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Investigating Torque with 3D Printed Waterwheels

Developed by Benjamin Hancock, Deva O'Neil, and Hannah McPherson - Published November 8, 2020

This exercise set is a follow up to the previous exercise set, "Efficiency of a Water Turbine (3D Printing Lab)" found at: https://www.compadre.org/PICUP/exercises/exercise.cfm?I=464&A=waterturbine. Using a wheel printed in this previous exercise set, students will test the factors that affect the maximum load that can be lifted by the water wheel for a given amount of water driving the wheel. In particular, students will investigate the relationship between lever arm and torque by designing and printing modifications to their waterwheel.
Subject Area Mechanics First Year * Identify lever arms and sources of torque (Exercise 1) * Determine how to reduce confounding variables (Exercise 2) * Use 3D printed attachments to experiment with maximum load for varying values of lever arm and plot the results (Exercise 2) * Experimentally probe the relationship between lever arm and torque (Exercises 2 - 3) * Explain deviations between theoretical and experimental data by analyzing confounding variables and assumptions (Exercises 3 - 4) * Develop a theoretical model to explain how the lever arm affects the maximum load that can be lifted (Exercise 4) 120 min

These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).

**Exercise 1: Predicting the Effect of Different Lever Arm Radii on the Maximum Load** In Part 1 of this lab (previous exercise set), a water wheel was 3D-printed in order to determine its efficiency in lifting a block. In this second part, we will study the effect of different values of lever arm on the maximum mass of the block. In order to do so, it is important to properly identify the lever arm of the system that attributes to lifting the mass, as this is the factor that will be manipulated. - In the figure below, two distances A and B are shown. Which has the same length as the lever arm for the torque due to the block? - Which has the same length as the lever arm for the torque due to the water? These two dashed lines are perpendicular to the lever arms on the wheel. ![](images/3DPWheel/Untitled_Diagram_16.png "") To develop a conceptual understanding of the role of the lever arm, answer the following: - If the wheel is in equilibrium, what must be true about the net torque on it? - Other than the block, what will be exerting a torque on the wheel? It would be helpful to draw a free-body diagram to identify the forces acting on the wheel. The free body diagram should lead to the observation that the amount of torque available to lift the block is determined by the torque due to the water. This torque is usually approximately constant. It relates to $R$ and the force of the block's weight, $F$, as follows: $\tau_{block} =RFsin\theta$. - For a wheel in equilibrium, do you think a small or large lever arm will maximize the weight to be lifted. Predict whether you want a small or large lever arm for both the torque on the wheel due to the water, and the torque on the wheel due to the block. **Exercise 2: Using 3D-Printed Water Wheels to Lift Different Masses** The goal of this experiment is to observe the result of changing $R$, the lever arm for the torque due to the block, to determine how it affects a wheel's maximum load. To do this, you will design and print attachments that have different values of $R$. This isolates the effect of $R$ without changing the amount of friction between the axle and the support, assuming that the attachments all have the same mass. - If you try to increase the radius of the entire wheel axle, it produces a very strong correlation between $R$ and the maximum load. However, much of this effect is not due to the increase in $R$. Why does increasing the radius of the entire wheel axle introduce a confounding variable? Utilizing a wheel that was printed in the previous exercise set, design and print 5 attachments with varying radii. Be sure to drastically change the radius between each attachment (at least 5mm). The mass of the attachments should also be as similar as possible. Sample attachments are shown below. A sample stl file for an attachment fit to a 6mm radius wheel is also included in this exercise set. In your attachment design, incorporate a hole for easy string attachment. ![](images/3DPWheel/IMG_20201022_111922431_2.jpg "") To examine the effect of changing the lever arm of the wheel, the mass of the wheel should be held constant. Increasing the lever arm size does not need to add mass, since the infill percentage can be changed when printing to adjust the density of the object. In the slicer program, adjust the infill density to keep the mass of the wheel about the same as in Exercise 4. It will most likely not be the exact same mass, but try to get the masses as close as possible using the infill options available. The image below is an infill menu for Qidi Print. Small differences may exist between programs; however, the overall concept remains similar. ![](images/3DPWheel/BenHancockInfill_page_001_1.jpg "") Using 100mL of water, positioned at a consistent height, $h$, above the wheel, experimentally determine the maximum load that each attachment is able to lift by finding the largest mass that gets lifted x distance by the water. Test many different masses for each wheel since the smaller the mass difference between trials, the more accurate the data will be. Before collecting data, test your wheel under the water reservoir to find the point that the wheel performs the best. Once found, it is important that the wheel remains in this position between each trial, as small deviations from this position could lead to large variations in data. It might be useful to mark the position of the stand holding the wheel with marker or tape for consistency. - Record and plot the maximum load of each wheel. (maximum load vs. lever arm) **Exercise 3: Interpretation of Observations** It should have been observed that the lever arm affects the amount of mass the wheel is able to lift. This trend could be the result of several variables, such as moment of inertia or friction. In what follows, we will derive this mathematical relationship that predicts the relationship between $R$ and maximum load. - At this point, you should have noted that $\tau_{net} = 0 = \tau_{water} + \tau_{block} + \tau_{friction}$. - What must be true in order to assume that $\tau_{block}$ is constant? - Thinking about the experiment that you just performed, to what extent do you feel these assumptions were justified? - Looking at your graph, what is the general relationship between $R$ and maximum load? In other words, what happens to the maximum load capacity as $R$ increases. **Exercise 4: Analysis** This relationship can be seen with a derivation starting with $\tau_{net} = 0 = \tau_{water} + \tau_{block} + \tau_{friction}$. We had made the assumption that friction is negligible and the angular acceleration of the wheel is 0, leaving: - $\tau_{block}=-\tau_{water}$ It will also be assumed that $\tau_{water}$ is for the most part constant. Using the torque formula, it can be shown that $\tau_{block} = RFsin\theta$. Since the Force of the block is perpendicular to the radius, $\theta=90$. The force of the block is constant at the weight: $F=mg$ The overall relationship is then $\tau_{block}=mgr$. Solving for mass it can be seen that $m=\tau_{block}/(g*R)$ - Do your results follow the predicted trend? - Explain any discrepancy between the calculated and observed pattern. It is important to be able to think as to why the data might be varying from the predicted. Go through these questions and explain how each could have affected the experiment. - What assumptions were the weakest that could explain any discrepancies? - What are some other sources of error in the experiment? **Exercise 5 (optional extension): Testing Different Wheel Densities** - What would be some different ways of varying the density of the wheel to determine what effect it has on the maximum load that can be lifted? Try to come up with at least 3 ways to do this. Hint: What was done in Exercise 2 to keep attachment mass constant? - Discuss what method would best reduce the effect of confounding variables. - Do an experiment to test the effect of the wheel's density and exhibit your results in a graph. What do you conclude about the optimum wheel density? - What would explain your results?

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### Credits and Licensing

Benjamin Hancock, Deva O'Neil, and Hannah McPherson, "Investigating Torque with 3D Printed Waterwheels," Published in the PICUP Collection, November 2020.

The instructor materials are ©2020 Benjamin Hancock, Deva O'Neil, and Hannah McPherson. 