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Geometry Expressions Newsletter
Geometry Expressions Newsletter
December 2013   
Mechanical Expressions Beta
Mechanical Expressions is a new program, based on the same geometry engine which allows you to create explicit mathematical models of mechanical phenomena.

We showed Mechanical Expressions at the ASME Convention in San Diego and got some interesting feedback.

Interested in checking it out? Want to give us your own feedback?

Mechanical Expressions is in open Beta, and a copy can be downloaded from here.


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How to... do kinematics in Mechanical Expressions

A crank slider has crank velocity w.  What is the velocity of the slider?

Here's how we do this in Mechanical Expressions
  1. Draw the triangle ABC, positioning A at the origin and C on the x axis.   
  2. Constrain the lengths of AB and BC and the angle BAC.  As the angle changes, AB will rotate around A while C slides back and fort
  3. h along the x axis.
  4. Select the Angle and use Mechanics Input / Velocity/A
    cceleration
    .  Specify a velocity of w and leave the acceleration zero.
  5. Select point C and use Mechanics Output / Velocity Acceleration.  You will get the velocity and acceleration of the point.
Problem of the Month
 
Points A, B and C lie on a smooth closed curve such that triangle ABC has sides with maximal sum of squares.  Show that the medians of ABC are normal to the curve.

   

An app which lets you explore this problem is here.


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Solution to October's Problem
 
  Unequal masses M and m are placed on the ends of a rod whose pivot point is movable.  We know that the beam balances if k=m/(M+m).  Hence the equivalent simple pendulum would have infinite length.

Find the value of k which yields the shortest equivalent simple pendulum.  Show that the length of this simple pendulum is the ratio of the geometric mean of m and M to their arithmetic mean.

   

An app which lets you explore this problem is here.


The acceleration of a simple pendulum is inversely proportional to its length. Hence the simple pendulum will have shortest length when its acceleration is greatest.  We need to find the value of k in our compound pendulum which maximizes the acceleration.

The acceleration of the angle can be computed using Mechanical Expressions.


This value may be copied into a CAS, differentiated and solved to find a maximum.

The result may then be inserted into Mechanical Expressions instead of k, as shown left.

The acceleration value at the maximum is automatically computed by Mechanical Expressions.  You may observe (below) that the length of the corresponding simple pendulum is indeed the length of the compound pendulum times the ratio of the geometric mean of the masses to their arithmetic mean. 













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