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Physics of a Sliding Ladder

Using results from related rate problems, some calculus books suggest that a ladder leaning against a wall and sliding under the influence of gravity will reach speeds that approach infinity. This Demonstration is built from the actual equations that govern the motion of the ladder as determined by the theory of rigid body mechanics. It shows that a sliding ladder never reaches very high speeds. The motion can be followed in two contrasting situations, with the top of the ladder either free to move away from the wall or constrained to be in contact with the wall. The forces are calculated for the falling ladder just before the top hits the floor.

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The ladder is represented as a uniform rod of mass and length . Friction is ignored. Let be the reaction at the top of the ladder and let be the reaction at the bottom of the ladder. Let denote the angle of the ladder with the vertical and the initial value of . The position of the center of mass is given by
,
.
Motion I: The ladder is simply leaning against the wall, so that the top of ladder can move away from the wall, giving
.
Moreover,
.
Thus the ladder leaves the wall when , that is, when
.
This result shows that the ladder leaves the wall as the top end falls one third of its original height.

After the ladder leaves the wall, it falls only under the action of gravity and the reaction force . For this second part of the motion, the position of is given by
where
and
and the time is measured from zero at the moment the ladder leaves the wall.
When the ladder hits the floor, the top end has a finite velocity
and a speed
.
Motion II: The ladder is constrained, for example by means of a frictionless ring, so that the top of the ladder is not allowed to leave the wall. Here
In this case, the reaction at the top end of the ladder originally points in the direction away from the wall. As the ladder falls, decreases to zero when the top has fallen one third of the original height. Then it reverses direction, increasing continuously.
The reaction at the bottom end first decreases then increases and, when the ladder hits the floor, it takes the nonzero value . Furthermore, the final velocity of the bottom end of the ladder is zero. More details can be found in the [1].
Reference
[1] S. Kapranidis and R. Koo, "Variations of the Sliding Ladder Problem," The College Mathematics Journal, 39(5), 2008 pp. 374–379.
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