Explore

Learn the power of Geometry Expressions by browsing through our many examples.
Explore > Newsletters > 2013-10
Previous Next
Geometry Expressions Newsletter
Geometry Expressions Newsletter
October 2013   
Mechanical Expressions Beta
Geometry Expressions allows you to create explicit mathematical models of phenomena which can be expressed geometrically.
Mechanical Expressions is a new program, based on the same geometry engine which allows you to create explicit mathematical models of mechanical phenomena.

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

How to... find a reaction force in Mechanical Expressions
A mass m is attached to a wall by two metal rods of length b, attached distance a apart.  We wish to determine the reaction force in the rods and the reaction at the points of attachment to the wall.

Here's how we do this in Mechanical Expressions
  1. Draw the metal rods AB and BC.  
  2. Constrain their lengths to be b.  
  3. Constrain the coordinates of the points of attachment A and C to be (0,0) and (0,a).
  4. Add a mass m at point B.
  5. Select one of the length constraints and use Mechanics Output / Reaction Force to compute the force in the bar. 
  6. Select one of the coordinate constraints and use Mechanics Output / Reaction Force to display the reaction at that connection.   

You'll notice that we measured the reaction force in constraints rather than in geometrical elements.  Mechanical Expressions assumes that it is the constraints which are responsible for holding the model together, and hence carry forces.  These constraint reactions can be interpreted as body forces, assuming your constraint scheme reflects the physics. 

Problem of the Month
 
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.



August Problem Solution
 
circle isotomic
Isotomic
 

Given a circle, and a point P, the isotomic of the circle with respect to P is the locus of the reflection of P in the tangents to the circle.

 

Let A be a point on the circle, and let B and C be points on the line through A and P a fixed distance d from A.

 

Then the locus of B and C as A runs round the circle is a Circle Conchoid.


For what locations of P and values for d is
circle conchoid
Conchoid
the conchoid the same as the isotomic?

   

An app which lets you explore this problem is
here.

Take the isotomic diagram and let B be the image of P under reflection in a line through O parallel to the tangent. As BP' is the composition of two reflections it is a translation of twice the distance between the lines.  Hence |BP'| is the diameter of the original circle.  B also lies on the circle OP.
Hence if P lies on the circumference of the circle OA, and if d = 2|OA|, the conchoid and the isotomic are identical.


eBooks

A breakthrough in the use of interactive diagrams for gaining new insight into mathematics.
Available on iTunes
common core book cover
Each group of apps consists of simple, short, and focused interactive demonstrations and explorations of a single standard.
Available on iTunes

L Van Warren has combined Geometry Expressions apps with the Kindle in his latest eBook available from amazon.com here
11 Geometry Expressions  eBooks are now available as a bundle for the recession-friendly price of $85.95

Quick Links
Euclid's Muse

Geometry Expressions

Books