**Instructions:** Do your work on the paper provided. Put
your **name, section number and PLA's name** on that paper. Work
neatly. Show your work. **JUSTIFY YOUR ANSWERS.**

- 1
*45 pts* - A spring is
suspended from a fixed support as shown in the figure. In the
coordinate system shown, an equation governing the motion of a mass
*m*attached to this spring is*my*'' +*ky*= -*mg*. The lower end of this particular spring settles to rest at*y*= -0.098 m when a mass of 0.8 kg is hung from it.- Suppose a mass of 0.2 kg is suspended from the end of this
spring. Write a model for the motion of the mass when it is
started by lifting it up to
*y*= 0.1 m and releasing it from rest. - Find the period of oscillation of the 0.2 kg mass. Would the period of oscillation of the 0.8 kg mass be larger or smaller?
- The governing equation has a steady state. Find it and explain carefully why it is physically reasonable.
- If the mass were immersed in a fluid, its oscillations would
be damped, and the governing equation would have the form
*my*'' +*py*' +*ky*= -*mg*. For what range of values of the damping coefficient*p*will the 0.2 kg mass oscillate? Would the 0.8 kg mass oscillate over the same range of*p*values?

- Suppose a mass of 0.2 kg is suspended from the end of this
spring. Write a model for the motion of the mass when it is
started by lifting it up to

- 2
*30 pts* - Consider the
system
- Find its general solution.
- Suppose
*y*(0) = 2,*z*(0) = 4. Use one step of Euler's method to estimate*y*(3),*z*(3).

- 3
*25 pts* -
- Find a general solution of
*y*'' - 16*y*= 0. - Find a general solution of .
- Solve the initial-value problem ,
*y*(0) =*y*'(0) = 0.

- Find a general solution of

© 1996 by Will Brother. All rights Reserved. File last modified on December 6, 1996.