# Combined Function Question (Dampened Sine Wave)

I am having trouble with a combined function task. It is a mass that is attached to a spring, and a spring to a wall (think, a doorstop). When it is pulled away from the wall it oscillates along the floor, however due to friction on the floor it slows down as it approaches 0. Along the lines of a dampened sine wave. The function we need is: d(t) = f (t)× g(t) + r It follows the following parameters: • The mass is at a resting position of r = 30 cm. • The spring provides a period of 2 s for the oscillations. • The mass is pulled to d = 50 cm and released. • After 10 s, the spring is at d = 33 cm.

I started with the equation for the model without friction (so as though it remained constant) and I believe it would be something like 20*cos(3x)+30 (feel free to correct me if I'm wrong). From this point, however, I am not sure how to determine the function when the friction is applied.

And, I'm not sure where to go from here. If you have any ideas, I would appreciate it! :)

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This sort of thing is usually done by setting up and solving a second order (linear homogeneous constant-coefficient) differential equation, but something about your presentation of the problem suggests that that's not what you want. So you've probably been given a general formula for the solution with lots of parameters ("coefficients") in it, and your job is to evaluate the parameters. So my advice is to search whatever notes and texts you have for something that looks a little like $$e^{-at}(b\cos\omega t+c\sin\omega t)$$ and then use the information you've been given (resting position, natural period of the spring, initial position of the mass, position when $t=10$) to help you to evaluate the $a,b,c$.