06-April-2015: Work-Kinetic Energy Theorem

     The purpose of this experiment was to find a relationship between work done by a spring on a cart and kinetic energy of a cart. There were three parts to this experiment. The first part involved work done by a non-constant spring force. The second involved the work-kinetic energy principle. The third part involved the work-kinetic energy theorem.

Part 1

     The first part of the experiment required the measure of the work done when a spring was stretched a certain distance and the finding of the spring constant. First, a cart, spring, force sensor, and motion sensor were setup. The cart was placed on a level ramp, with a spring attached to one end of the cart. The other end of the spring was attached to a force sensor. The motion sensor was placed at the opposite end of the track. Logger Pro was opened so that the data could be recorded. The force sensor was calibrated and zeroed. The trial began by slowly, but steadily, moving the cart down the track towards the motion sensor. This would stretch the spring, which would pull the force sensor. The motion sensor and force sensor provided a force vs position graph.

setup of the cart, spring, and force sensor

Motion sensor at opposite end of the track

Here is the resultant graph of force vs position.

Since it is known that force of a spring is F=kx, the slope of the best fit line of the graph is going to be k, the spring constant. The spring constant was found to be 2.785 N/m. 

Since work done is force*distance, the area under the curve of the graph is equal to the work. Using the integration feature of Logger Pro gave a number of 0.4519 N*m. 

So, from this part you can see it is possible to find the work done using a graph.

Part 2

     The second part of the experiment asked to plot a graph of force vs position and a graph of kinetic energy vs position, and to then analyze the graphs to see any relationship between them. The setup was exactly the same as in part 1. In Logger Pro, a new calculated column was added to the table for the calculation of kinetic energy (KE). The formula input was KE=(1/2)mv^2, where m was the mass of the cart and v its velocity. The mass was found using an electronic balance. 

     For this part, instead of starting the cart next to the force sensor where the spring would be in a relaxed position, the cart started off near the motion sensor so that the spring was stretched and the force sensor had an initial force. When the cart was released, the spring pulled the cart all the way back. The motion sensor recorded the cart's velocity so that the kinetic energy was able to be calculated. 

     Here is the resultant graph. The purple points represent the force vs position graph and the orange points represent the kinetic energy vs position graph.

     The work the spring does on the cart can be found the same way as before, by using the integral feature of Logger Pro to find the area under the curve. The kinetic energy is simply the point on the graph at a certain position. If the work done up to a certain position is compared with the kinetic energy at that position, it shows the two are equal. For example, here is a portion of the graph which compares the two:

     It shows the integral to be 0.3044 N*m and the KE of the cart at that point to be 0.298. The two numbers are very close to each other. This analysis was done at several points to ensure the relationship is consistent, which it was. So, this shows that the work done by the spring on the cart is equal the the change in kinetic energy. 

Part 3

     This final part involved watching a video of a professor performing an experiment. The professor uses a machine to pull back on a large rubber band. The force being exerted on the rubber band is recorded by an analog force transducer onto a graph. A cart of unknown mass is then attached to the rubber band. The cart is released and passes through two photogates a certain distance apart. The photogates were used to calculate the final speed of the cart, which would allow for the calculation of the final kinetic energy of the cart. A rough sketch of the graph produced by the professor in the video looks like:

     From the first two parts of this experiment, it is known the work can be found by calculating the area under the curve of a force vs position graph. So, for this graph it can be done by breaking the graph up into shaped we know how to find the area of and then adding up the areas. The first section is a triangle, the second is a rectangle, and the last two are trapezoids. The areas of those shapes are easy to calculate, and so they were. The final number for the total work was 25.675 J.

     The video provided data for the mass of the cart, the distance between the photogates, and the time when the cart passes through the photogates. With that data the velocity of the cart was found to be 3.33 m/s and the mass was 4.3 kg. So, the final kinetic energy of the cart was calculated to be 23.88 J. We expected a value close to 25.6 from the above graph, and we got pretty close. The difference between the numbers might have been caused by the sketch of the graph. Since the actual graph was not perfectly straight, but instead was quite squiggly, the actual area under it would be different. Also, the experiment the professor performed was all done by hand with no help of computers. The accuracy depended on the consistency of the person performing the experiment. Thank science for computers!