Characteristics of a Two-Body Holonomic Constraint Mechanical System

Haiduke Sarafian,Nenette Hickey

doi:10.4236/wjm.2017.76014

Abstract

Two massive blocks are connected with a massless unstretchable line of 2l. One of the masses is placed on a horizontal frictionless table, l distance away from the edge of the table the other one is held horizontally equidistance from the edge along the extension of the line. The latter is released from rest. As it falls under gravity’s pull, it drags the one on the table. It is the interest of this investigation to analyze the kinematics of the system. Because of the holonomic constraint of the system, analysis of the problem encounters complicated super nonlinear coupled differential equations. Utilizing Mathematica we solve the equations numerically. Applying the solutions we quantify numerous kinematic quantities; most interestingly we evaluate the run-time, and the trajectory of the falling block. Analysis is robust allowing us to address the “what if” scenarios.

Highlights

The investigation of the proposed scenario outlined in the abstract stems from questioning, “For two identical blocks which one reaches the vertical leg of the table first?” [1] [2]
The challenge stems from the fact that the solution of the needed equations being a set of coupled super nonlinear ordinary differential equations analytically is unsolvable
An interested reader may extend the scope of the investigation considering 1) non-identical masses 2) determining masses making the blocks reach the edge simultaneously and 3) replacing the massless line with a massive one

Summary

Motivation and Goals

The investigation of the proposed scenario outlined in the abstract stems from questioning, “For two identical blocks which one reaches the vertical leg of the table first?” [1] [2]. It is an easy task without quantifying. Quantification of the run-time, requires analytic analysis It requires developing either expressions for the horizontal displacements of the blocks or their numeric plots vs time.

Physics of the Problem

Conclusions and Suggestions