Question

Critical Damping (repeated real roots) Ifb2=4mk then the term under the square root is 0 and the characteristic polynomial has repeated roots,−b/2m,−b/2m.Now we use the roots to solve equation (1) in this case. We have only one exponential solution, so we need to multiply it bytto get the second solution.

Basic solutions:e−bt/2m,te−bt/2m.
How we gathered the part we need to multiply it by t to get the second solution. I

Answer 1

In a damped mass-spring system, if the term under the square root is zero, the characteristic polynomial has repeated roots. To find the second solution, we multiply the first exponential solution by t.

Step-by-step explanation:

When solving for the equation of motion in a damped mass-spring system, if the term under the square root is zero, then the characteristic polynomial has repeated roots. In this case, we have one exponential solution, so we need to multiply it by t to get the second solution.

The basic solutions are e^(-bt/2m) and te^(-bt/2m).