Washington University in St. Louis Introductory Physics Lab Statics Lab Summer 2015 Statics: Bikes, Forces, and Torques Pre-­‐Lab: About Our Bikes A Bit of History “The bicycle has been with us in various forms for quite a long time and it has had a far greater impact on human society than one might guess. Today it is assuming even greater importance in the affairs of man. In many countries, particularly in Asia and Africa, it is by far the most important means of transportation. Even in our own affluent society, we find people relying more and more on bicycles for basic transportation needs. Bikes are relatively cheap and readily available. In many cities, traffic problems are so severe that bicycles offer the quickest means of travel from point-­‐to-­‐
point. They are non-­‐polluting and actually healthful to the rider. And perhaps most importantly, they are the most energy-­‐efficient means of conveyance known. Not only do they not consume fossil fuels, electricity or nuclear power, but they use human-­‐produced energy so well that a person on a bicycle is a more efficient transporter of mass than even fish or a horse, not to mention automobiles or airplanes.” The above quote seems like something that you might read today on http://sustainability.wustl.edu/, but it’s actually almost 40 years old. In his 1978 tour de force, The Bicycle, Philip DiLavore outlines the impact that this simple machine has had on societies across the globe. If we can learn anything from history, it is that the bicycle, which started as a design by Leonardo da Vinci, has had and will continue to have a lasting impact on human life. Getting to Know Our Bikes When you get to lab, you will find a bicycle which has been mounted on a stand. Typically these stands are used by cyclists who want to turn their standard bicycle into a stationary bike which they can use to exercise indoors. Other than the stand, the only modification to the bicycle is the replacement of the pedal with a pedal bar. This pedal bar will allow us to easily attach weights to the bicycle where the pedal would normally be located. This week’s experiments will involve applying a force to the pedal bar and reading the corresponding output force the rear wheel would exert on the ground. Before we can do these experiments, we should get familiar with some of the terminology we will use throughout this lab. Figure 1 shows a photo of the front part of the drive train of our bicycle. The key components of our bike are labeled. 1 Washington University in St. Louis Introductory Physics Lab Statics Lab Summer 2015 About the Crankset The pedal bar, crank arm, and front gears are all rigidly fastened together. All of these components together will be referred to as the crankset. If an appropriate force is applied to the pedal bar, the crankset will rotate as a single unit. About the Chain The way that the chain interacts with the crankset is important to our problem. From the viewpoint shown in Figure 1, the chain is looped over one of the front gears and pulls straight to the left. The force is exerted at the top of whatever gear the chain is on. Thus, the chain produces a torque that is directed out of the page (or counterclockwise). Use the right hand rule to verify that you get this direction for the torque. Figure 1: The front part of the drive train of our bicycle. Applying an appropriate force to the pedal bar causes it, the crank arm, and the front gears to rotate. All of these components together will be referred to as the crankset. The bottom part of the chain is not under any tension. There is therefore no torque associated with the bottom of the chain. You can check this out if you have a bike at your disposal. Firmly activate the rear brakes and place some of your weight on the pedal as if you are trying to pedal forward. While doing this, if you reach down and wiggle the chain, you will find that the top part of the chain is under very high tension while the bottom part of the chain is very easy to wiggle. Crankset Problem Consider the scenario shown in Figure 2. There is a disk with a mass of 2.0 kg attached to the pedal bar. The crank arm is horizontal. The crankset is static, indicating that all torques on the crankset are balanced. The distance from the crank bolt to the pedal bar is 17 cm. For all of the problems in this Pre-­‐Lab, all torques are understood to be about the crank bolt. That is, the crank bolt is the axis of rotation. Figure 2: For use with PL1-­‐PL5. PL1. What is the direction of the torque produced on the crankset by the 2-­‐kg mass attached to the pedal bar? PL2. What is the magnitude of the torque produced on the crankset by the 2-­‐kg mass attached to the pedal bar? PL3. What is the direction of the torque produced on the crankset by the chain? PL4. What is the magnitude of the torque produced on the crankset by the chain? PL5. If the chain is on gear #1, which has a radius of 5.7 cm, what is the tension in the chain? 2 Washington University in St. Louis Introductory Physics Lab Statics Lab Summer 2015 About the Rear Wheel For a bicycle to be useful, the tension in the chain must produce a torque on the rear wheel. This is what we examine next. Figure 3 shows the essential components of the rear wheel. When the rear wheel is static, we can think of the gears as being part of the wheel: the gears, spokes, rim, and tire are a rigid body. This time, the chain exerts a force to the right, producing a torque that is into the page (clockwise). Just like we saw at the crankset, the chain exerts its force on the gear such that 𝑟 and 𝐹 are perpendicular. When a bike is being used realistically, the rear tire will be in contact with the ground. There is a frictional force between the ground and the tire that produces a torque that opposes the torque produced by the chain. In a static situation, these torques are balanced. Rear Wheel Problem Figure 3: The rear wheel with gears and chain (left). A close-­‐up of the gears and chain (right). For use with PL6-­‐PL11. Refer to Figure 3 for the directions involved in these problems (PL6-­‐PL11). Consider a scenario where the tension in the chain is 125 N. The chain is on gear #3 on the rear wheel where the radius of the gear is 4.5 cm. The diameter of the wheel is 26 inches. The wheel is in contact with the ground. The wheel is static. The rear axle is understood to be the axis of rotation for all torques. PL6. What is the direction of the torque that the chain exerts on the rear wheel? PL7. What is the magnitude of the torque that the chain exerts on the rear wheel? PL8. What is the direction of the torque that the ground produces on the rear wheel? PL9. What is the magnitude of the torque that the ground produces on the rear wheel? PL10. What is the magnitude of the frictional force that the ground exerts on the rear wheel? (We are not interested in the normal force that the ground exerts on the wheel.) PL11. What is the direction of the frictional force that the ground exerts on the rear wheel? (We are not interested in the normal force that the ground exerts on the wheel.) Read This: Is your response to PL11 consistent with the direction that bikes like to travel? PL12. The normal force exerted by the ground on the rear wheel points straight up, directly toward the rear axle. What is the torque produced on the rear wheel by the normal force? (Assume the normal force has a magnitude of 250 N.) End of Pre-­‐Lab 3 Washington University in St. Louis Introductory Physics Lab Statics Lab Summer 2015 Part I: Bicycle Statics This lab will begin with a couple of statics problems involving a bicycle. We refer to these as statics problems because nothing will be moving. The rear wheel will be static. The crankset will be static. All torques and forces are balanced. Equipment •
•
•
Bicycle (Appendix A gives values for the radii of the gears) Force sensor with string attached Disk masses •
•
•
•
Protractor Measuring tape Digital balance Oven mitt (for use as a brake) Read This: Your bicycle can be dangerous if you are not careful. Please keep your hands and arms away from any moving parts. Wear the oven mitt to help you stop the rear wheel. The steel disks STOP can also be potentially dangerous. Please avoid using the portion of the table that is below the crankset. Do This: Inspect your bicycle and make sure that the chain is passing over the front gear with the smallest diameter (gear #1) and the rear gear with the largest diameter (gear #1). This particular configuration is termed the lowest speed. If the chain is not passing over the appropriate gears, you should adjust it by using the gear levers on the handle bars until it is in the correct configuration. (Note: In order to change the gears, you must turn the pedal as if you were pedaling the bicycle forward. The front gear is changed using the gear levers on the left side of the handle bars and the rear gear is changed using the gear levers on the right side of the handle bars.) 1. Statics with Right Angles The first statics problem will follow the setup shown in Figure 4. Figure 4: The static setup for Section 1. The crank arm is horizontal and holds disk masses on the pedal bar. The rear wheel is held stationary. A carabiner is attached to a zip tie that is zipped around the rear wheel, directly above the rear axle. The carabiner is attached to a string that is pulled horizontally to the left. The other (unseen) side of the string is attached to a force sensor. The force sensor measures the tension in the string. 4 Washington University in St. Louis Introductory Physics Lab Statics Lab Summer 2015 Read This: The next few instructions will direct you to make your station look like the one shown in Figure 4. Keep in mind that everything is static. Nothing is moving. All of your force measurements today will be on static systems. Setting up the system and keeping it static will likely take some teamwork. Do This: Attach the force sensor to the rear wheel by attaching the carabiner to the zip tie that is zipped around the rear wheel. The zip tie should be located directly above the axle. Things should look a lot like Figure 4. You should always be pulling horizontally. Do This: While holding the rear wheel, rotate the crank arm counterclockwise until it is horizontal, pointing toward the front of the bike as shown in Figure 4. The bubble level can be helpful in determining whether the crank arm is horizontal. Place three (3) disk masses on the pedal bar and secure them in place with the nut. Read This: Is the experimental setup clear? When you have reproduced the static system shown in Figure 4 you may relax and move on to the Checkpoints. Checkpoint 1.1: Your bike should be in the smallest gear in the front and the largest gear in the back. Further, there should be three disk masses on the pedal bar with the crank arm horizontal. What is the magnitude of the torque that the disk masses are applying to the crankset? Checkpoint 1.2: Knowing that the crankset is not undergoing angular acceleration, what is the magnitude of the torque that the chain exerts on the crankset? Checkpoint 1.3: Using your response to Checkpoint 1.2 along with a value shown in Appendix A, calculate the tension in the chain. (The chain should be on the smallest gear in the front, gear #1.) Checkpoint 1.4: Using your response to Checkpoint 1.3 along with a value from Appendix A, what is that torque that the chain exerts on the rear wheel? (The chain should be on the largest gear in the rear, gear #1.) Checkpoint 1.5: Knowing that the rear wheel is in static equilibrium, what is the magnitude of the torque that the string exerts on the rear wheel. Checkpoint 1.6: What should be the magnitude of the force recorded by the force sensor? Read This: Now that you have made your prediction, you will set up the software in order to test the prediction. Read This: Many measuring devices have multiple ranges that you can select under different circumstances. Your Vernier force sensor is one such device. Notice that it has a switch that can either be set to “±10 N” or “±50 N”. When set in the “±10 N” range, the force sensor can reliably measure forces with magnitudes up to 10 N. If you try to measure a force of 20 N, the sensor would give you an incorrect reading. When set in the “±50 N” range, the force sensor can 5 Washington University in St. Louis Introductory Physics Lab Statics Lab Summer 2015 reliably measure forces up to 50 N. In this range, the force sensor can also measure forces under 10 N, but such a reading would be more accurate if the sensor were set to the “±10 N”. For this specific sensor, the uncertainty is 0.01 N when using the “±10 N” range and 0.05 N when using the “±50 N” range. For forces greater than 50 N, this is not the right force sensor to use. Checkpoint 1.7: Based on your response to Checkpoint 1.6, to which setting should you switch the force sensor? Do This: Open Logger Pro. Read This: In the lower left corner you should see a white box that displays the instantaneous force reading from the force sensor (Figure 5). You will make a measurement while the force sensor is horizontal. While horizontal with nothing attached, the reading of should be within about .05 N of zero. Figure 5: Example force reading, an If the reading is much different from that, you can zero the sensor by clicking instantaneous value. , the zero button. Read This: When measuring the force, you can use that instantaneous value or you can average the force over some time. To average the force, click the Start Collection button (Figure 6). Values will start appearing in the data table on the left of the screen, and points will start appearing on the graph to the right. After 10 seconds have elapsed data collection stops. Highlight a portion of the plot that looks smooth. Then click the “Statistics” button (Figure 6). At this point a box shows up. The “mean” value displayed on the third line of this box is the force measurement we are interested in. Figure 6: Start Collection (left) & Statistics (right) Checkpoint 1.8: Test your theoretical force value from Checkpoint 1.6 with an experiment. Checkpoint 1.9: Was your prediction in Checkpoint 1.6 accurate? Discuss what may account for any differences between your prediction and your experimental value. That is, what are the sources of uncertainty? Might the uncertainty in your force reading be larger than the uncertainty quoted above (0.01 N or 0.05 N depending on the setting)? S1
STOP Synthesis Question 1 (35 Points): Compare your experimental and theoretical values for the force recorded by the force sensor. As part of your response, give a very clear derivation of the theoretical value that includes a theoretical value for the tension in the chain. Do This: Disconnect the carabiner from the zip tie on the rear wheel. You are about to spin the wheel. In addition, please remove any masses from the pedal bar. 6 Washington University in St. Louis Introductory Physics Lab Statics Lab Summer 2015 2. Statics with Ugly Angles The setup for the second experiment is a little uglier. The force sensor will still be pulling horizontally, but the zip tie will no longer be directly above the axle. Further, the crank arm will not be horizontal. The setup is shown in Figure 7. The bike is still in the lowest speed. Figure 7: Three (3) disk masses are attached to the pedal bar which is at an angle of 𝜃!" = 30° . The carabiner is attached to the zip tie such that 𝜃!"# = 30°. The string is pulling horizontally to the left. The string is attached to a force sensor. The chain is passing over the front gear of smallest diameter and the rear gear of largest diameter. S2
STOP Synthesis Question 2 (35 Points): For the setup depicted in Figure 7, compare your experimental and theoretical values for the force recorded by the force sensor. As part of your response, give a very clear derivation of the theoretical value that includes a theoretical value for the tension in the chain. If you follow the same Checkpoints that appeared in Section 1, you will put yourself in very good shape. Do This: Disconnect the carabiner from the zip tie on the rear wheel. You are about to spin the wheel. In addition, please remove any masses from the pedal bar. 7 Washington University in St. Louis Introductory Physics Lab Statics Lab Summer 2015 Part II: Changing Gears The Story You find yourself on a bike in Forest Park’s Great Basin, riding toward the Art Museum where you’ll take in some culture with a friend for free. There’s only one problem: Art Hill! This steep slope separates you from works by Picasso and Van Gogh. Luckily, rather than riding a trendy fixie, you’re atop a 21-­‐speed marvel of engineering. 3. Low Speed vs. High Speed Up to now, we’ve been using bicycles to study physics. Now we change gears (literally) and use physics to understand how we use bicycles. Different combinations of gears are referred to as “speeds.” In Part I, you were working with the bikes in the lowest speed, which is the combination where the chain is on the smallest gear in the front (#1) and the largest gear in the rear (#1). The highest speed is the combination where the chain is on the largest gear in the front (#3) and the smallest gear in the rear (#7). Read This: In order to change the gears, you must spin the pedal as if you were pedaling the bicycle forward. The front gear is changed using the gear levers on the left side of the handle bars and the rear gear is changed using the gear levers on the right side of the handle bars. Please make sure that the carabiner is disconnected before you pedal the bike. 3 S3
Synthesis Question 3 (30 Points): When riding up a hill, it is common practice for a biker to shift into a low speed. Gather some data and use it to explain why a biker would be likely to use the lowest speed rather than the highest speed when biking up a hill. Your response should include the following. •
•
•
•
Force measurements with the bike in the highest speed and the lowest speed A free-­‐body diagram of a bicycle on a hill A free-­‐body diagram of a bicycle on a flat Explanation of how the three bullet points above support the biker’s choice 8 Washington University in St. Louis Introductory Physics Lab Statics Lab Summer 2015 Appendix A: Speeds and Gears Figure 1 shows the front part of the drive train of our bicycle, including the front set of gears. In addition to these gears, there is a set of seven gears that are attached to the rear wheel of the bicycle. The front and rear sets of gears are connected by a chain. Any particular front gear-­‐chain-­‐rear gear combination is termed a speed. Our bike has 21 possible gear combinations (3×7) so we say it is a 21-­‐speed bicycle. By adjusting the combination of front and rear gears the chain is connected to, the rider is able to control his or her riding experience. Table 1 lists the specifications of both the front and rear sets of gears which will be useful during this week’s experiments. These radii have an uncertainty of approximately 2 mm. Gear # 1 2 3 -­‐-­‐-­‐-­‐ -­‐-­‐-­‐-­‐ -­‐-­‐-­‐-­‐ -­‐-­‐-­‐-­‐ Front Gears Radius [cm] # of Teeth 5.7 28 7.7 38 9.7 48 -­‐-­‐-­‐-­‐ -­‐-­‐-­‐-­‐ -­‐-­‐-­‐-­‐ -­‐-­‐-­‐-­‐ -­‐-­‐-­‐-­‐ -­‐-­‐-­‐-­‐ -­‐-­‐-­‐-­‐ -­‐-­‐-­‐-­‐ Gear # 7 6 5 4 3 2 1 Table 1: Data for the front and rear gears for our bicycles. Rear Gears Radius [cm] # of Teeth 2.8 14 3.2 16 3.6 18 4.0 20 4.5 22 4.9 24 5.7 28 9