Question

12. A spring-mass-dashpot system is released from rest with an initial displacement given by $y(0) = y_0$. Consider the following question: ""What happens to the displacement $y(t)$ if we keep the values of mass $m$ and spring constant $k$ fixed but increase the damping constant $gamma$?"". In particular, select an arbitrary but fixed time $t > 0$, think of the solution $y(t)$ as being a function of the damping constant $gamma$, and determine $lim_{gamma o infty} y(t)$. Do you have any intuitive insight as to what the answer should be? Develop the answer with the following steps. (a) Show that the roots of the characteristic polynomial are $lambda_1 = frac{-gamma - sqrt{gamma^2 - 4mk}}{2m}$, $lambda_2 = frac{-gamma + sqrt{gamma^2 - 4mk}}{2m}$. Solve the initial value problem, assuming the system to be overdamped. Express the solution in terms of the two roots $lambda_1$ and $lambda_2$. (Since we are interested in the system's behavior for large values of $gamma$, the overdamped assumption is appropriate.) (b) Show that $lim_{gamma o infty} lambda_1 = -infty$ and $lim_{gamma o infty} lambda_2 = 0$. (c) Use the results of parts (a) and (b) to determine $lim_{gamma o infty} y(t)$ (with $k$, $m$, and $t > 0$ fixed). What is the physical meaning of your answer? Does it agree with your intuition? Does it make physical sense in retrospect?

          12. A spring-mass-dashpot system is released from rest with an initial displacement given by $y(0) = y_0$. Consider the following question: ""What happens to the displacement $y(t)$ if we keep the values of mass $m$ and spring constant $k$ fixed but increase the damping constant $gamma$?"". In particular, select an arbitrary but fixed time $t > 0$, think of the solution $y(t)$ as being a function of the damping constant $gamma$, and determine $lim_{gamma 	o infty} y(t)$. Do you have any intuitive insight as to what the answer should be?
Develop the answer with the following steps.
(a) Show that the roots of the characteristic polynomial are
$lambda_1 = frac{-gamma - sqrt{gamma^2 - 4mk}}{2m}$, $lambda_2 = frac{-gamma + sqrt{gamma^2 - 4mk}}{2m}$.
Solve the initial value problem, assuming the system to be overdamped. Express the solution in terms of the two roots $lambda_1$ and $lambda_2$. (Since we are interested in the system's behavior for large values of $gamma$, the overdamped assumption is appropriate.)
(b) Show that $lim_{gamma 	o infty} lambda_1 = -infty$ and $lim_{gamma 	o infty} lambda_2 = 0$.
(c) Use the results of parts (a) and (b) to determine $lim_{gamma 	o infty} y(t)$ (with $k$, $m$, and $t > 0$ fixed). What is the physical meaning of your answer? Does it agree with your intuition? Does it make physical sense in retrospect?

$12. A spring-mass-dashpot system is released from rest with an initial displacement given by y(0) = y0. Consider the following question: ""What happens to the displacement y(t) if we keep the values of mass m and spring constant k fixed but increase the damping constant gamma?"". In particular, select an arbitrary but fixed time t > 0, think of the solution y(t) as being a function of the damping constant gamma, and determine limgamma o infty y(t). Do you have any intuitive insight as to what the answer should be? Develop the answer with the following steps. (a) Show that the roots of the characteristic polynomial are lambda1 = frac-gamma - sqrtgamma^2 - 4mk2m, lambda2 = frac-gamma + sqrtgamma^2 - 4mk2m. Solve the initial value problem, assuming the system to be overdamped. Express the solution in terms of the two roots lambda1 and lambda2. (Since we are interested in the system's behavior for large values of gamma, the overdamped assumption is appropriate.) (b) Show that limgamma o infty lambda1 = -infty and limgamma o infty lambda2 = 0. (c) Use the results of parts (a) and (b) to determine limgamma o infty y(t) (with k, m, and t > 0 fixed). What is the physical meaning of your answer? Does it agree with your intuition? Does it make physical sense in retrospect?$

Added by Robert M.

Question

Please give Ace some feedback