Determining the Equation for an Underdamped Oscillating System

[trautlein]

Homework Statement

I am trying to figure out the equation to represent the motion of a weighted car attached to an ideal spring that is on an inclined plane (represented by Figure 1. Note, the car in my problem is attached to a spring , whereas in that picture it is attached to a string.)

Using a motion detector and the program Logger Pro, I graphed the position function of the car against time and got the graph that is represented here in Figure 2.

For my equations I determined that my initial amplitude was .1145 by taking maximum and minimum values of the hight of two points of the sinusoidal function.

In an earlier experiment we had determined the 'k' value of the spring to be 27.43, giving me an $$\omega_{0}$$ value of 4.278.

The mass of my car is 1.499kg.

Homework Equations

For my equation I was using a $$\tau$$ of 1.69 calculated from the $$\omega_{0}$$

$$\tau\,=\frac{m}{b}$$

The Attempt at a Solution

I have tried to fit the curve using the curve fitter in Logger Pro, however none of the equations match mine, and when I try to define a function the program says:

Equation must be an equation in "t".

I am assuming that the position function will be given by $$x(t)=A^{2}_{0}\ast\,e^{-t/\tau}\astcos(\omega\,t+\delta)$$

This is assuming that $$\tau\,=\frac{m}{b}$$

This is the equation that I am trying to find and get to fit to my graph.

Does $$\textbf{F}\,_{d}\,=-bv$$ ?

Is there anyway that we can graph the theoretical position with the variables that we have here?

 

[ehild]

The car oscillates around a position X₀, different from zero, so the position function should be

x(t)=X₀+A*exp(-t/τ)*cos(ωt+δ),

where τ=2m/b, if F_D=-bv.

ehild

 

[trautlein]

ehild, thank you for the response! I actually just came to that same conclusion about twenty minutes ago, but to see your response really excited me - it means everything ended up okay for me.

Again, thank you.

I am supposed to do anything with this thread once my question has been answered?

 

[ehild]

I am pleased that you found it out by yourself. Leave the post as it is, so as other people can learn from it. The post was interesting, with very good pictures.

ehild

 

[trautlein]

I also learned a ton about LaTeX in the process of writing this too...