Page 91 - Advanced engineering mathematics

2.4 Spring Motion 71

This means that solutions of the spring equation immediately translate into solutions of the circuit
equation with the following identifications:

Displacement function y(t) ⇐⇒ charge q(t)
Velocity y (t) ⇐⇒ current i(t)

Driving force f (t) ⇐⇒ electromotive force E(t)
Mass m ⇐⇒ inductance L
Damping constant c ⇐⇒ resistance R
Spring modulus k ⇐⇒ reciprocal 1/C of the capacitance.

SECTION 2.4 PROBLEMS

1. This problem gauges the relative effects of initial per centimeter. The mass in turn is adhered to a dash-
position and velocity on the motion in the unforced, pot that imposes a damping force of 10v dynes, where
overdamped case. Solve the initial value problems v(t) is the velocity of the mass at time t in centimeters
per second. Determine the motion of the mass if it is

y + 4y + 2y = 0; y(0) = 5, y (0) = 0

pulled down 3 centimeters from equilibrium and then
and struck upward with a blow sufficient to impart a veloc-
y + 4y + 2y = 0; y(0) = 0, y (0) = 5. ity of 1 centimeter per second. Graph the solution.

Solve the problem when the initial velocity is (in turn)
Graph the solutions on the same set of axes.
2,4,7, and 12 centimeters per second. Graph these
2. Repeat the experiment of Problem 1, except now use solutions on the same axes to visualize the influence
the critically damped, unforced equation of the initial velocity on the motion.
y + 4y + 4y = 0. 11. How many times can the mass pass through the equi-

3. Repeat the experiment of Problem 1 for the under- librium point in overdamped motion? What condition
damped, unforced equation can be placed on the initial displacement to ensure that
it never passes through equilibrium?

y + 2y + 5y = 0.
12. How many times can the mass pass through equi-
Problems 4 through 9 explore the effects of changing the
librium in critical damping? What condition can be
initial position or initial velocity on the motion of the placed on y(0) to ensure that the mass never passes
object. In each, use the same set of axes to graph the solu- through the equilibrium point? How does the initial
tion of the initial value problem for the given values of velocity influence whether the mass passes through
A and observe the effect that these changes cause in the the equilibrium point?
solution.
13. In underdamped, unforced motion, what effect does
4. y + 4y + 2y = 0; y(0) = A, y (0) = 0; A has values the damping constant have on the frequency of the

1,3,6,10,−4and −7. oscillations?
5. y + 4y + 2y = 0; y(0) = 0, y (0) = A; A has values 14. Suppose y(0) = y (0) = 0. Determine the maxi-

1,3,6,10,−4and −7. mum displacement of the mass in critically damped,
6. y + 4y + 4y = 0; y(0) = A, y (0) = 0; A has values unforced motion. Show that the time at which this

1,3,6,10,−4and −7. maximum occurs is independent of the initial dis-
placement.
7. y + 4y + 4y = 0; y(0) = 0, y (0) = A; A has values

1,3,6,10,−4and −7. 15. Consider overdamped forced motion governed by

8. y + 2y + 5y = 0; y(0) = A, y (0) = 0; A has values y + 6y + 2y = 4cos(3t).

1,3,6,10,−4and −7. (a) Find the solution satisfying y(0) = 6, y = 0.

9. y + 2y + 5y = 0; y(0) = 0, y (0) = A; A has values (b) Find the solution satisfying y(0) = 0, y (0) = 6.
1,3,6,10,−4and −7.
Graph these solutions on the same set of axes to com-
10. An object having a mass of 1 gram is attached to the pare the effects of initial displacement and velocity on
lower end of a spring having a modulus of 29 dynes the motion.
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