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# Classroom Tips and Techniques: The Sliding Ladder

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Classroom Tips and Techniques: The Sliding Ladder

Robert J. Lopez

Emeritus Professor of Mathematics and Maple Fellow

Maplesoft

The Mathematical Model

Model a ladder of length  as the line segment between  and  in the first quadrant of the -plane. (See Figure 1.)

Let  determine the position of a point on the ladder by writing

so .

 > p1:=plot([[1,0],[0,2]],style=line,color=black,thickness=3): p2:=plots:-textplot({[.5,.1,typeset(a)],[.6,1,typeset(L)],[.2,.8,typeset(sqrt(L^2-a^2))]}): plots:-display(p1,p2,scaling=constrained,tickmarks=[0,0]);

 >

Figure 1   Sliding ladder of length

Eliminate  to discover that the orbits of the fixed point lie on the ellipse . This result is obtained by solving  for , and substituting for  in  to obtain the first entry in Table 1. The remaining steps in the calculation are annotated in Table 1.

 Set  in Square both sides Add  to both sides Divide both sides by Table 1   Calculations showing that the trajectory of a point on a sliding ladder is part of an ellipse

The midpoint of the ladder, characterized by , traces the first quadrant portion of the circle

Table 3 in the Appendix provides an interactive and annotated derivation of the results in Table 1.

Visualization

 • Figure 2 provides a version of Adri van der Meer's solution for the sliding ladder. The trajectory of the midpoint traces the first-quadrant portion of a circle of radius 1.
 • The -coordinate of the bottom of the ladder is given by the value of the parameter .
 • Clicking the button "Figure 2" re-initializes the animation.
 • Click on the graph to access the animation toolbar, in the center of which is a slider that can be used to move through the frames of the animation.

Doug Meade's suggested applying the animate command to a function whose output was a single frame of the animation. The code in Table 2 defines , a function whose output is the single frame of an animation in which an arbitrary point on the ladder is determined by the value of .

 Table 2   Definition of the function

There are several ways to utilize the function . The older interactiveparams command can be applied to generate a Maplet pop-up with either one or two sliders. With two sliders, the "animation" is executed by moving the slider for ; with one, the animation is auto-executing.

Alternatively, the revised Explore command can be applied, this time resulting in embedded components with again, either one or two sliders.

Figure 3 illustrates the Maplet pop-up with two sliders; Figure 4, with one slider.

 • In Figure 3, the slider controlling , the location of the bottom of the ladder, implements the animation. The slider controlling  selects a point on the ladder.

 • The output from this command is deliberately suppressed for several reasons, not the least of which is that the Interactive Parameter Maplet has a tendency to crash Maple when the sliders are moved too quickly or too often.
 • In Figure 4, the single slider controls , and hence the location of the fixed point on the Ladder tracing the curve shown in red. The Start button initiates the animation of the sliding ladder.

 • As for Figure 3, the output of the interactiveparams command is suppressed.

 Figure 3   Interactive Parameter Maplet with two sliders

 Figure 4   Interactive Parameter Maplet with one slider

The equivalent of Figure 3, but with embedded components, is produced by the following application of the (Maple 17) revised Explore command. (The function  must first be defined before the sliders below will work.)

 : :

The equivalent of Figure 4, but with embedded components, is produced by the following application of the (Maple 17) revised Explore command. (The function  must first be defined before the sliders below will work.)

 :

Appendix

Table 3 provides an interactive and annotated derivation of the implicit form of the trajectory of an arbitrary point on the sliding ladder.

 • Enter the parametric equations for the trajectory and press the Enter key.
 • Context Menu: Solve_Eliminate a Variable_

 • Context Menu: Conversions_Equate to 0

 • Select ; add  to both sides via Smart PopUp.

 • Context Menu: Manipulate Equation Use Equation Manipulator to square both sides. Return result.

 • Context Menu: Move to Right

 • Select  and use the Smart PopUp to factor this to .

 • Context Menu: Expand_ Holding Unexpanded_

 • Context Menu: Conversions_Equate to 0

 • Select ; add  to both sides via Smart PopuUp.

 • Select ; divide by . via Smart PopUp.

 • Context Menu: Expand_ Holding Unexpanded_.

Table 3   Derivation of the implicit form of the trajectory of an arbitrary point on the sliding ladder

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