MA2051 - Ordinary Differential Equations
Second Exam - Originally Given 1996 A Term
Sample Exam #3 - B96
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?
- 2 30 pts
- Consider the
- 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'' - 16y = 0.
- Find a general solution of .
- Solve the initial-value problem ,
y(0) = y'(0) = 0.
© 1996 by Will Brother.
All rights Reserved. File last modified on December 6, 1996.