We define volumetric strain rate as the rate of increase of volume of a fluid element per unit volume. In Cartesian coordinates we write the volumetric strain rate as $rac{1}{V} rac{D V}{D t}=rac{\partial u}{\partial x}+rac{\partial v}{\partial y}+rac{\partial w}{\partial z}$. Write the primary dimensions of each additive term, and verify that the equation is dimensionally homogeneous. Show all your work.

Determine the bending strain energy in the A-36 steel beam. $I=99.2\left(10^6\right) \mathrm{mm}^4$.

Should the rate of return of a call option on a long-term Treasury bond be more or less sensitive to changes in interest rates than the rate of return of the underlying bond?

Determine the torsional strain energy in the A-36 steel shaft. The shaft has a radius of $30 \mathrm{~mm}$.

Indicate the statement as true or false The loss coefficient represents the amount of energy lost per radian relative to the strain energy.

A member of an investment committee interested in learning more about fixed-income investment procedures recalls that a fixed-income manager recently stated that derivative instruments could be used to control portfolio duration, saying, “A futures-like position can be created in a portfolio by using put and call options on Treasury bonds.” a. Identify the options market exposure or exposures that create a “futures-like position” similar to being long Treasury-bond futures. Explain why the position you created is similar to being long Treasury-bond futures. b. Explain in which direction and why the exposure(s) you identified in part (a) would affect portfolio duration. c. Assume that a pension plan’s investment policy requires the fixed-income manager to hold portfolio duration within a narrow range. Identify and briefly explain circumstances or transactions in which the use of Treasury-bond futures would be helpful in managing a fixed-income portfolio when duration is constrained.

An element of aluminum is subjected to triaxial stresses. Calculate the strains in the element in $x, y$, and $z$ directions if the stresses $\sigma_x, \sigma_y$, and $\sigma_z$ are $-20\ \mathrm{MPa}, 28\ \mathrm{MPa}$, and $-18\ \mathrm{MPa}$, respectively. Assume $E=70\ \mathrm{GPa}$ and $\nu=0.33$. Also, Find the strain energy density of the element.

Find the temperature of a cavity having a radiant energy density at 2000 A that is 3.82 times the energy density at 4000 A.

Should the rate of return of a call option on a long-term Treasury bond be more or less sensitive to changes in interest rates than the rate of return of the underlying bond?

If you buy a 13-week T-bill with a maturity value of $5,000 for$4,922.15 from the U.S. Treasury Department, what annual interest rate will you earn?

a. Distinguish between specific energy and energy density of a fuel. b. Estimate the energy density of water that falls from a waterfall of height 75 m and is used to drive a turbine.

The compound solid steel rod shown in is subjected to a tensile force $P$. Assume that $E=29,000 \mathrm{ksi}, d_1=$ $0.50$ in., $L_1=18$ in., $d_2=0.875$ in., $L_2=27$ in., and $P=5.5 \mathrm{kips}$. Determine (a) the elastic strain energy in rod $A B C$. (b) the corresponding strain-energy density in segments (1) and (2) of the rod.

Determine the vertical displacement of point $C$ of the simply supported 6061-T6 aluminum beam. Consider both shearing and bending strain energy.

A slightly tapered bar AB of rectangular cross section and length L is acted upon by a force P. The width of the bar varies uniformly from $b_{2}$ at end A to $b_{1}$ at end B. The thickness t is constant. (a) Determine the strain energy U of the bar. (b) Determine the elongation $\delta$ of the bar by equating the strain energy to the work done by the force P.