# ma012 - Design of an Oscillating Sprinkler Bart Braden...

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Design of an Oscillating Sprinkler Bart Braden Northern Kentucky University Highland Heights, KY 41076 Mathematics Magazine, January 1985, Volume 58, Number 1, pp. 29–38. T he common oscillating lawn sprinkler has a hollow curved sprinkler arm, with a row of holes on top, which rocks slowly back and forth around a horizontal axis. Water issues from the holes in a family of streams, forming a curtain of water that sweeps back and forth to cover an approximately rectangular region of lawn. Can such a sprinkler be designed to spread water uniformly on a level lawn? We break the analysis into three parts: 1. How should the sprinkler arm be curved so that streams issuing from evenly spaced holes along the curved arm will be evenly spaced when they strike the ground? 2. How should the rocking motion of the sprinkler arm be controlled so that each stream will deposit water uniformly along its path? 3. How can the power of the water passing through the sprinkler be used to drive the sprinkler arm in the desired motion? The first two questions provide interesting applications of elementary differential equations. The third, an excursion into mechanical engineering, leads to an interesting family of plane curves which we’ve called curves of constant diameter. A serendipitous bonus is the surprisingly simple classification of these curves. The following result, proved in most calculus textbooks, will play a fundamental role in our discussion. LEMMA. Ignoring air resistance, a projectile shot upward from the ground with speed v at an angle from the vertical, will come down at a distance (Here g is the acceleration due to gravity.) Note that gives the maximum projectile range, since then Textbooks usually express the projectile range in terms of the “angle of elevation’’, but since the range formula is unaffected when the zenith angle is used instead. The sprinkler arm curve In Figure 1, a (half) sprinkler arm is shown in a vertical plane, which we take to be the xy plane throughout this section. Let L be the length of the arc from the center of the sprinkler arm to the outermost hole, and let be parametric equations for the curve, using the arc length s , as parameter. Let denote the angle between the vertical and the outward normal to the arc at the point We’ll see that the functions and which define the curve are completely determined (once L , and have been chosen) by the requirement that streams passing through evenly-spaced holes on the sprinkler arm should be uniformly spaced y s 0 d a s L d y s s d x s s d s x s s d , y s s dd . a s s d 0 s L , y 5 y s s d x 5 x s s d , sin 2 s p y 2 2 u d 5 sin 2 u , p y 2 2 u ; sin 2 u 5 1. u 5 p y 4 s v 2 y g d sin 2 u . u

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when they strike the ground. If there were a hole at the point on the sprinkler arm, the direction vector of the stream issuing from this hole would be and this stream would reach the ground at a distance The condition that evenly-spaced holes along the arm produce streams which are evenly spaced when they reach the ground is that be proportional to s :
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