Physics 311 - Fall 97

Problem set 9 - PLEDGED

Due Thursday, Nov. 20

1.

Ultracentrifuges. The Physics Building is officially known as the ``Jesse W. Beams Laboratory of Physics.'' Professor Beams was one of the inventors of the ultracentrifuge: one of the problems he had to face was mechanical failure. Suppose that part of an ultracentrifuge consists of a hollow steel cylinder, of height h, inner radius R₁, and outer radius R₂; the steel has a density $\rho$ and a yield stress S_m. Obtain a formula for the maximum allowable angular velocity for the centrifuge. Now put in some numbers. Suppose R₁=4 cm, R₂=5 cm, and h=10 cm. What is the maximum number of revolutions per minute before failure?

2.

Hammering. If you try to drive a wood nail into a cinder block, it will buckle. But even driving a nail into wood requires some skill.

(a)

Explain why a nail buckles, qualitatively.

(b)

Assume the nail is 3 cm long and has cross section 1mm². How large an applied force can cause it to buckle?

(c)

Nails also bend if you do not drive them straight. Explain why this happens, qualitatively.

(d)

The nail of part (b) has been driven part way into a horizontal surface, but makes an angle $\theta$ with the perpendicular. You are hammering straight down. How large an applied force will cause the nail to bend?

(e)

Sketch a bent nail and a buckled nail, still in the nail hole.

(f)

Describe how a concrete nail differs from a wood nail.

3.

Energy in earthquakes. How much elastic energy is released in an earthquake?

(a)

Find an estimate for the Young modulus of rock (granite, say). Rock is brittle; assume that the strain at breaking is about 10^-4. From this estimate the elastic energy density which is released when the fault slips (assume that the stress required to cause the fault to slip is about the same as the stress required to break the rock; this is probably an overestimate).

(b)

To calculate the energy, we need an appropriate volume. I've seen pictures of a fence which straddled a fault, and became displaced about 1 m across the fault after a moderate quake. Therefore, if $\Delta l\approx 1\ {\rm m}$, and the strain is 10^-4, an appropriate length seems to be about 10 km. What is the energy? You should get a large number, which may not be particularly meaningful to you; compare it to some other large energies which are meaningful.

(c)

The Richter scale is an empirical scale which provides some measure of the destructiveness of an earthquake. According to geology textbooks, a rough formula is

$$m\approx {\frac{1}{2.4}}\log _{10}E-2.4,$$ (1)

where m is the ``magnitude'' of the quake and E is the energy released, measured in ergs. What is the magnitude of our model earthquake? How does the energy released in a magnitude 6 quake compare to the energy released in a magnitude 8 quake?

4.

True or false. Justify your answers.

(a)

Sound waves in air are transverse waves of compression and rarefaction.

(b)

The speed of sound in air at 20${{}^\circ}\,$C is twice that at 5${{}^\circ}\,$C.

(c)

The speed of sound in air is greater than in water.

(d)

The attenuation of sound in air generally increases with frequency.