EML 5060Analysis in Mechanical Engineering Exam 1 10/1/92
Exam 1 Van Dommelen 2:45-4:10pm

1.

A mass spirals outward, the distance from the center being given by $\rho = 4 \sqrt{\theta}$. If the rate of change in angular position is $\dot \theta = 3$, find the radial and azimuthal components of velocity $v_\rho$ and $v_\theta$, and those of the acceleration $a_\rho$ and $a_\theta$.

2.

Find the moment of inertia $I_x = \int y^2 {\rm d} A$ for the inside of the polar curve

$$\rho = \sin \theta \qquad 0\le\theta\le\pi$$

Note: for even integer p

$$\int_0^{\pi/2} \sin^p x {\rm d} x =\int_0^{\pi/2} \cos^p x {... ...t 5 \ldots (p-1) \over 2 \cdot 4 \cdot 6 \ldots p} {\pi\over 2}$$

3.

For the truss sketched below, the tension forces T₁, T₂, T₃, and T₄ in the four bars satisfy the five equations:

T₁ + T₂ = 0

-T₁ + T₂ + 2 T₄ = 200

T₂ - T₃ = 0

- T₃ = 2 P

T₃ + 2 T₄ = 200

Write the augmented matrix of the system and determine for which value(s) of P a solution exists, and whether it is unique.

4.

A coupled set of pendula as shown above has modes of the form

$$\theta_1 = A \cos(\omega t +\phi)\qquad \theta_2 = B\cos(\omega t + \phi)$$

For certain values of the spring constant, masses, and pendulum lengths, the amplitudes satisfy

$$- \omega^2 \pmatrix{A\cr B\cr} = \pmatrix{-3&1\cr 1&-3\cr} \pmatrix{A\cr B\cr}$$

Find the natural frequencies $\omega$.For each frequency, compare $\theta_1$ and $\theta_2$.

5.

The kinetic energy of a certain thin plate due to rotation around its center of gravity has the form

$$T = 2 \omega_1^2 - 2 \omega_1 \omega_2 + 2 \omega_2^2 + 4 \omega_3^2$$

In principal axes, there would be no $\omega_1 \omega_2$ term. Find the direction of the principal axes of this body by diagonalizing the quadratic form.