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1. Systems are unstable if their characteristic equations have a positive root (solution). Determine whether each of the following characteristic equations represents a stable or unstable system.

(i) s^3 + 6 s^2 + 11 s + 6 = 0
(ii) s^3 + s^2 - 8 s - 12 = 0

2) In a Hooke's law experiment, equal increments of force, w, are applied to a metal rod and the length of the rod is recorded. Within the elastic limit, the increases of length, x, are found to be equal.

After the application of the 10th increment of force, the length of the rod is found to be 50.03 mm and after the 15th increment of force the rod's length is found to be 50.08 mm. Determine the original length of the rod, l, and evaluate

SUMMATION[n=1 to 10] (l + n x)

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This solution shows how to determine if the given systems are stable of unstable.

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** Please see the attached file for the complete solution response **

1. Systems are unstable if their characteristic equations have a positive root (solution). Determine whether each of the following characteristic equations represents a stable or unstable system.

(Note that for a linear system the characteristic equation of A is given by (pleases see the attached file)

(i)

Solution: Descartes's Rule of Signs tells us that the number of positive real zeros of (pleases see the attached file) is equal to the number of variations in sign of (pleases see the attached file) or less than that number by an even integer. Since there are no variations in sign (every coefficient of f is positive), we see that the characteristic equation (pleases see the attached file) has no positive solutions. Therefore, it represents a stable system.
Note that we can also use ...

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