12. Translational and Rotational Equilibrium*

Transcription

1 12. Translational and Rotational Equilibrium* This experiment will involve a rigid object in equilibrium, particularly a crane boom at a dock yard. Before coming to lab, read the sections in your text on torque and rotation. The definition of equilibrium is as follows. Learning Objectives: 1. Learn the definition of torque and two practical ways to calculate torque about an axis (a) using a moment arm (b) using the force perpendicular. 2. Practice applying two conditions for equilibrium that come from the 1 st law of motion to solve several problems in context. When an object is in equilibrium: 1. The vector sum of all forces acting on the object is zero. 2. The algebraic sum of torques on the object around any axis is zero. (These conditions prevent linear and angular acceleration, as you learn from the book.) Reading Assignment: Knight, Jones & ield (161): 7.1 The rotation of a Rigid Body, 7.2 Torque, 7.3 Gravitational Torque and the Center of Gravity Seway and Vuille (211): 8.1 Torque, 8.2 Torque and Two Conditions for Equilibrium, 8.3 The Center of Gravity Serway and Jewett (251): 10.6 Torque, 10.7 Rigid Object Under a Net Torque Review on Torque: Imagine trying to turn a bolt with a wrench. The effectiveness of the force you apply to rotate the bolt depends on the distance out to where the force is applied, measured from the axis of rotation. Of course it also depends on the magnitude and the direction of the force. Even when the same amount of force is applied to an object, whether or not the object will rotate, or how much, depends on two other factors. An extreme example is the case when you try to turn a nut, but the force is directed straight inward along the handle of the wrench. That is, a stab the wrench straight at the nut. This would be useless. Such a force, however strong it might be, does not help to rotate the nut. It just pushes on it. On the other hand the most efficient way to rotate the nut is apply the force perpendicular to the handle of the wrench, as far from the nut as possible. Therefore the magnitude of torque, which is the measure of ability of a force to rotate something around a certain axis is defined as follows: Magnitude of Torque = force times moment arm d sin( ) Axis of rotation (Points out of the page) d Point of application of the force The point on the object where a particular force acts is called the point of application of. A force can exert a torque around a particular axis. You need to choose the axis before I can tell you what the torque is, even after you already told me what the magnitude, direction and point of application of the force are. In the work we do here, the axis will always be assumed to stick out of the page, i.e. perpendicular to the plane of the beams and/or the strings defining the apparatus. (Let s just call this the plane.) There are two handy ways to compute torques in such a case. * William A Schwalm

2 Notice that d sin ( dsin ) d( sin ) Just two different groupings So there are two ways to calculate the magnitude of torque. Place origin at the axis so that d = r. Method #1 axis of rotation h r Line of action point of application Here r is the displacement vector from the axis to the point of application. The line of action is a line in the direction of the force through the point of application. The moment arm denoted by h is the perpendicular distance from the line of action to the axis. The torque around the axis is h rsin + Counterclockwise - Clockwise Method #2 axis of rotation r point of application Here, is the component of the force perpendicular to r. In other words, it is the perpendicular part of the force. Then we have r r sin + Counterclockwise - Clockwise PRE-LAB EXERCISES (Bring solutions with you to lab.) 1. Define the following: a. Point of application of force b. Line of action of force c. Leaver arm for torque about a certain axis due to force 12-2

3 b d Translational and Rotational Equilibrium 2. (Physics 251 only) Write down the definition of torque (a vector) as a cross product. Explain in detail how this leads to the two methods described above. 3. ive forces are applied to a door, as shown. or each force, is the torque about the hinge positive, negative or zero? a c Hinge e b d 4. Six forces, each of magnitude either or 2, are applied to a door as shown. ind the six torques 1 through 6 and rank in order, from smallest to largest. 1 L /4 L /4 L /4 L /

4 The Crane Boom: A crane boom is used to load cargo onto ships. Your group is employed by a freight company to analyze the forces acting on the boom, the tension in the cables, and the forces of attachment to the vertical mast under various load conditions. Equipment: Pulley, mass set meter stick hook collar clamp, protractors (2), support rod clamp Problem 1 Your team is to report on the following: To what extent can you use a laboratory model to study the conditions for the equilibrium for an actual crane boom when the boom is horizontal? To what extent can you verify that these conditions hold? m 1 pulley pivot point A end point B m 2 You need to build and study a laboratory model. In this case we suggest how to set up, since this set-up defines the problem. The apparatus consists of a meter stick that is mounted on a hook collar clamp, a protoractor with a plumb bob, a mass set, a support rod, a pulley, a clamp and a paper clip. The pulley is attached to a vertical support rod, or mast, which is mounted on the bench top by means of a clamp. One end of the meter stick is supported by a string attached at the 30cm mark. This string passes over a pulley, and is fastened at its other end to a 0.50 kg mass labled m 1. Use a piece of clear tape to attach the protractor to the meter stick at the 50cm mark as shown. Position a 0.20 kg mass labeled m 2 and the protoractor so that the meter stick is horizontal and balanced (protoractor reading 90 degrees). Prediction and Method questions: 1. How many forces are acting on the boom when it is horizontal? Make a free-body diagram (BD) showing all forces for the equilibrium configuration described above. 2. Which of the forces you found in step 1 will tend to rotate the meter stick (boom model) about the pivot point A? 12-4

5 3. Which forces found in step 1 want to make the meter stick rotate about point B when the boom is horizontal? 4. List the quantities you need to measure for the calculation for the equilibrium conditions (both torques and forces) when the axis is through point A. 5. How do you find the moment arm about A for the tension in the string representing the cable that supports the boom? Plan: Work out a measurement plan for the analysis treating A as the axis of rotation. You should design a data table and explain how measurements will be taken. Describe how you would use the meter stick to measure the position of the point of application of each of the applied forces with respect to the point marked A. Implementation: Measure each force that would give a torque about A. 12-5

6 Analysis: igure out whether the equilibrium conditions are satisfied, and the extent to which you can actually verify this, given your measurements error. Can you tell if the net force zero? Remember there are two components. Do x-components of forces contribute to the torque about A? How come? Calculate the torque about the pivot point A for the each force you calculated. Does the net torque nearly equal zero? If not, how much extra torque is needed to balance the system? Show your work. Measure the mass of the meter stick including the protractor and the plumb bob with the balance provided. Determine the center of mass of this by setting it on a sharp edge. The edge of a ruler would be fine for this purpose. Compute the percent difference between the net torque and the net positive (or net negative) torque. Is there force acting on the meter stick at the pivot point A? How much y-component of the force is necessary to balance the system? How much x-component? Including the force from the pivot you just calculated, apply the condition for rotational equilibrium about some other point (choose a point, for example, at the center of mass of the meter stick, or the opposite edge of meter stick marked B in the picture). Calculate the net torque about this new point. Calculate the % difference between the net torque and the net positive (or net negative) torque for the step 8. Conclusions: What is the point of these measurement activities? Are there forces that do not contribute to the calculation of torque? Is there any force that you can t measure directly? To get the equilibrium condition recall that you can select any point through which the axis of rotation passing. Which point in the plane defined by the meter stick and the string would you choose? Why? To check the validity of your result you could change another point, say point B in the set up, and can reapply the equilibrium conditions. What would be a practical disadvantage of choosing the point B for the axis of rotation in terms of what you can measure? Was your calculation in agreement with the textbook description of the static equilibrium conditions? If not what caused discrepancies? In other words, are they within what you estimate to be your experimental measurement error or not? (Better refer to some actual numbers 12-6

7 here. Your supervisor will be ticked if you can t even estimate quantitatively how much experimental error there is.) Problem 2 Now we get serious. Suppose the problem changes. A load is attached to point B. Then the boom is raised up to a 45 o angle. We need to know how the torques and forces change. 1. Prediction question: The base of the vertical mast is planted in the dock. It can withstand only a certain maximum torque before it snaps. When the boom is up at a 45 o angle, a load (hanging from B) is lifted that puts a torque of 75% maximum on the base of the mast. Including both the load and the weight of the boom, can the facility withstand the torque when the boom is lowered to straight-out horizontal without snapping the mast? Show details of your calculation. (Yes, you have enough information.) 2. Method question: If the length of the actual boom at the dock yard is 40 feet, and the boom weighs.65 tons. (a) What mass should you hang from point B in your lab model to simulate a 5.85 ton load for the actual facility? (b) A one-newton force for the lab model should correspond to how much corresponding force on the real facility in order to make things to scale? (c) A distance of one meter on the model should correspond to a distance of what at the shipyard? 12-7

8 Plan: Set up a measurement plan to study the operation of the crane boom at a 45 o angle. You want to know what the torque about the base of the boom will be, due to both the boom weight and the load hanging from point B. You also need the tension in the cables and the force components x and y acting at the hinge point. Thus of course you need to move the suspended mass m 1 out to point B at the end, and you need to make it the right size to represent 2.50 tons. (You may need to figure out another suspension method.) Make a plan stating what needs to be measured, how you will figure what you need to calculate from the measurements, etc. Explain too how you determine the scaling ratios to get the model to correspond to the problem at the shipyard, whose going to do what in your team, and so on. Implementation: Make all the measurements and record the data here in a neat table, with enough additional comments to make sense of things. Analysis: igure out the torque that will result at the base of the mast, both for the lab model and for the crane boom at the yard. Show your analysis and explain. Show also a calculated error estimate, so that the company can know to what extent to rely on these figure, and how much safety margin to allow. 12-8

9 Conclusions: What did you learn by doing this? 1. In particular, what kinds of things do you learn about the analysis by actually building a model that you would not have learned just by solving a problem in the textbook? Explain in some detail. 2. Restate the two conditions for equilibrium, both translational and rotational, in your own words, being careful of the details, and then explain how these were applied in problem

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