9 Torque. Experiment objectives: Experiment introduction:

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1 9 Torque Experiment objectives: 1. Achieve an understanding of how to calculate torque 2. Achieve an understanding of how to determine moment arm when calculating the torque of a force 3. Achieve an understanding of torque as a vector 4. Cultivate the habit of keeping all experimental data in a well-organized manner Experiment introduction: Torque In rectilinear motion, Newton s 2 nd Law of motion, F net = ma, tells us how net force F net on an object of mass m drives its acceleration a. In angular motion, Newton s 2 nd Law still works but in a different form as seen in Equation (1) τ net = Iα, (1) where τ net is the net torque on an object of moment of inertia I and α is the angular acceleration of the object. Torque is the tendency of a force to rotate an object about an axis. For the situation depicted in Figure 1, where a rod is pivoted at the left end and two forces, its own weight W and an external force T, are applied; the rod may rotate around the pivot under the two torques from the two forces. Line of action for force T Moment arm l T for force T T Moment arm l W for force W Figure 1: A schematic of a rod pivoted at the left end. Two forces, the rod s weight W and an external force T, are applied on the rod. The line of action for force T is also shown. The moment arms for force T and W are represented as dashed lines in the schematic. The torque τ F caused by a force F is determined by the following formula W τ F = Fl, (2) Page 1

2 where l is the moment arm of the force, which is defined as the shortest length from the pivot to the line of action of the force. Figure 2 singles out the force T from Figure 1, and uses it as an example to show how one can calculate the torque of a force. Line of action for force T Moment arm l T for force T T θ Figure 2: Example to show how one can calculate the torque of a force. L in the figure measures the length from the pivot to the point on the rod where the force T is applied; θ measures the angle between the force T and the rod. In many cases, the moment arm of a force can be intangible and hard to measure directly, for example l T for force T in Figure 2. If one can measure the length L from the pivot to where T applies on the rod and the angle θ from T to the rod, one can determine using trigonometry that l T = L sin(θ). Therefore, the torque of force T is τ T = Tl T = TL sin(θ). Just like force, torque is a vector and has directions in the x, y and z directions of a Cartesian coordinate system. However, in simple situations, such as this experiment, we can just use clockwise or counterclockwise to represent the direction of a torque. By convention, a counterclockwise torque is defined as positive; a clockwise torque negative. For the rod in Figure 1, with the force T alone (neglecting the weight W for now), the rod will rotate counterclockwise; therefore, force T has a counterclockwise torque on the rod. Similarly, with the weight W alone (neglecting the force T for now), the rod will rotate clockwise; therefore, weight W has a clockwise torque on the rod. Equilibrium L In rectilinear motion, when the net force on an object is zero, Newton s 2 nd Law tells us that the object undergoes no linear acceleration. Similarly, in angular motion, when the net torque on an object zero, the object undergoes no angular acceleration. When both linear and angular accelerations are zero for an object, it is in equilibrium state. Please note that when the net torque on an object is zero, it may refer to one of the following two situations: 1. There is simply no torque on the object at all. 2. There are both positive and negative torques on the object, but they add up to be zero, which can be summarized in Equation (3). τ 1 + τ 2 + τ 3 = 0 (3) Page 2

3 Exploration: During the Exploration, you have a horizontal meter stick pivoted at one end, attached to a force sensor at the other end. The force sensor measures how much force is needed to keep the stick in equilibrium when various masses are attached to the meter stick at various spots, as shown in Figure 3. Conduct the following investigation: 1. Derive the formula to calculate the force sensor reading when various hanging masses are attached to the meter stick at various spots and the force sensor is held perpendicular to the meter stick; you may pick 2-4 hanging masses 2. Measure the actual force using the force sensor 3. Compare your calculation and experiment results 4. Repeat steps 1-3 for different positions of hanging masses 5. Repeat steps 1-3 after you tilt the force sensor to an angle with the meter stick. Meter stick Force sensor Hanging mass Figure 3: Schematic of experiment setup. A horizontal meter stick is pivoted at one end, and attached to a force sensor at the other end. Various hanging masses can be attached to the meter stick at various spots. Exploration grade: 20 points Please answer the following questions with one or two sentences. Some of the questions may appear in the post-lab quiz. Your instructor will randomly check your answers. 1. The weight of the meter stick must be considered. It is not accurate to simply assume the weight applies at exactly the middle of the meter stick. To determine the exact location of the weight, one can balance the meter stick on a pivot object as shown below. Where does weight apply on your meter stick? 2. The smallest scale of the meter stick is millimeter. If recording position readings in unit of meter, to which decimal place should your numbers be? 3. In this experiment, you will need to adjust a few interconnected things (please check out the section of Exploration notes: for details). It happens to everybody that after your group adjusted one thing to perfection, it may throw off the previous adjustments. So it is a good idea to go back and double check all adjustments. Have you double checked all adjustments? Please also present the following to your instructor for a grade: 1. The derived formulas 2. Hanging masses your group used and positions they are placed at Page 3

4 3. Calculation and experiment results of the force sensor reading, and the percent difference (use the percent difference formula below) 4. Relevant calculation %difference = experiment value calculation value 100% calculation value Please note that points will NOT be marked down if any of the above is wrong; however, points will be deducted based on the following guideline. More than half of the materials are missing, illegible and/or poorly organized; results cannot be understood. Exploration notes: No deduction Some but less than half of the materials are missing, illegible and/or poorly organized; efforts have to be made to understand the results. Everything is legible and well organized; instructors can easily understand the results. The picture below shows the experiment setup. The successful run of this experiment depends on correctly adjusting equipment to the working order. You will need to move the clamped post at the table edge. Force sensor Before measurement, a force sensor must be zeroed by pressing the ZERO button with no force on its hook. The meter stick is pivoted through a rod. Make sure that the rod is perpendicular to the meter stick. Page 4

5 To obtain the correct force reading, the string MUST be lined up with the axis of the force sensor. In addition to checking string alignment from the front, please also check from the side to make sure that the force sensor, the string and the meter stick all fall into the same plane. Application: For the Application part of this lab, your instructor has a new setup on the front table as shown in Figure 4. Ask your instructor for the values of m 1, m 2, θ, the new position of the pivot, and the new position where the meter stick s weight applies. Your instructor will also randomly set the positions of m 1 and m 2. Please use your calculation to determine the force reading from the force sensor. θ m 1 m 2 Figure 4: A schematic of experiment setup for the Application part. Bring your results to your instructor once you have confidently calculated the force reading Your instructor will place m 1 and m 2 at pre-assigned positions and take a reading from the force sensor. Application Grade: 20 points Please present the following to your instructor for your Application grade. 1. The values of m 1, m 2, θ, the positions of the pivot, m 1, m 2, and the position where the meter stick s weight applies 2. Relevant derivation steps 3. Critical calculation steps Page 5

6 4. The calculated force reading. Comparison of calculated force sensor reading to the actual reading Calculated force sensor reading is within of ±0.1N ±0.3N ±0.5N ±0.7N ±0.9N > ±0.9N the actual reading. Points Additionally, points will be deducted based on the guideline below No deduction Some but less than half of the materials are missing, illegible and/or poorly organized; efforts have to be made to understand the results. More than half of the materials are missing, illegible and/or poorly organized; results cannot be understood. Everything is legible and well organized; instructors can easily understand the results. Page 6

7 Lab 9 Report Rubric writing abstract The abstract of a technical document serves as a summary, which presents the work done, the results achieved in the complete document. It is usually one short paragraph, and does not include mathematical equations, footnotes, references, graphics, or tabular material. In other words, an abstract briefly summarizes what has been achieved, what method has been used, and how it was done; and the language of an abstract should not be technically specific so that readers of general background can easily understand it. In the report for this lab, please write an abstract, which will be graded using the following rubric (The rubric is adapted from the abstract requirement of a physics journal, Applied Physics Letter). The sample abstract attached may be helpful. Format Item 0 points 2 points Paragraph spacing The abstract is not double spaced. The abstract is double spaced. Abstract length The abstract is either too short or too long. The abstract is one paragraph of no more than 250 words. Font Non-standard font used for report. Regular Times New Roman font of size 12 is used for abstract. Grammatical tense Lab activities are not reported in past tense. All lab activities are reported in past tense. Content Item 0 points 3 points Lab topic The lab topic is not summarized. The lab topic is briefly summarized. (What physics did you work on in this experiment?) Lab results The results are not summarized. The results are summarized. (What quantities did you compare in this experiment, how did they compare?) Lab method The lab method is NOT described. The lab method is described. (How did you determine the results?) Language Informal language with more than 8 grammar errors Formal language with fewer than 3 grammar errors Don t forget to attach a copy of this rubric to your lab report, otherwise 5 points will be marked down. Page 1

8 Sample with comments: (The abstract below came from the scientific journal Applied Physics Letters.) What is the new technique? What is the method to verify the new technique? the results obtained with the new technique were consistent with those of other approaches. The mechanical properties of ultrathin films synthesized by atomic layer deposition (ALD) are critical for the liability of their coated devices. However, it has been a challenge to reliably measure critical properties of ALD films due to the influence from the substrate. In this work, we used the laser acoustic wave (LAW) technique, a non-destructive method, to measure the elastic properties of ultrathin Al 2 O 3 films by ALD. The measured properties are consistent with previous work using other approaches. The LAW method can be easily applied to measure the mechanical properties of various What has been achieved? There is challenge for traditional approaches, an easy alternative technique is found to solve the challenge. Page 2