Our unforced spring mass model is mx00 + βx0 + kx = 0 with m, β, k > 0. We know physically that our spring will eventually come to rest no matter the initial conditions or the values of m, β, or k. If our model is a good model, all solutions x(t) should approach 0 as t → ∞. For each of the three cases below, explain how we know that both roots r1,2 =−β ± Sqrt(β^2 − 4km)/2m will lead to solutions that exhibit exponential decay. (a) β^2 − 4km > 0. (b) β^2 − 4km =0. (c) β^2 − 4km >= 0.
Our unforced spring mass model is mx00 + βx0 + kx = 0 with m, β, k > 0. We know physically that our spring will eventually come to rest no matter the initial conditions or the values of m, β, or k. If our model is a good model, all solutions x(t) should approach 0 as t → ∞. For each of the three cases below, explain how we know that both roots r1,2 =−β ± Sqrt(β^2 − 4km)/2m will lead to solutions that exhibit exponential decay. (a) β^2 − 4km > 0. (b) β^2 − 4km =0. (c) β^2 − 4km >= 0.
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Our unforced spring mass model is mx00 + βx0 + kx = 0 with m, β, k >
0. We know physically that our spring will eventually come to rest no
matter the initial conditions or the values of m, β, or k. If our model
is a good model, all solutions x(t) should approach 0 as t → ∞. For
each of the three cases below, explain how we know that both roots
r1,2 =−β ± Sqrt(β^2 − 4km)/2m
will lead to solutions that exhibit exponential
decay.
(a) β^2 − 4km > 0.
(b) β^2 − 4km =0.
(c) β^2 − 4km >= 0.
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