MA2051 - Ordinary Differential EquationsSample Makeup Exam - B96

Second Exam - Originally Given 1996 A Term

1 45 pts
Recall the differential equation that governs the (small) oscillations of a mass m pivoted on a rod of length L: .

1. Write a model for the motion of a mass of 1.2 kg suspended from a rod of length 2.45 m that is started in motion from a vertical position with a push to the left of 0.1 radians/s.
2. What is the maximum angular displacement of the mass?
3. The mass was started in motion from the vertical position. How much time will pass before it returns to vertical again? If the mass is increased to 2.0 kg, will that time increase or decrease?
4. Suppose the rod stretches slightly. Will the period of the pendulum's oscillations increase or decrease?
5. In reality, friction in the pivot eventually stops the motion of the pendulum. A governing equation is ; p is the (small) damping coefficient.

You want to design a pendulum that swings for as long as possible. You can't change the pivot, but you can change both the length of the rod and the size of the mass it supports. What choice(s) will minimize the effects of damping?

2 10 pts
Use one step of Euler's method to estimate the position and velocity at t = 0.5 s of the forced pendulum modeled by , , .

3 20 pts
Find a general solution of

4 25 pts

1. Find a general solution of y'' + 4y' - 96y = 0.
2. Find a general solution of y'' + 4y' - 96y = 24t.

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

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