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
 Draw the metal rods AB and BC.
 Constrain their lengths to be b.
 Constrain the coordinates of the points of attachment A and C to be (0,0) and (0,a).
 Add a mass m at point B.
 Select one of the length constraints and use Mechanics Output / Reaction Force to compute the force in the bar.
 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

 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
 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 = 2OA, 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 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 recessionfriendly price of $85.95


