similarity with the previous case in a silly way: the diagram is the same one, just upside down, even the letters! Again, the horizontal component of the force is constant, but the vertical force component is the total weight below the point we're thinking of. If you think about what's suspended in a suspension bridge, the heavy part is almost entirely the roadway, not those thin cables holding the roadway. The roadway is uniform, so the weight from the center out to some point is just proportional to the length of the roadway, the horizontal distance from the center to any point. So again, the vertical force is proportional to the horizontal distance from the center, and so is the cable slope. Thus we have shown that the cable traces a parabola! (since the slope variation determines the curve)

Read that again and think about it.

If you followed that, congratulations! You've just solved the differential equation for the suspension bridge, a topic in advanced calculus! If not, don't worry: it is very advanced.

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