# Differential equation: Spring/Mass system of driven motion with damping

1. ### TeenieBopper

29
1. The problem statement, all variables and given/known data
A 32 pound weight stretches a spring 2 feet. The mass is then released from an initial position of 1 foot below the equilibrium position. The surrounding medium offers a damping force of 8 times the instantaneous velocity. Find the equation of motion if the mass is driven by an external force of 2cos(5t).

2. Relevant equations
F=kx
m=W/g

m$\frac{d^{2}x}{dt^{2}}$+$\beta$$\frac{dx}{dt}$+kx=f(t)

3. The attempt at a solution

I found that k=16$\frac{lb}{ft}$ and m=1 slug. This gets me the following equation:

$\frac{d^{2}x}{dt^{2}}$+$\beta$$\frac{dx}{dt}$+16x=2cos(5t)

I'm at a loss for how to determine $\beta$, which is the damping force of 8 times the instantaneous velocity. I don't know how to determine instantaneous velocity. I know that once I have $\beta$, I can just use a LaPlace transform to find x(t). But $\beta$ is my stumbling block right now.

As I was writing this, it occured to me that $\frac{dx}{dt}$=instantaneous velocity and that would make $\beta$=8. That in turn makes the problem very easy to solve. Am I correct in this thinking?

We kind of rushed through this application in class the other day. Thanks in advance for any help you're able to provide.

2. ### rock.freak667

6,219
Yeah I would agree that β=8 here. Just make sure your initial conditions are correct with the proper signs.

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