5 10/03 W

1. Write down the worked-out mathematical expressions for the integrals requested in question 5.1. This is a good exercise in identifying various surface and volume integrals in integral conservation laws. Explain their physical meaning, if any. Don't worry about actually doing the integrations. However, show integrands and limits completely worked out.

   Take the surfaces $S_{I}$, $S_{II}$, $S_{III}$, and $S_{IV}$ to be one unit length in the $z$-direction. (To figure out the correct direction of the normal vector $\vec n$ at a given surface point, note that the control volume in this case is the right half of the region in between two cylinders of radii $r_0$ and $R_0$ and of unit length in the $z$-direction. The vector $\vec n$ is a unit normal vector sticking out of this control volume.)

2. 5.14. Find both the horizontal and vertical components of the force. Make sure that you clearly define what control volume you are using, as there is no unique choice.

3. 5.11. Clearly define what control volume you are using.