^{1}

^{2}

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.

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?” [

_{2}(t),0) and (x_{1}(t),y_{1}(t)), respectively.

Block 1 is released from rest, as it falls pulls on block 2. The falling block sets a time-dependent tension, T(t), in the line. The massless line holds the same tension in the line. Tension, T, and angle, θ, with vertical are shown.

Applying Newton’s law gives the equations of motion for the blocks,

And

The holonomic constraint is formulated as,

From Equations (1) and (3) and substituting for

On the other hand from Equations (2)-(4) and substituting for

we have,

Equation (6) depends only to the coordinates of block 1.

By the same token Equation (5) is reduced to the coordinates of the block 1 provided

Now Equation (7) depends only on the coordinates of block 1.

Equations (6) and (7) form a set of coupled ordinary differential equations. Hopelessly we were unable to solve the set of Equations (6) and (7) analytically. We apply Mathematica’s numeric solver NDSolve; we were stunned with its power! Coordinates of the blocks are shown in

For practical reasons the half-length of the line is set ℓ = 1.0 m. Because the blocks are identical, derived equations are mass independent. To set the time

scale we apply the free-fall run-time, namely

The run-time of the actual case not being a free fall is a bit longer.

By eliminating parameter t between the coordinates of block 1, it enables us to display its trajectory shown in

As shown in

tion,

As shown, θ, begins at 90˚ and at the end of run-time plunges to about 11˚.

Next we utilize the solutions evaluating kinematic quantities such as speed and acceleration of the individual block. These are displayed in

As depicted the falling block begins with gravity acceleration, g, gains acceleration as it falls,

Applying Equation (1) we also depict the tension in the line, T; this is shown in

We complete the analysis by displaying one of many phase diagrams, namely the plot of horizontal speed of block 1 vs. its horizontal coordinate, i.e.

We analyze a problem that on its face is trivial. Although true, its detailed analysis is proven to be challenging. 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. This is another example that one needs to apply a Computer Algebra System (CAS). By applying one of the most powerful CASs, Mathematica, amazingly we were able to deduce the needed solutions. Having the solutions on hand, various information, e.g. trajectory of the falling object, dynamic quantities e.g. tension in the line is evaluated. For the majority of the quantities of interest we also applied the superb graphic capabilities of Mathematica displaying the results. 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. Reference [

Sarafian, H. and Hickey, N. (2017) Characteristics of a Two- Body Holonomic Constraint Mechanical System. World Journal of Mechanics, 7, 161- 166. https://doi.org/10.4236/wjm.2017.76014