Physics 311 - Fall 97

Problem set 8

Due Thursday, Nov 13

1.

Compression waves in a thin elastic bar. An elastic bar made of a material with a Young's modulus E and a density $\rho$ has a cross-sectional area A. In this problem you will derive the wave equation for compression waves in the bar.

(a)

Let the displacement of an element of the bar from its equilibrium position be u(x), where x labels the position along the bar. Show that in this situation the stress s(x) at a point x is

$$s(x)=E{\frac{\partial u}{\partial x}}.$$ (1)

(b)

Now consider a small segment of the bar of length $\Delta x$. There is a force F(x) acting on the left end of this piece and a force $F(x+\Delta x)$ acting on the right end. Show that the net force acting on this segment is

$$\Delta F=F(x+\Delta x)-F(x)\approx E{\frac{\partial ^{2}u}{\partial x^{2}}}A\Delta x.$$ (2)

(c)

Now apply Newton's Second Law to this segment and show that the displacement satisfies the wave equation; i.e.,  

$${\frac{\partial ^{2}u}{\partial x^{2}}}={\frac{1}{c^{2}}}{\frac{\partial ^{2}u}{\partial t^{2}}},$$ (3)

where the wave speed is $c=\sqrt{E/\rho }$.

(d)

As the wave travels down the bar, what happens to the diameter of the bar?

2.

Wave speeds. Look up the Young's modulus, shear modulus, and Poisson's ratio for three different materials (your choice). Calculate the speed of torsional and compressional waves in these materials. Any conclusions?

3.

Dislocations.

(a)

On page 30-16 of Feynman, volume II, are three pictures of bubble rafts. Make a photocopy and mark the positions of the dislocations. Find their Burgers vectors.

(b)

Take a few paragraphs to explain why the existence and motion of dislocations explains the strength (i.e., the yield stress) of ductile materials. Why are alloys usually harder than pure metals?

4.

Bursting pressure. Here we will calculate the bursting pressure of a steel container. Assume that the container is a cylinder of length L, radius R, with wall thickness t ($t\ll R$); the pressure of the gas inside the tank relative to the outside pressure is $\Delta p$.

(a)

By considering a small portion of the wall of the tank, calculate the stress s in the wall in terms of t, R, and $\Delta p$ (and anything else which might be important). The tank will burst when this stress exceeds the yield stress $s_{{\rm yield}}$ of the tank walls. Use this fact to calculate the bursting pressure of the tank.

(b)

A scuba tank has typical dimensions L = 24", R = 5", t = 1/4". By using a typical value of the yield stress for steel (see PQRG), calculate the bursting pressure for this tank.

(c)

Now you probably don't want to fill the tank to the bursting pressure--the actual bursting pressure may be lower than the theoretical value for any number of reasons (what might they be?). So assume that the tank is pressurized with air to 1/2 the theoretical bursting pressure. How long a dive could you make with such a tank?