Last assignment, Phys 311

1.

Second moments. The quantity

$$I=\int y^{2}dA$$

as defined in Feynman is called the ``second moment of area'' by mathematicians, and the ``moment of inertia'' by most people.

(a)

For a simple cross section of a rod or beam, sketch the trace of the neutral plane and indicate the coordinate y to be used in the above formula. Justify Feynman's assertion that I is the moment of inertia of a slice with unit mass per unit area (about which axis?).

(b)

Compute I for a beam having rectangular cross section of sides a and b. If b = 2a, how much more force is required to bend the beam in the easy direction than in the other?

(c)

Compute I for a tube having inner radius a, outer radius b. The math is easier if you show first that, in this case,

$$2I=\int \left( x^{2}+y^{2}\right) dA$$

Is a hollow tube harder to bend than a solid cylinder of the same weight and length?

2.

Breaking columns. Galileo's treatise Two New Sciences introduced scaling, along with many other new concepts. Please read Prof. Fowler's lecture notes for Phys 109 at
http://www.phys.virginia.edu/classes/109N/lectures/scaling.html
and answer the following:

(a)

Exactly how much bigger must the rope be to hold up the chandelier that is eight times heavier? How much bigger must the radius be, how much bigger the cross-section?

(b)

A marble column laying on the ground has a tendency to break if it supported at two points only. Which is more likely to break, a small column or a larger one of the same shape? If the small column has length l and radius r and the big column has length L, what must its radius be to make it as unlikely to break as the small column? (The answer is in Two New Sciences.)

(c)

If the size of a building is doubled, all lengths being doubled, how much stronger must the materials be to make the bigger building as safe as the original? Assume that the strength can be increased without changing the density.

3.

The catenary. Suppose you hang a rope with a constant mass per unit length. as shown below. What is its shape? For an unstretchable rope or chain, this is a famous problem, often given in calculus courses.

(a)

Consider a small segment of the rope, as shown, which has a mass dm. Show that  

$$d\left( {\frac{dy}{dx}}\right) ={\frac{(dm)g}{T_{1}}}.$$ (1)

What is the meaning of T₁? Why is it a constant?

(b)

Let $\lambda$ be the mass per unit length of the rope, so that $dm=\lambda \,ds$, with dm the arclength of the segment. By using your result from part (a), show that the shape y(x) satisfies the differential equation  

$${\frac{d^{2}y}{dx^{2}}}={\frac{\lambda g}{T_{1}}}\sqrt{1+\left( {\frac{dy}{dx}}\right) ^{2}}.$$ (2)

(c)

This differential equation is nonlinear, and generally difficult to solve; the solution is  

$$y(x)=a\cosh (x/a).$$ (3)

Substitute this trial solution into the differential equation, Eq. 2, and verify that it is indeed a solution, if a is chosen properly. This solution is known as the catenary.

(d)

Suppose you want to build an arch whose members are only in compression. What shape should it take? Why? This is the shape of the famous Gateway Arch in St. Louis.

4.

Humerus allometry. Allometry is the study of the growth of part of an organism relative to the entire organism. Allometric studies often reveal scaling relations between the size of a limb and the mass or height of an animal.

(a)

The length l versus the thickness d of the humerus for different antelopes is plotted on a log-log scale in the book On Size and Life. From the plot we see that $l\propto d^{2/3}$. Derive this relation from what you know about elasticity and buckling.

(b)

In old horror movies Godzilla is depicted as an overgrown iguana, the size of a Tokyo skyscraper. Is this consistent with allometric requirements?