Motion System Design

# Fun with Fundamentals: Problem 171

Bar nonsense and Solution to Problem 170. Three rights can sometimes stack up to one wrong, as this month's problem by Dave Ahmad of Metuchen, N.J., demonstrates.

### Bar nonsense

Problem 171 — Three rights can sometimes stack up to one wrong, as this month’s problem by Dave Ahmad of Metuchen, N.J., demonstrates.

“... just because I screwed up once before ...,” muttered Lucius Bluff. “Of course I ordered 3.5-in.- diam. rods! You can see for yourself!”

Bluff had interspersed the circular rods with some three-cornered rods.

“There is no waay that those circular rods are that large,” retorted Theodore McSnibb. “Why, they fit snugly between the three-cornered rods, which are 5 in. on each side.

To settle the argument, Bluff and McSnibb took a random sample of one circular rod surrounded by three 3-cornered rods. They found that this configuration forms a large equilateral triangle that is 10 in. on a side. The inner triangle that holds the circular rod is also equilateral, and its sides are tangent to the circular cross-section of the rod. What is the diameter of the rod?

Technical consultant: Jack Couillard, Menasha, Wis.

Solution to last month’s problem 170 — You’re no name dropper if you answered 60 deg. Here’s how the publicity shot became an action photo.

Let:
E = Rod’s modulus of elasticity, given as 30 × 106
R = Radius of curvature at the neutral axis, in.
I = Moment of inertia of a section of rod, lb-in.
M = Bending moment, lb
σ = Bending stress, given as 49,000 psi
C = Distance from the beam’s neutral axis to extreme fiber, given as 1/32 in. 42, or 1/64 in.
θ= Angle of circular arc, deg
L = Length of arc, given as 10 in.

From the mechanics of beams, we know that, for beams with symmetrical sections in pure bending:

or,

Set (1) and (2) equal to each other and solve for R.

Knowing R, we can now solve for θ.

Contest winner — Congratulations to 8th grader Bogdan Tache of North Canton, Ohio, who won our February contest by having his name drawn from the 253 contestants who answered correctly out of a total of 271 for that month. A TI-68 calculator is in the mail to him.

The TI-68 Advanced Scientific Calculator by Texas Instruments can solve five simultaneous equations with real and complex coefficients and has 40 number functions that can be used in both the rectangular and polar coordinate systems. Other functions include formula programming, integration, and polynomial root finding. The calculator also features a lastequation replay function that lets you double-check your work.