A body with mass m is connected to a spring in 1D and is at rest at X = A > 0. For the region X > 0, the only force acting on the mass is the restoring force of the spring. For the region X < 0, a viscous fluid introduces damping into the system.
a) Find the speed of the particle at X = 0.
b) Find the speed of the particle at X = 0, as it emerges from the X < 0 region.
c) Find the maximal position D the body reaches in the positive region after having left the negative region
F = -kx for x > 0
F = -kx-bv for x < 0
The Attempt at a Solution
a) x(2) +k/m x = x(2) + ω02x = 0
The characteristic equation for free oscillations:
x = x0cos(ω0t + δ)
where x0 = A and δ = 0
x = Acosω0t
From here its easy enough to find the the speed at x = 0 is Aω0
b)x(2)+ 2βx(1)+ ω02x = 0 when x < 0
The characteristic equation for this differential is:
x = Ae-βtcos(ω1t) where ω1 is √(ω02-β2)
The period of oscillation is ∏/ω1
My confusion at this point is how to incorporate the initial velocity into this equation.
Once I know that I can easily plug in the period into the velocity equation and find the new velocity at x = 0