A thin tubular shaft of circular cross section with inside diameter 100 mm is subjected to a torque of $5000\ \mathrm{N} \cdot \mathrm{m}$. If the allowable shear stress is 42 MPa, determine the required wall thickness t by using (a) the approximate theory for a thin-walled tube, and (b) the exact torsion theory for a circular bar.

Correct each of the following misstatements from Dalton’s atomic theory.

a. Elements are made of tiny particles called molecules.

Indiana University professors investigated first-year college students’ knowledge of astronomy ( Astronomy Education Review, Vol. 2, 2003). One concept of interest was the Big Bang theory of the creation of the universe. In a sample of 148 freshmen students, 37 believed that the Big Bang theory accurately described the creation of planetary systems. Based on this information, would you be willing to state that more than 20% of all freshmen college students believe in the Big Bang theory? How confident are you of your decision?

Correct each of the following misstatements from Dalton’s atomic theory.

b. All atoms of a given element are very similar.

Apply collision theory to explain why collisions between two reacting particles do not always result in the formation of a product.

The theory of evolution by natural selection requires variability of a given trait. Why is variability necessary and where does it come from?

A hydrogen atom is in its first excited state (n = 2). Using the Bohr theory of the atom, calculate the potential energy.

Which of these species has a longer bond, $\mathrm{B}_2$ or $\mathrm{B}_2^{+}$? Explain in terms of molecular orbital theory.

What is the neutral theory of molecular evolution as proposed by Kimura (1968)? What determines the frequency of neutral alleles in a population?

Conduction of an electric current is a general property associated with metals. How does the band theory for metallic bonding explain conductivity?

Which scientist developed the atomic theory of matter?

(a) Antoine Lavoisier

(b) John Dalton

(c) Aristotle

(d) Joseph Proust

Using the VSEPR theory, predict the molecular structure of the following polyatomic ions.

b. perchlorate ion, $\mathrm{ClO}_4{ }^{-}$

A capillary tube of 1.2 mm diameter is immersed vertically in water exposed to the atmosphere. Determine how high water will rise in the tube. Take the contact angle at the inner wall of the tube to be $6^\circ$ and the surface tension to be 1.00 N/m.

According to the correspondence principle, (a) new theory must agree with old theory where they overlap. (b) Newton's mechanics is as valid as Einstein's mechanics. (c) relativity equations apply to high speeds while Newton's equations apply to low speeds. (d) special relativity and general relativity are two sides of the same coin.