Abstract
In a recent publication the author derived and experimentally tested several theoretical models, distinguished by different boundary conditions at the contacts with horizontal and vertical supports, that predicted the forces of reaction on a fixed (i.e. inextensible) ladder. This problem is statically indeterminate since there are 4 forces of reaction and only 3 equations of static equilibrium. The model that predicted the empirical reactions correctly used a law of static friction to complement the equations of static equilibrium. The present paper examines in greater theoretical and experimental detail the role of friction in accounting for the forces of reaction on a fixed ladder. The reported measurements confirm that forces parallel and normal to the support at the top of the ladder are linearly proportional with a constant coefficient of friction irrespective of the magnitude or location of the load, as assumed in the theoretical model. However, measurements of forces parallel and normal to the support at the base of the ladder are linearly proportional with coefficients that depend sensitively on the location (although not the magnitude) of the load. This paper accounts quantitatively for the different effects of friction at the top and base of the ladder under conditions of usual use whereby friction at the vertical support alone is insufficient to keep the ladder from sliding. A theoretical model is also proposed for the unusual circumstance in which friction at the vertical support can keep the ladder from sliding.
Highlights
The objectives of the present paper, designated Part II, are twofold: 1) to demonstrate quantitatively that the forces of reaction measured in Part I are consistent with the law of static friction [6] for the usual prevailing condition that friction at the wall alone cannot keep the ladder from sliding, and 2) to clarify several important points of confusion regarding the differences in the relations governing friction at the wall and at the ground
The fixed ladder modeled as an E-B beam has routinely served in the pedagogical literature to illustrate the condition of static equilibrium, the worked models were always simplified by neglect of friction at the wall in nearly every mechanics textbook known to the author in use from the early 20th Century [13] to the present time [14]
In the paper [1] designated Part I, the author derived theoretically and confirmed experimentally a solution to the long-standing “ladder problem”, i.e. the problem to predict the 4 forces of reaction on a fixed ladder resting on horizontal ground and inclined against a vertical wall
Summary
The ladder was modeled as an Euler-Bernoulli [E-B] beam [3] [4] [5] in which the force of friction between the ladder and the wall constituted a critical part of the solution. The objectives of the present paper, designated Part II, are twofold: 1) to demonstrate quantitatively that the forces of reaction measured in Part I are consistent with the law of static friction [6] for the usual prevailing condition that friction at the wall alone cannot keep the ladder from sliding, and 2) to clarify several important points of confusion regarding the differences in the relations governing friction at the wall and at the ground
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