8/5/2016 - Work-Kinetic Energy Theorem Activity (Lab 11)

PURPOSE:
Prove that the work done when you stretch a spring through a measured distance is equal to the change in the kinetic energy of the spring.

THEORY:
If there are no external forces acting on the system, then ideally, the total work done to the system should be equal to the change in the kinetic energy of the system. For a non-constant force, such as a cart+spring on a ramp, the total work done can be calculated by finding the area under the force vs. distance graph. In a perfect world, the calculated area and the change in KE should be equal. In the world we live in, however, the accuracy of our calculations will depend on a variety of factors, such as the accuracy of our assumptions, the preciseness of our equipment, and the diligence of the people who perform the lab. In this experiment, we will explore how well our theoretical model of energy fits with reality.

APPARATUS:
We set up a ramp, cart, motion detector, force probe, and spring as shown in the diagram. One end of the spring is attached to a metal rod which is clamped to the table.

cart+spring system

EXPERIMENTAL PROCEDURE:
Part 1: Force vs. Distance

1. Calibrate the force sensor.
2. Set up the apparatus as shown.
3. Open the apt experiment file to display a force vs. distance diagram in Logger Pro.
4. Zero the force probe, verify the motion sensor is measuring toward the detector as the positive direction.
5. Sketch/Capture an image of the graph. Find the spring constant. Find the work done in stretching the spring using the integral function on Logger Pro.

Part 2: Kinetic Energy vs. Distance

1. Measure the mass of the cart.
2. Create a New Calculated Column for the KE of the system with respect to position.
3. Zero the force sensor and the motion detector at the desired starting position. 
4. Pull the cart back about 0.6 m, let it go, and begin graphing your KE data.
5. Compare the ΔK from points a,b to the area under the F vs. x graph through points a,b.

DATA/GRAPHS:
mass of cart = 0.756 kg
spring constant, k = 5.794 N/m

Force vs. Position Graph

KE vs. x and ∫ F(x)dx through points (a,b)

KE vs. x and ∫ F(x)dx through points (a,c)

KE vs. x and ∫ F(x)dx through points (a,d)

Δx - Work - Kinetic Energy - Data Chart

ANALYSIS:
We calculated the work done by the force of the spring by integrating the force function with respect to position. This method works because it is the equivalent of calculating the area under the force vs distance curve, which yields W = force*distance. After comparing it to the change in kinetic energy, our group found that the KE at the final position for a given range of values was approximately equal to the work done over the same set of values. For example, between the position values x = 0.51 m to x = 0.17 m, we found the work done between those points to be about 0.63 J, while the change in KE was about 0.83 J. In other words, we found that the change in KE was relatively close to the total work done on the system between those two points. On average, however, there was about a 24% difference between the two values, indicating a few sources of uncertainty in our experiment.

CONCLUSION:
Our data clearly illustrates that the change in the KE of our system did not entire coincide with the total work done over two points. However, while the reality of our calculations did not totally match up with their theoretical counterparts, this does not mean that the theory we tested is discredited. In reality, our data showcases how small changes in our methodology can influence irregularities in our calculations. In other words, there are several reasons why there was such a large disparity between the calculated work and the change in KE of the system. For instance, the lab equipment we used is not the best in the world. If we decided to buy and use more expensive and precise lab equipment, then we could have measured the mass, velocity, and force of the system with minimal error, which undoubtedly could have greatly increased the accuracy of our calculations.

GROUP MEMBERS: Xavier C., Billy J., Matthew I.