EML 5060 Analysis in Mechanical Engineering 9/22/95
Closed book Van Dommelen 9:00-9:50 EST

Show all reasoning and intermediate results leading to your answer. One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used.

1.

(33 points). Six points P₁, P₂, P₃, P₄, P₅, and P₆ in a certain structure have to be grounded. The person in charge did not do a good job. The person connected point P₁ to P₂ with an electrical wire, and the person also connected point P₂ to P₃. The person connected P₄ to P₅, P₅ to P₆, and P₆ to P₄. But the person did not connect any point to the ground! In view of the points that the person connected, we can say the following about the electric potentials V₁, V₂, V₃, V₄, V₅, and V₆:

$$V_1 - V_2 = 0 \qquad \hbox{wire 1}$$

$$V_2 - V_3 = 0 \qquad \hbox{wire 2}$$

$$V_4 - V_5 = 0 \qquad \hbox{wire 3}$$

$$V_5 - V_6 = 0 \qquad \hbox{wire 4}$$

$$V_6 - V_4 = 0 \qquad \hbox{wire 5}$$

Show that the rank of this system of 5 equations in 6 unknowns is one less than what you would expect. What does this mean in terms of the wires that the person has used? Do the potentials have to be zero at all six points? Do they all have to be the same? Solution

2.

(33 points). You are designing a machine to automatically polish the surface

$$z = \sqrt{x^2 + y}$$

Of course, the plane of the polishing disk needs to be tangential to the surface at all times. So what is the equation of the plane of the polishing disk to polish the point where x=2 and y=5? Solution

3.

(34 points). The kinetic energy of a rotating plate is given by

$$T= 3 \omega_x^2 + 4 \omega_x \omega_y + 6 \omega_y^2$$

where $\omega_x$ and $\omega_y$ are the components of the angular velocity vector. Suppose that it is known that T=1. What is the maximum possible magnitude of the angular velocity vector $\vert\vec\omega\vert$ in that case? Hint: simplify the quadratic form. Solution