Question

Suppose that the mass in a spring-mass-dashpot system with m=1,c=2, and k=8 is set in motion (at t=0 ) with x(0)=2 and x' (0)=−2. When x(t)>0 the spring is stretched, and the spring is compressed when x(t)<0. a) Find the position of the mass for all t>0. b) List all the times the mass changes direction (hits a local max or min). Which of these times represents when is the mass most compressed?

Answer 1

a) The position of the mass for all \
`(t > 0\) is \(x(t) = 2e^(-t) \sin(3t) - 2e^(-t) \cos(3t)\). b`) The mass changes direction, hitting a local max or min, at
`\(t = (\pi)/(6), (5\pi)/(6), (3\pi)/(2)\).`The time when the mass is most compressed is
`\(t = (5\pi)/(6)\).`

Step-by-step explanation:

For part a), the position equation
`\(x(t) = 2e^(-t) \sin(3t) - 2e^(-t) \cos(3t)\)`was obtained by solving the differential equation representing the spring-mass-dashpot system with given initial conditions. The term
`\(2e^(-t) \sin(3t)\)`represents the stretched portion of the spring, while
`\(-2e^(-t) \cos(3t)\)`represents the compressed part. This solution illustrates the oscillatory motion of the mass, showing its position for all
`\(t > 0\).`

Part b) identifies the times when the mass changes direction or hits a local max/min. These occur when the derivative of the position function
`\(x(t)\)`is zero or undefined. Solving for
`\(x'(t)\)`and finding its roots yields
`\(t = (\pi)/(6), (5\pi)/(6), (3\pi)/(2)\)`as the times the mass changes direction. Comparing the values of
`\(x(t)\)`at these points,
`\(t = (5\pi)/(6)\)`represents the time when the mass is most compressed, as the absolute value of \(x(t)\) is smallest at that point among the given times.