So we have a elastic slingshot which slides a block across the floor until coming to a stop but we've been facing a couple difficulties in determining our spring constant. We have a table of forces required to pull the rubber band as a slingshot at an angle, but we also have a table of the forces required to pull it as if it was a linear apparatus.
I am more than confused as to which set of values determine the appropriate spring constant when I will determine energy values of the system. I would assume that the linear force has no relevance because the slingshot's motion is constricted. Instead, I would assumed that I should use the angular data which behaves like a thicker linear band in the same direction. Any ideas?
Best Answer
I think the simplest thing is to calculate the energy directly from the data given - using simple numerical integration of the data corresponding to the configuration you are actually using (I am assuming that's the angular configuration).
First, a graph of your data:
The data is quite linear but with a fair bit of noise on it. The best fit quadratic through the "angular" data, forced to go through (0,0), has a small quadratic component. If you just integrated that expression, you would have the expression for energy as a function of displacement:
$$E = \int F\cdot dx = \int (38.5 x^2 + 27.7 x) dx = 12.8 x^3 + 14.4 x^2 + C$$
As long as $x$ is quite small, the cubic term will be small compared to the linear term.
Alternatively, you can use the linear fit through the data - this gives a slope (spring constant) of 32.7 and you would ignore the cubic term.
Here is the calculated energy as a function of displacement using the two methods (over the range 0 - 0.16 units; note that you really should always quote your units when describing an experimental setup...)
As you can see, over this range the linear assumption gives almost the same result.
This is not a complete answer to your question - but I hope it helps your thought process.