Abstract
A loose rod of mass m1 and length l leans against one of the faces of a cube of mass m2 and side length a. The assembly is placed on a horizontal table with one end of the rod touching the table and its other end leaning against the edge of the cube. We set the rod and the center of mass of the cube on the same vertical plane, and then we release the assembly from the rest. For frictionless contacts, we calculate the separation runtime of the rod from the cube as a function of m2/m1 and a/l. This entails forming the equation describing the motion of the system. The equation of motion is analytically unsolvable nonlinear differential equation. Applying a Computer Algebra System, specifically Mathematica [1] [2], we solve the equation numerically. Utilizing the solution, in addition to evaluating the separation runtime, we quantify a list of dynamic quantities, such as the time-dependent interface forces, and, geometric quantities, such as the trajectory of the loose end of the rod. A robust Mathematica code addresses the “what if” scenarios.
Highlights
Motivation of suggested investigation stems from the fact that the proposed assembly is composed of a sliding cube that acts as a point-like object with only linear kinematics in contrast to the rod that in addition to the former possesses rotational degrees of freedom
In addition to evaluating the separation runtime, we quantify a list of dynamic quantities, such as the time-dependent interface forces, and, geometric quantities, such as the trajectory of the loose end of the rod
The problem appears to be a 2D two-body problem with numerous degrees of freedom ; the restricted motion of the rod practically reduces to a problem with only one degree of freedom
Summary
Motivation of suggested investigation stems from the fact that the proposed assembly is composed of a sliding cube that acts as a point-like object with only linear kinematics in contrast to the rod that in addition to the former possesses rotational degrees of freedom. Utilizing the numeric solution in hand, we quantify a list of kinematic information primary as noted the separation runtime of the rod-cube system. Additional quantities such as speed, acceleration and contact forces of the bodies are quantified as well. We utilize Mathematica numeric utilities solving the needed equations conducive to quantifying kinematic and dynamic quantities of interest. We close the report with concluding remarks suggesting ideas furthering the investigation
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