Lab 11 - April 8, 2015 - Work-Kinetic Energy Theorem Activity

Lab 11:
Brandon Elder
April 8, 2015

Work- Kinetic Energy Theorem Activity

Purpose:

This experiment is three-part, but the main purpose is to be able to measure the work done by a spring on a system by stretching a spring through a measured distance and then graphing the data to determine the relationship between work done and the kinetic energy. In theory, the work done on the spring should equal the amount of kinetic energy of the cart. Analyzing the graph and the area under the curve will give the amount of work done.

Apparatus Set-Up:

Fig. 1 The apparatus set up with the cart connected to the force sensor and the motion
detector placed on the ramp at the far end to observe the cart.

Set up a ramp on the counter top with a force probe, motion detector, and spring. A cart will be placed on top of the ramp and connected to the force probe with the spring. The motion detector will be placed at the opposite end of the ramp (see Fig. 1). Connect the force probe and motion detector to Logger Pro as usual. Some things to note would be that the spring needs to be horizontal when in it's relaxed state in between the cart and the force sensor with no obstruction. If the spring is not horizontal, then some of the force measured when the cart is pulled will be given in the y-direction and will not result in an accurate reading. The file to open in LoggerPro is the L11E2-2 (Stretching Spring), as this will display the force vs. position axes. Finally, the last important things to accomplish during the set-up would be to ensure that the force probe and motion detector are zeroed properly with the spring in the loose state and not stretched. Verify the motion detector is set to "reverse direction" so that as the cart is moved toward the detector, the position info will be recorded in the positive direction. You are ready to perform the experiments.

Considerations: 

Fig. A Graphs of constant vs non-constant forces
being applied to the cart on a horizontal surface.

A constant force applied will result in a linear graph of F vs. x (position or stretch of the spring) while a non-constant force applied will result in a non linear graph (see Fig. A). Work is equal to the force multipled by the displacement which is normally multiplied by cosine of the angle between. However, since we are using a flat surface, the factor of cosine multiplied by 180 results in 1, thereby leaving us with the work being equal to the force times the displacement.

      Fig. 2 Equation for Kinetic Energy.      

Experiment # 1 & 2:

Collect data in logger pro while someone pulls the cart slowly towards the detector until the spring has been stretched approximately 1 meter. The data will then be ready for viewing on the graph and table in logger pro. From observing the table, it is evident that the velocity, position, force, and time have all been calculated.  The next step is to add a calculated column that will define the kinetic energy. Kinetic energy is equal to k=.5*m*(v^2) (see Fig. 2). Since Logger pro automatically calculates the velocity, we create a calculated column for KE, using the formula for KE and inputting the meausred mass of the system, so that we can graph the KE vs position. We then added the KE to the y-axis of the graph so that force and KE were plotted on the y-axis were plotted against the position on the x-axis at the same time. The graph of the force needs to be integrated so that we can compare that value to the value of the KE graph at any point along the graphs. These values should be equal because the KE is equal to the area under the force graph (amount of work done). Fig 2a and 2b show different data points and the values for both the KE and the integrated values or amount of work.

Fig. 2a The integral under the force graph is equal to 0.3493 m*N and the KE at the corresponding point on the KE graph is equal to 0.354 J.

Fig. 2b The integral under the force graph is equal to 0.3241 m*N and the KE at the corresponding point on the KE graph is equal to 0.331 J.

Experiment #3:

A video was shown in class that plotted the Force vs the stretch of a rubber band. We hand sketched a graph similar to the video graph (See Fig. 3). In the last few seconds of the video we are given the time, the change in position, and the mass of the system. The purpose here is to analyze the area under the graph to determine the work done. Then to compare this to the kinetic energy of the cart from the video using the formula for KE to show that the KE is equal (or very close) to the amount of work done on the cart. The calculated KE is shown in Fig 4. The amount calculates out to 23.88 Joules while the area under the graph equals 25.675 Joules. The area under the graph was an estimation from quickly observing the video but it is evident here that the theorem for work and kinetic energy holds true.

Fig. 3 The amount of work done in the system is the area under the graph of work vs stretch. The total work is equal to 25.675 Joules.

Fig. 4 The calculated KE here is equal to 23.88 Joules. This amount was calculated using the variables given to us from the video. This amount of KE is compared to the amount of work from above, under the graph, and it is evident that work is in fact equal to the amount of kinetic energy.

Conclusion:

The spring constant was established by analyzing the force applied to the spring and by evaluating that with the change in position of the spring. The formulas are in Fig. 5. The integral was taken of the graph to find the work done in stretching the spring. In experiment 2 we saw that the work done on the cart by the spring equaled its change in kinetic energy. The experiments proved the work-energy principle that relates work to kinetic energy change by stating that the work done by the net force on a system equals the change in kinetic energy. 

Fig. 5 The formulas used to calculate the spring constant.