In this lab activity our primarily goal was to find the spring constant of a car’s internal spring. We were able to accomplish this by measuring the velocity, while changing the mass through multiple trials, to ultimately find k or the spring constant using our mathematical model we derived.
II. Theory
According to Hooke’s Law, the force required to compress a spring is directly proportional to the distance that it was stretched/compressed. As stated by Hooke’s Law:
Comparing the law of conservation of energy and Hooke’s Law we are able to relate the spring constant to the mass and velocity. Thus, we are able to further simplify to find the velocity in terms of mass and the spring constant. To first begin deriving our mathematical model, we set the potential energy and kinetic energy equal to each other due to the lack of friction which in return means that energy is conserved.
We are able to set our potential energy from the spring equal to the kinetic energy because we are working with a “frictionless track”. Due to the fact that it is frictionless, no energy is dissipated and energy is conserved. From our mathematical model we are able to obtain the independent and dependent variable. The independent variable is the mass(kg) and the dependent variable is the velocity(m/s). Through the usage of these variables from our graph we can determine the general equation for the best fit line of our data. The equation is as follows:
y=Ax+B
The relationship between the mathematical model and the general best fit line are directly correlated. The slope of the mathematical model is equal to the slope of the general best fit line. Through the slope of both the mathematical model and general best fit line we are able to extract the slope that will be used to find the spring constant of the internal spring of the smart car.
II. Procedure
To begin our experimental setup, we had to determine what we needed to include in order to find the velocity of the car in order to find the spring constant in the end.
As shown above, our experimental setup included a smart car with an internal spring, a “frictionless track”, and a motion sensor. To begin using the experimental setup, we leveled the track using a digital angle gauge. Once that was completed, we next stepped forward to attach the motion sensor next to the pole at the end of the track. After we had the track fully set up and functional, we moved on to weighing the car as well as measuring the compressed and non-compressed spring in order to find x. To measure our x, we subtracted the non-compressed and compressed and found it to be 0.02818m.
To begin the experiment in order to find the spring constant, we compressed the spring and placed the smart car (with no mass placed on top of it at the beginning) on the track. Next, we released the smart car and measured the position vs. time to obtain the velocity. Following that, to obtain more data we added 50g masses for each of the following trials for at least 5 trials. Once we added the masses, we repeated the compression and releasing portion.
To find our velocity we used a program called CAPSTONE. CAPSTONE allows us to make a position vs. time chart and graph that will give us the opportunity to find our velocity. To find our data we highlight the multiple points the motion sensor was able to retrieve in order to find a more suitable value for that trial. Once we were able to retrieve all of our data, we solved for the spring constant using the data we collected in CAPSTONE.
From the graph above, we are able to extract the slope from the equation y=0.1526x + 0.0217. The slope being 0.1526. Using this slope we are able to find k or the spring constant. As shown below we solve for the spring constant using the slope found and divide it by our x.
Solving for spring constant using graphed data and table data:
Our spring constant of 192.2 is a reasonable answer. As defined by the graph the data followed a positive linear path. Thus, we can easily concur that our result was reasonably good. A spring constant is unable to be zero or negative. First, it cannot be zero because if the spring constant was zero you basically do not have a spring. A spring always attempts to go back to its equilibrium position. If you stretched the spring and the constant was zero, there would be no force acting on it trying to return it to its equilibrium position. If the result was negative, it would mean that when the spring is stretched all of the forces will go towards the displacement. Thus, with both of those statements we can believe that a spring constant of 192 is reasonable.
Experiments always face the fact that they may have errors come up and in the end skewing their results. Some errors that could have skewed our experimental setup could be friction, malfunctioning motion sensor, and the masses on top of the car. For our experiment we were informed to treat the track as “frictionless”. However, due to this not being possible to fully make sure of it. There could still be some present friction done by the track that would cause our data to be skewed. Having friction present would cause a systematic error as it would cause our data to go down. Another source of error could be our motion sensor that we used. As we released the car, the sensor was in charge of measuring the velocity. Although, the motion sensor could have a malfunction and cause for it to report the wrong velocity, thus skewing our data and ultimately our final spring constant. The last source of error that could be present is that the masses that were placed on top of the car could have increased the velocity of the car as the masses were not fully placed in the car. The masses were simply placed on top and they could move about at the top. This could cause for our data to vary greatly and causing random error in our data.
The experiment could have many errors arise as trials are being done. Some of the improvements that could be made to the equipment/procedure is to clarify whether the track is frictionless and be sure on which it is. Another improvement that could easily be made is to find a way to set masses on top of the car without having them move around while being released.
VII. Conclusion
At the beginning of this experiment we were instructed to create an experiment that would give us the ability to retrieve the spring constant from an internal spring of a smart car. My team decided in order to find the spring constant we set the spring potential energy equal to kinetic energy. Using this as our mathematical model we only needed to find the velocity. Thus, we were able to construct our experimental setup which involved us doing multiple trials with differing masses in order to find the velocity. Using this data, we were able to graph it in order to find the slope. The slope was then divided by our x in order to find our spring constant. Through our analysis, we were also to determine that our final spring constant of 192.2 was a reasonable answer.