# Energetic Evaluation of Frictional Effects in Motion Experiments

This lab is designed to align with AAOT science outcome #1: Gather, comprehend, and communicate scientific and technical information in order to explore ideas, models, and solutions and generate further questions.

## Materials:

• writing utensil
• calculator
• digital device with spreadsheet program
• digital device with internet access

## Objectives

1. Apply conservation of mechanical energy to model the speed of a multi-component system accelerated by the gravitational force.
2. Compare the model predictions with the experimental results to evaluate the size of model inaccuracy caused by neglecting friction in the model.
3. Analyze trends in the difference between model prediction and experimental results to narrow down which system components are the highest contributors to dissipation of mechanical energy.
4. Discuss methods for reducing the friction and associated energy dissipation in the system.
5. Identify additional analysis of existing data or additional experiments or that would provide further evidence to narrow down which system components are the highest contributors to dissipation of mechanical energy.

## Experimental MEthods

The following video shows a cart accelerated by a falling mass attached to the cart by a string running over a pully.  The experiment was repeated for 5 different cart masses, starting with the empty cart and successively adding four additional masses. The velocity of the cart was measured by the cart’s optical sensor. All of the necessary setup data is available in the video and the recorded velocity data is available in an online spreadsheet. You can copy and paste the data into your own sheet for analysis.

## Modeling

We will build a model for the maximum speed reached by the cart. The maximum speed is reached as the mass hits the floor, after which point the string becomes slack. The model will assume that the work done by the forces of friction and air resistance is negligible compared to the total mechanical energy of the system. In other words, we will assume the mechanical energy of the mass-cart system does not change during the experiment.

1) Write down an equation representing the conservation of mechanical energy for this system. Use the system at rest as the initial state. Use the moment just as the mass hits the floor, after falling a height of \$Delta h\$, as the final state. Remember that both the mass and the cart are moving at that point.

2) Symbolically solve your equation for the final speed of the system. Show your work.

## Data Analysis

3) Create a single graph containing plots of the the velocity vs. time data for all 5 trials. Examine your graphs to make sure you have a complete dataset that includes the maximum speed reached by the cart for each trial.

4)  Be sure to name your graph and add axis labels with correct units.

5) Use the MAX function in your spreadsheet program to find the maximum speed for each trial. (The data sheet contains an empty row at the top for you to place this function).

6) In a new area of your spreadsheet create a table that looks like this:

 hanging mass (kg) cart mass (kg) accessory mass (kg) height (m) accessory masses system mass (kg) predicted max speed (m/s) measured  max speed (m/s) percent difference in speed initial energy (J) final energy (J) percent difference in energy tension (N) 0 1 2 3 4

7) Fill in the values of the experimental parameters, hanging mass, cart mass, accessory mass, and fall height.

8) Use your spreadsheet to calculate the total system mass based on the cart mass, hanging mass, and number of accessory masses for each trial.

9) Use your spreadsheet to apply your model equation to calculate the predicted max speed for each trial. (Don’t forget that that the hanging mass was falling. How does that affect your value for the change in height of the hanging mass?)

10) Enter the measured max speed values for each trial that you found using the MAX function.

11) Use your spreadsheet to calculate the percent difference in the measured and predicted speeds.

12) Use your spreadsheet to calculate the initial mechanical energy of the system.

13) Use your spreadsheet to calculate the final mechanical energy of the system.

14) Use your spreadsheet to calculate the percent difference in the initial and final mechanical energy of the system.

## conclusions

15) Do the data suggest that the effects of air resistance and friction are significant? Explain.

16) According to your mechanical energy model, all of the potential energy in the hanging mass was transferred to kinetic energy in the moving cart and falling mass. The model neglected friction, air resistance, and the transfer of kinetic energy to the pully. Do you expect your model overestimate or underestimate the amount of kinetic energy in the cart + hanging mass system? Do you expect your model your model overestimate or underestimate the speed of the cart? Explain.

17) According to your predicted and measured values, does your model over/under estimate the energy and speed as you expected? Explain.

## Further Questions

18) Our data contains additional information that we have not analyzed. We can and actually try to narrow down which of the elements neglected by our model was most likely to have contributed the most to differences between predicted and measured values. Let’s add a few hypotheses that we can test with our data. Fill in the blanks with the correct relation between values (increase or decrease).

• If  transfer of mechanical energy to thermal energy by air resistance was the dominant contributor to the % difference between predicted and measure values, then % difference should clearly ______________ with the increasing max speed.
• If  transfer of mechanical energy to thermal energy by friction in the axles of the cart wheels is the dominant contributor to the % difference between predicted and measure values, % difference should clearly ____________ with increasing the cart weight.
• If  transfer of potential energy to kinetic energy in the rotating pulley is the dominant contributor to the % difference between predicted and measure values, then % difference should clearly ______________ with increasing max speed.

19) To test your hypotheses, create the following graphs shown in the table, fit linear trendlines to each, then fill in the table, except for the last row.

 Hypothesis Graph Slope R2 Hypothesis supported? (Explain) Air resistance % difference vs. max speed Cart Axle Friction % difference vs. total cart mass Pulley Kinetic Energy % difference vs. max speed Pulley Axle Friction % difference vs. string tension

20) We have not yet tested the likelihood that friction in the pulley was a main contributor to the percent difference. Let’s add another hypothesis:

• If  transfer of mechanical energy to thermal energy by friction in the axle of the pulley is the dominant contributor to the % difference between predicted and measure values, then % difference should clearly ______________ with increasing tension on the string.

21) To test this hypothesis we need to determine the string tension. The tension in the string does work over some distance to increase the kinetic energy of the cart. We can apply the work-energy theorem to model to the cart to model the tension in the string.

Based on your results, what can you say about how the various elements neglected by your model contribute to the % difference in predicted and measured values? [Hint: You hypothesis is basically saying that if the first half of your hypothesis is correct, then the second half should be observed. That does not imply that if you observe the second half, the first half is necessarily correct: If P then Q does not imply if Q then P. Assuming it does is a logical fallacy know as affirming the consequent (sometimes called converse error, fallacy of the converse, or confusion of necessity and sufficiency).

22) Notice that this model assumed equivalence of inertial and gravitational mass. The cart mass, hanging mass, and  accessory mass values were all measured using scales, so those were gravitational masses, but we used them in calculating the total kinetic energy. Additionally, the same hanging mass value was used in calculating gravitational potential energy and in contributing to the total kinetic energy of the system.  Therefore, similar to previous two labs, this experiment was also a test of the equivalence principle. Does this new experiment add evidence that supports or refutes the equivalence principle?