Static Equilibrium. Torque - also known as moment of force. (Serway Sec. 11.1) Rigid objects in static equilibrium. (Serway Secs. 12.1, 12.

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1 Physics Topics Static Equilibrium If necessary, review the following topics and relevant textbook sections from Serway / Jewett Physics for Scientists and Engineers, 9th Ed. Torque - also known as moment of force. (Serway Sec. 11.1) Rigid objects in static equilibrium. (Serway Secs. 12.1, 12.3) Torque (or moment of force) is a quantity which measures the tendency of an applied force to cause a rotation. The magnitude of torque can be calculated as τ = r F sin θ (1) where r is a vector pointing from the pivot point to the position at which the point is applied, and θ is the angle between the vectors r and F. For a rigid object (or system of objects) in static equilibrium, both the net external force and the net external torque applied to the system must be zero. Fext = 0 τext = 0 (2) Introduction You are designing a new building with a horizontal observation deck which is a cantilevered section jutting out from the main structure (as an example, consider the graduate house at U of T, on the corner of Harbord and Spadina). This section of the building is to be supported by a vertical cable some distance along it, and by the main structure at the other end. For a safe design, you need to know how much force these supports must exert when people are on the observation deck. As a first step, you model this observation deck as a rod with supports shown below. Apparatus Support frame Ruler pivot Ruler mounts (2) Hanging masses Vernier LabPro + Logger Pro software Vernier Dual Range Force Sensor (2) Clamps Page 1 of 8

2 Figure 1: A model of a cantilevered observation deck. The deck is modeled as a rigid rod (ruler), the attachment to the rest of the structure occurs at the pivot. The mass of people on the observation deck is represented by the hanging mass M. Pre-Lab Questions Please complete the following questions prior to coming to lab. At the beginning of lab, you will be given a short quiz which is heavily based on one (or more) of these questions. 1.) Read through the entire lab writeup before beginning 2.) What is the specific goal of this lab? Exactly what question(s) are you trying to answer? Be as specific as possible. ( To learn about topic X... is not specific!) 3.) What specific measurements or observations will you make in order to answer this question? Answer this question in words. Then, on a separate sheet, use your answer to create a data table which you will fill during lab. 4.) Draw a free body diagram for the hanging ruler shown in Fig 1. Label all forces acting on the ruler and be sure to indicated where these forces are applied. Don t forget the force of gravity on the ruler! Gravity acts at the center of mass (aka center of gravity); label this point a distance x CM from the pivot. 5.) Choose a coordinate system. Write the equation for the balance of forces Fext = 0, making sure to include all of the forces acting on the ruler. 6.) Choosing the pivot point at the left-most part of the ruler, write an equation for the torque from each of the forces acting on the rod. Forces which would cause a counterclockwise rotation are treated as positive torques. Forces which would cause a clockwise rotation are treated as negative torques. Page 2 of 8

3 7.) Using the results of the previous part, write the equation for the balance of torques τext = 0 in terms of the measurable quantities M, d, l, F, x CM, and other known constants. Procedure I (M = 0) 1.) Force Sensor Configuration (a) Connect the dual-range force sensor to Channel 1 of the LabPro interface. Set the range switch on the force sensor to 10N. Mount the force sensor vertically on the support frame using a clamp. (b) Open Logger Pro on your computer (c) Choose Experiment Calibrate CH1: Dual Range Force Sensor (d) Click Calibrate Now. (e) With no mass hanging from the sensor, enter 0 in the Reading 1 field. Click Keep. (f) Hang the 300g mass from the sensor (this applies 2.94N of force). Enter 2.94 in the Reading 2 field. After the reading shown stabilizes, click Keep. (g) Click Done. (h) Hang the metal ruler mount from the force sensor. Once the ruler mount is properly hanging, select Experiment Zero. Select the sensor from the list and click OK to zero the force sensor. (i) Select Experiment Data Collection. Set the duration to 2 seconds and the sampling rate to 100 samples/second. (j) Set up the program to report the average of the samples by selecting Analyze Statistics, and select the relevant sensor. 2.) Measurements (a) Measure and record the mass of the ruler. (b) Hang the ruler horizontally with the ruler mounts as shown in the figure. The left end of the ruler should be resting on the pivot. Make any adjustments to ensure the ruler is level. For this part, do NOT hang any additional mass from the ruler. (c) Measure and record the length d. (d) Use Logger Pro to measure the force F by clicking Collect. The program will take 200 samples over 2 seconds, plot the data, and report the average/standard deviation in a box on the plot. Record the average value, and the standard deviation (a measurement of uncertainty) in your notebook. Page 3 of 8

4 (e) Go back to step 2c and repeat the procedure for at least three more values of d. When you finish, you should have at least 4 data points with measurements of d and F. Record all of your data in a table. Analysis I 1.) Using your data along with the predictions you made in the pre-lab questions, compute a value of the center of mass of the ruler x cm. Include uncertainty in your result. (You may need to review the rules for propagating uncertainty from Lab 1). 2.) Is your result for x cm consistent with your intuition? Does it make physical sense? If not, you may need to revisit the equation you used to calculate x cm. Procedure II - (M 0) 1.) Using a mass hanger, hang a mass of 300g from the ruler as shown in Fig 1. Record the values of d, l and M in your notebook. Don t forget the mass of the hanger! 2.) Use Logger Pro to measure the averaged force F as was done in Procedure I. Record the average force and the standard deviation (a measure of uncertainty). 3.) Repeat this procedure twice more by varying either M, l or both. Record all of your results in a table. Analysis II - (M 0) 1.) Using your predicted equation(s) from the pre-lab, for each set of values M, d, l, what would you predict for the value of F? Compare the predicted values with your measured values and calculate a % error for each. 2.) Choose one of your data points (a particular M, d and l), what do you predict will be the force exerted by the pivot on the ruler? Show your calculations and explain your reasoning. 3.) Test your prediction with a second dual-range force sensor. (a) Make sure this second force sensor is connected to Channel 2 of the LabPro interface. (b) Set the range switch on the force sensor to 50N. (c) Calibrate and zero the force sensor by following the directions to Force Sensor Configuration in Procedure I. Note that since the range is 50N, you may want to use a heavier mass for the calibration. (d) Position this second force sensor at the same position as the pivot. Remove the pivot and instead hang the end of the ruler on the second force sensor. Page 4 of 8

5 (e) Use the second sensor to measure and record the force that the pivot was exerting. 4.) Compare your predicted pivot force with your measured pivot force. Calculate the % error. Wrap Up The following questions are designed to make sure that you understand the physics implications of the experiment and also to extend your knowledge of the physical concepts covered. Each member of your group should be able to answer any/all of these questions. Your TA will check that this is the case; please check out with your TA before exiting lab. 1.) A uniform rod pivots about a frictionless, horizontal axle through its center. It is placed on a stand, held motionless in the position shown in Fig. 2 and then gently released. Figure 2: Explain in words what will happen to this rod. Figure 3: Dual Force Sensor Configuration 2.) For the situation shown in the Fig 3, which sensor will read the larger force? Explain in words. Page 5 of 8

6 (a) Both sensors read the same force, which is half the weight of the load (Mg/2). (b) Both sensors read the same force, but it is not necessarily equal to Mg/2. (c) The left sensor measures the larger force. (d) The right sensor measures the larger force. 3.) Again consider the setup shown in Fig 3, where both force sensors are connected. In Analysis II, you examined a particular data point with a given M and d. For these values, is there some place you could hang the mass M such that the left force sensor measured no force? If so, find that position. If not, explain why not. Informal Report At the end of the lab, you will be asked to submit an informal report on the work you did. The informal report is a simple record of the data/analysis of the experiment, but does not need a lengthy introduction, conclusion, or procedure section. This informal report should include: Identifying Info - Your name, your partner(s) names, the date, course and section, TA name, and the name of the experiment. Data - Lists and/or data table(s) showing all measurements taken, including units and uncertainties. If you used any techniques not listed in the lab manual or encountered unexpected issues, briefly comment on these in your report.. Analysis - Sample calculations to demonstrate to your TA/instructor how your analysis was completed. If you propagated uncertainty in any calculations, a sample of each type of calculation should be included. Include properly labeled graphs, including error bars to indicate uncertainty (if applicable). If you used any non-standard techniques, explain your methodology in words. Results - Your main results. If necessary include percent error calculations and/or uncertainty. Briefly comment on any unexpected results. Wrap up - A brief (no more than a few sentences) response to each of the wrap-up questions. Page 6 of 8