Listro,+Fihma,+Bloom

=Lab: Elastic Potential Energy= Members: Ryan Listro, Evan Bloom, and Sam Fihma Date: March 7, 2011 Class: Period 2

__//Part 1: Finding the Spring Constant//__


 * Purpose:** To determine the spring force constant of the spring we were given.


 * Hypothesis:** We can find the spring force constant by adding mass to the spring, and then measuring the change in distance from the equilibrium position of the spring. Once this data is collected and graphed, the result will display the change in force, which is mass times gravity, on the y-axis and the change in distance on the x-axis. If we take the slope of the graph, it will equal the spring force constant.

F=mg ||
 * [[image:xtcvygubhindjf.png]] || F=w


 * Materials:** Spring, spring stand, masses, ruler, mass hanger


 * Procedure:**
 * 1) ** Set up spring stand as follows: place the spring on the stand and move the ruler so that the end of the spring is lined up with zero centimeters. **
 * 2) ** Place a mass hanger on the spring and note the change in distance of the spring. Record both the change in mass and distance in an excel spreadsheet. **
 * 3) ** Add more mass and continue to record the changes in mass and distance. Make sure to repeat each mass trial about three times to get an average distance for each mass. **
 * 4) ** Once all data is recorded, multiply each mass, which should be in kilograms, by 9.8 in order to convert them to force. Then, graph your findings by using the changes in distance for the x-values and the changes in force for the y-values. **
 * 5) ** Once graphed, find the slope of your data points to find the spring force constant. **

A picture of our setup is here:


 * Data:**

The equation of this part that we are focusing on is **F=-kx**. Since we made x as positive in our table (x should be negative because the displacement of the string is point down), the negative in front of kx disappears, so we have **F=kx**. The force takes the y value and the displacement takes the x value in this graph. Because we did this, our graph displays the relationship between the change in force and the change in distance of the spring**.** The slope of our line is what x is being multiplied by, and in this case that is k, our spring force constant. In our graph, k is 3.3455.
 * Graph:**



__//Part 2: Finding the Relationship between Spring Constant and Velocity//__

**Purpose:** Find the relationship between spring force constant and velocity.

**Hypothesis:** The further the spring is stretched back, the faster the paper flag will travel past the photo-gate timer. The relationship between spring force constant and velocity is seen below as they are directly related to each other.

**Materials:** Cart, spring, photo-gate timer, ramp, ruler, data studio application and connection wires

**Procedure:**
 * 1) Set-up the experiment. Make sure the ramp is perfectly level, attach the cart to the spring, and place it at the end of the ramp. Then measure the width of the paper flag attached to the cart (Note: This value will be used when determining velocity). Next, set up data studio by selecting the photo gate timer experiment.
 * 2) Once set-up, pull back the cart away from the spring while measuring the stretched distance.
 * 3) As the cart is released, start the data studio experiment. Record the time that data studio measured. Then divide the previously recorded flag width by this recorded time. The end result will be the velocity of the cart.
 * 4) Repeat steps 2-3 multiple times for each distance. Also, be sure to vary the distance the cart is being pulled back. In the end, there should be about five recorded distances– each with three trials.
 * 5) With the final data, make a graph using velocity as the y-value and the change in distance as the x-value. This graph will reveal the relationship between the two variables.

A video of our procedure is here: media type="file" key="Movie 9jdhfgjdgg.mov" width="300" height="300" This is the result of a typical trial on DataStudio:

= =

**Data:**


 * Sample Calculations:**

This is for how we calculated the theoretical velocity.

This is for how we calculated the actual velocity. We used .009 as our distance because the the photo-gate timer, along with Data Studio, calculated the time using a small strip of paper attached to the mass. This paper was only .9 cm in width.

This is for how we calculated the Percent Error. How we got the theoretical value is shown underneath the graph.

**Graph:**

This is our graph of our observed velocity vs. how stretched the spring was. Velocity is the y-value and the distance is the x value. The equation for velocity, which is shown above is simplified. It is shown below. This makes what is shown below the slope. Since the square root of k over m is just a number that is being multiplied to x, the graph is a linear fit. Below shows how we calculated our theoretical slope that was used in percent error above.

**Overall Conclusion (of 1 and 2):** From the results we accumulated, it is apparent that the hypotheses for both parts 1 and 2 are correct. For part one, we proved that we could find the spring force constant of a particular spring by using the direct relationship between force (graphed on the y-axis) and the x-value (graphed on the x-axis). The slope of the line is our k value, or the spring force constant. We calculated a 10.28% difference for our k value compared to the class average. The main reason for this, in addition to human error, is that each group used a different spring. No two springs were alike, so the class value could not be the same as our calculated value. Our hypothesis for part 2 was also correct. The further we stretched the spring back, the less time the flag spent in the photo-gate timer. This relationship was shown through a direct, linear relationship between the stretched distance of the spring (x value) and the velocity. The graph proves to be a good linear fit as proved by our calculations. The equation shows that velocity is equal to the x value times the square root of k over m. The square root of k over m is actually just the slope of the graph, and therefore nothing but a multiplier. This shows how x and v are directly related. For this part of the lab, we got a very low, constant value for our error. We had error around 3% for most of our trials. A very trivial source of error would be friction. Since the coefficient of friction between the tires of the cart and the track is extremely close to zero, friction accounts for an almost negligible portion of the error in this lab. A source of error that is a little more prominent, however, is the k value that we chose to use in this part of the lab. When conducting our theoretical calculations, we used our own calculated k value rather than the class average. Using the class average k value could have changed our results. Also, a source of human error came when we pulled the cart back. We would pull the car back to a certain length, and record multiple trials at a particular distance. However, since we are only human we could not replicate the exact spot every single time, so this may have caused our results to be off a little bit.