Linear Motion Lab Essay Example

Linear Motion Lab Paper 2. LINEAR MOTION In this experiment you will study the motion of an object in one dimension from a number of points of view. You will demonstrate how the variables of motion are related by differentiation and integration and investigate the relationship between potential and kinetic energy. Theory Why Study Motion? Motion is everywhere in the universe. Only at a temperature of absolute zero is the motion in any body truly absent. If motion exists then so also does energy. To the delight of the modern-day physicist the tools that were invented by Galileo Galilei, Isaac Newton and others 200 years ago to describe motion apply everywhere in the universe, from electrons in our own bodies to the farthest galaxy. The study of motion and of energy is at the heart of physics. This experiment deals with motion of the simplest kind, motion in one dimension or motion in a straight line. Kinematics and Dynamics The subject of motion is divided for convenience into the subtopics of kinematics and dynamics. Kinematics is concerned with the aspects of motion that exclude the forces that cause motion. In a manner of speaking, kinematics is focussed on the development of definitions: position, displacement, velocity, acceleration and on the relationships that exist between them. Dynamics widens the study of motion to include the concepts of force and energy. Definitions Position Kinematics begins with the idea of position. Suppose that we photograph an object moving to the left along a horizontal path at two instants of time and superimpose the images for study (Figure 1). We will write a custom essay sample on Linear Motion Lab specifically for you for only $16.38 $13.9/page Order now We will write a custom essay sample on Linear Motion Lab specifically for you FOR ONLY $16.38 $13.9/page Hire Writer We will write a custom essay sample on Linear Motion Lab specifically for you FOR ONLY $16.38 $13.9/page Hire Writer We examine one image with a ruler and mark off the number of units that separate the object from the ruler’s zero. The zero is a reference or origin at a position of zero units by definition. The position of the object at any another place is, say x units. x is an instantaneous quantity since it applies to a specific clock time—the instant the photograph was taken. Position like length is a basic quantity and is dependent only on the unit used. But position involves direction also. In principle the object could be to our right or to our left. To include the information of direction we use a vector. The magnitude or length of the vector, say r, is r (or perhaps x), while the direction is to the right, meaning the object is to the right of the reference point. We could also agree that, by convention, the sign of x is positive in this particular case. Elapsed Time The two positions of the object in Figure 1 must be described with different vectors and different clock times. The photographs can be said to show two events, an initial â€Å"i† event and a final â€Å"f† event. There is now an elapsed time between the events equal to the simple difference: ?t = t f – t i , †¦[1] unit seconds, abbreviated s). Keep in mind that the concepts of clock time and elapsed time are different; an elapsed time is the difference between two clock times. L2-1 L2 Linear Motion 0 rf clock time tf object ri displacement ? r = rf – ri clock time ti object ? r = v ? t Figure 1. This drawing illustrates an object moving toward the origin (left) â€Å"photographed† at two positions. The corresponding clock times are indicated. Position, displacement and velocity vectors are given different head styles to emphasize their different natures. Displacement Displacement differs from position. In the elapsed time between the events the object moves from one position to another. The displacement is the difference between the two vectors describing the two positions: d. Eq[3] then becomes what is known as the instantaneous velocity ? dr ? =v. dt †¦[4] ? ? ? ? r = rf – ri , †¦[2] (unit meters, abbreviated m). Displacement, being the difference between two vectors, is also a vector. The displacement is negative in this case (according to our convention) since it points towards the origin. Velocity Average Velocity. Another quantity in kinematics is the average velocity. This is the displacement an object undergoes in one second of elapsed time. It is the ratio ? ? This quantity is abstract and tricky to imagine: it can be thought of as the average velocity that might be measured with a superior detection system over an infinitely short elapsed time (or the velocity at a specific clock time). In practice, with equipment available in a first year physics lab, it can be measured only approximately. If the displacement is known as an analytical function of time, r(t), then the instantaneous velocity at some clock time t0 is the tangent to the function at t0, or the first derivative of r(t) at t0. The finding of tangents is one of the objectives of this experiment. Acceleration The velocity of the object in Figure 1 may change with time. The velocity might decrease as a result of a force of friction between the object and the path. Or the velocity might increase if the path were not horizontal and a component of the force of gravity acts on the object. The time rate of change of the average velocity is called the average acceleration and the time rate of change of the instantaneous velocity is called the instantaneous acceleration. Both types of acceleration are defined as in eqs[3] and [4] with â€Å"v† subsituted for â€Å"r â€Å"and â€Å"a† substituted for â€Å"v†. ? ? r rf – ri ? = =v, ? t ? t †¦[3] (unit meters per second, abbreviated m. s–1). The average velocity, being a vector divided by a scalar, is a vector. The average velocity is negative here, too, since it points towards the origin. The magnitude of the average velocity is the speed. The elapsed time in eqs[1] and [3] is a finite interval. What would happen if this interval were infinitely small? Mathematically speaking, this amounts to taking the limit of eq[3] as ? t>0. The increments ? ust be replaced by the differentials L2-2 Linear Motion L2 Motion of an Object Whose Velocity is Constant In this experiment you will mostly be studying the motion of an object whose velocity is changing. However, for purposes of completeness we first consider motion at constant velocity. The case of an object moving t owards the origin on a horizontal plane is drawn in Figure 2. We suppose that the data pairs (t, r), where t is the clock time and r is the position are measurable at regular intervals by some detection system. Two such points when plotted on a graph might appear as shown in the upper half of Figure 3. A computer could be programmed to calculate the â€Å"average velocity† as the slope between the two datapoints and plot it as a point on a graph (lower half of Figure 3). The result is negative, the sign indicating the direction of the velocity vector. The computer software used in this experiment does something similar by finding the average velocity by averaging over the slopes between a number of datapairs (7 by default). Thus if a number of datapoints were measured and the results plotted on a graph, the result might resemble Figure 4. As the glider approaches the origin here the position decreases but always remains positive. The velocity remains at a constant negative value. The velocity is therefore just the derivative or the slope of the displacement versus clock time graph (or the slope of the position versus clock time graph here in one dimension). The velocity is seen to change little (if at all) with clock time and so the acceleration (decceleration) is very small. Motion Detector 0 clock time: tf rf clock time: ti ri positive displacement ? r = rf – ri v = ? r also to the left ? t Figure 2. An object is shown at two positions (events) while moving toward a detector on a horizontal plane. ti , ri ) Position ( tf , rf ) clock time Velocity ( tf , vf ) Figure 3. A graph of the two position-clock time datapoints described in Figure 2. Shown also is a point on the velocity graph as it might be generated from the slope between the two datapoints multiplied by the sign of the velocity vector. L2-3 L2 Linear Motion Figure 4. Typical position and velocity graphs as might be produced for an object mo ving as shown in Figure 2. Can you see how these graphs are consistent with Figure 3? Motion of an Object Whose Velocity is Changing with Time In this experiment you will mostly be ignoring the effects of the force of friction. However, for purposes of understanding it is useful to consider friction briefly. A small force of friction must exist between the glider and the layer of air on which it moves because the glider is seen to slow down. Friction acts opposite to the direction of motion (to the right in Figure 2) and therefore produces an acceleration also toward the right. This acceleration is often described as a decceleration in the sense that it is opposite to the velocity and describes a velocity decrease. (The object is slowing down. The velocity and acceleration versus clock time graphs in this case will resemble Figure 5. It is known from other experiments (â€Å"Simple Measurements†) that the force of friction, though small, has a complicated functional form giving rise to a decceleration that depends on the first (and sometimes the second) power of the speed. Gravity, unlike friction, is a constant forc e and is therefore much easier to deal with; the effect of gravity on motion we consider in the next section. Figure 5. Velocity and acceleration graphs for an object moving as shown in Figure 2 while subject to a small force of friction. Remember, graphed here are the magnitudes of the vectors multiplied by the sign corresponding to the direction of the vectors. Motion of an Object Whose Acceleration is Constant and Nonzero An object moving with constant non-zero acceleration down an inclined plane (neglecting friction) is shown in Figure 6. Its free body diagram is drawn in Figure 6b. Starting from the assumption that the acceleration is constant and nonzero we can develop the mathematics of the object’s motion using successive integration. If friction is negligible, then the L2-4 Linear Motion L2 agnitude of the force giving rise to the object’s movement is the component of the force of gravity along the incline, that is, where C2 is another constant of integration. C2 = r(0), the position at a clock time of 0 seconds; thus F = ma = –mgsin ? , †¦[5] r(t) – r(0) = 1 2 at + v(0)t . 2 †¦[9] where the minus sign means that the force vector points towards the origin. The glider†™s instantaneous speed at some arbitrary clock time t is obtained by integrating a from eq[5]: 1 v(t) = ? adt = at + C1 , where C1 is a constant of integration. C1 is the instantaneous speed at a clock time of 0 seconds, i. . , the quantity v(0); thus We can now make the following correspondence. If at event i we let ti = 0, then ri = r(0) and vi = v(0). Then we can also write for the event f, tf = t, rf = r(tf) and vf = v(tf). Then equations [9] and [7] take on the more familiar forms: rf – ri = and 1 2 at – v it 2 †¦[10] †¦[11] v f = vi + at . v(t) = v(0) + at . †¦[7] The glider’s position at the clock time t is found by integrating v(t) from eq[7]: r(t) = ? v(t)dt , = ? (at + v(0))dt , = 1 2 at + v(0)t + C2 , 2 †¦[8] In this experiment you can test these equations with the detection system and software several ways. Firstly, you can test if the position, velocity and acceleration data you collect really is described by functions of the form of eqs[10] and [11]. In other words, you can perform polynomial â€Å"fits† to the data, finding in the process the value of a. You can test the differential and integral relationships between the variables by invoking the options of slope and integral. These are some of the objectives of this experiment. ti , vi tf , vf Motion Detector rf m g sin ? 0 ? ? r = rf – ri (a) (b) ri posi tive N mg Figure 6. (a) A object is shown at two positions while moving down an inclined plane. The object’s free body diagram is shown in (b). The object is â€Å"pulled† toward the origin by the component of the force of gravity mgsin?. The acceleration is negative here according to the sign convention employed by the detection system. L2-5 L2 Linear Motion Work and Mechanical Energy In Figure 6 the object is shown moving down an inclined plane. It is therefore accelerating. Any time an object is accelerating, work is being done upon it. The concepts of force and energy take us in this section into the area of dynamics. The concepts of work and energy are closely related. The work done on a point mass2 by a net constant force F is defined Work-Energy Theorem the work done on an object goes to increase the object’s mechanical energy. This energy can be either kinetic energy, potential energy, or both. Both types of energy can be defined with reference to Figures 7 where an object of mass m is shown responding to an applied force in two ways. In Figure 7a the object is being moved to the left in our field of view along a horizontal, frictionless surface. In Figure 7b the object is being moved vertically upwards. Figure 7a enables us to define kinetic energy most easily. Figure 7b enables us to define potential energy. We first consider the former in the next section. ? ? W = F †¢ r = Frcos ? , †¦[13] (unit joules, abbreviated J). Here r is the object’s displacement vector and ? is the angle between the vectors F and r. (The †¢ symbol indicates the scalar or â€Å"dot† product of vectors. ) According to the F m hf Fg ? v = 0 F speed vf m ? h = 0 F speed vi m ? h reference position m Fg Figure 7. An object is shown responding to a force in two ways. In (a) the surface is horizontal and frictionless, so the work done by the source of the force F goes entirely into increasing the object’s kinetic energy. In (b) F is greater then the weight | F g | of the object by some infinitesimal amount so the work done by the source of the force goes entirely into increasing the object’s potential energy. F (a) hi (b) L2-6 Linear Motion L2 Kinetic Energy Suppose (Figure 7a) that we are able to measure the instantaneous speed of the object at the clock times ti and tf when the object is at positions ri and rf, respectively. The object is receiving a force F and is being accelerated. In general, F is not constant. The work done on the object between ri and rf is, from a generalization of eq[13] 3 1 1 = m? v 2 – vi2 ? . f ? 2 ? The work done is therefore the difference between two terms: W= 1 2 1 2 mv – mv . 2 f 2 i †¦[14] ? ? ? ? W = ? F †¢ dr = m? a †¢ dr = m? adr , since the vectors a and dr are parallel. Changing the variable of integration from r to v gives 4 If we call the expression mv2/2 the kinetic energy Ek, or the energy of motion, eq[14] can be written W = Ekf – Eki = ? Ek . †¦[15] dv dr ? , W = m? ? ? dt dt dt ? = m? v vdv , i vf In this particular case W is a positive number, i. e. , the kinetic energy of the object is increasing. The work done on the object is therefore equal to this increase in kinetic energy. This is one-half of the Work-Energy Theorem. Potential Energy If we assume in Figure 7b that the magnitude of the applied force F is insignificantly greater than the weight of the object acting downwards then the object moves upwards a distance ? h = hf – hi in some elapsed time at zero speed (thus acquiring negligible kin-etic energy). From the definition, eq[13], the work done on the object is W = E pf – Epi = ? Ep . †¦[17] ? ? W = ? F †¢ dr , = ? h mgdr . i hf Once again the work done is seen to be equal to the difference between two terms: W = mghf – mghi . †¦[16] If we call the expression mgh the potential energy of the object at the position h relative to the position 0 and denote it Ep we can write eq[16] as This is also a positive number. Thus the work done on the object is equal to the increase in the object’s potential energy. This is the second half of the Work-Energy Theorem. The change in kinetic energy is zero here. This energy is potential energy, gravitational potential energy to be exact. This is the energy an object possesses by virtue of its position relative to some reference position in a gravitational field. This energy is the work required (by us) to move the object (very slowly) to the final position from the initial, reference position. Now Figures 7 describe two idealized situations where the object is moving either horizontally or vertically. Should the object be moving on an inclined plane it will be moving at an acute angle relative to the horizontal and will therefore undergo changes in both kinetic and potential energy. One of the objects of this experiment is to study this general case. L2-7 L2 Linear Motion The Experiment Exercise 0. Preparation First Steps Identify the glider stowed on top of your air blower. Find the spacer. Identify the detector (the blue box) mounted just beyond the near end of the track) and the universal interface (ULI) board mounted on the front of the blower box. Simple Measurements You are going to log the glider’s mechanical energy so you need its mass. You can weigh it on one of the electronic balances in the lab. Measure the thickness of the spacer with the vernier caliper. The Air Track The object whose motion you will study is a glider on an air track (Figure 8). An air track is an ideal device for studying the motion of an object like a glider since it supplies a thin film of air on which a glider, having just the right shape, can move with reduced friction. Identify the air track’s two rear feet and its single front foot. Confirm that elastic bands are in place to serve as bumpers for the glider. The detector and associated electronics in the ULI board log the glider’s position at clock times you can select. The detector emits pulses of ultrasound at a frequency of 50 kHz. These pulses travel out from the detector through the air, reflect from the reflector on the glider and then return. Since the speed of ultra-sound in air at room temperature is known, the computer is able to calculate the distance of the glider from the detector (the position) by timing the round trip of the pulse. 5 To get velocity and acceleration, the computer makes several measurements and calculates averages and rates of change. Motion Detector near bumper far bumper single front spacer goes under front footpad ? X = (2. 29  ± 0. 01) m spacer Y Figure 8. The air track shown in an inclined position (scale is exaggerated here). X is the distance between the single front foot and the midpoint of the two rear feet. For all tracks X = (2. 29  ± 0. 01) m. L2-8 Linear Motion L2 Checks to Minimize Friction To ensure that your glider moves with minimum friction check the following: Motion setup should run. The Opening Screen The opening screen shows three narrow graph areas (one of which is shown in Figure 10). The position versus clock time graph should be currently selected (a box should be drawn around it). To select any graph move the mouse to position the cross-hair pointer over the graph and then click the mouse button once. Notice on the clock time axis of the graphs that the program is set by default to record for 10 seconds. The Expected Value of Acceleration Locate the spacer you will be placing under the front footpad of your track. Based on the geometry of Figure 8 what value of acceleration should you expect for your glider? You need this number in the formula for potential energy (next section). Entering Formulas LoggerPro automatically computes velocity and acceleration from the basic values of position it records, but it must be instructed to calculate energy (since energy depends on your glider’s mass). You have to enter the formulas into LoggerPro’s spreadsheet. To do this select Data New Column Formula. Figure 9 shows a fragment of the screen which then appears. On separate calls to New Column enter formulas for kinetic energy, potential energy and total energy as is shown in the figure. Pay attention to the common spreadsheet syntax (â€Å"*† means multiply, â€Å"^† means to the power of, etc. ) Notice that you can select variables and functions from the popdown lists. You enter velocity as â€Å"Velocity† which you can either type in yourself or select from the Variables popdown list. Take your time here. Your TA will help you with these tasks. The formula you will likely have the most difficulty with is potential energy. Convince yourself the formula can be written ? Ep = m x (expected value of a) x position. Or, in LoggerPro notation: ? If ON, turn any and all equipment OFF. ? In order to run properly, the â€Å"V† angle of your glider must match the â€Å"V† angle of your track. Read and follow the instructions on the blower box. Turn the air blower ON. In order to run properly, your glider must be supported by air at sufficient pressure. Read and follow the instructions on the blower box. ? ? Level Check Your air track must be level when you start this experiment otherwise your glider will experience an unaccounted for acceleration. To check for level first remove any spacer that might have been left under the front footpad of the track. Place your glider on the midpoint of the track and look for any drifting right or left. If your glider does drift adjust the screw on the front foot of the track to compensate. When the track is level your glider should not drift appreciably left or right. Setting up the ULI Board Turn the ULI board ON by means of the switch located in its lower right hand corner. A LED on the board should glow red or green. ? Booting LoggerPro To boot the program follow these steps: ? With the computer ON log into your account n the FISSURE server as you learned to do at the Orientation Workshop. (If for any reason you cannot log in you can save your work temporarily in the â€Å"Student Temp Save† folder on the local hard drive. In any case, you can log in and out at any time. ) On the local harddrive â€Å"Macintosh HD†, locate and open folders in this order: â€Å"Physics† †PHYA10 †02. Linear Motion†. Inside â€Å"02. Linear Motion† double click the icon Linear Motion. LoggerPro and the Linear ? ? L2-9 L2 Linear Motion m * (expected value of a) * â€Å"Position†. What is the expected value of â€Å"a† based on a 2 cm spacer? If later, you find you have made a mistake in typing and wish to modify or remove the column go back to Data Modify Column and correct your mistake. When ready click OK. Changing the Graph in an Area By the time you have finished entering these formulas you may observe that the label down the left hand side of the graph has changed (look ahead to the left hand side of Figure 10). You can change what is displayed in a graph area. To do this place the pointer over the label and hold down the mouse button. You can then select the graph you wish from the list of Columns to Plot. Do this now to change the label on the first graph back to r (m). Figure 9. A fragment of the screen which appears when you select New Column. Exercise 1. First Launch, More Options and First Observations The objective of this exercise is to obtain some data from a first launch, to explore a few options of the program and to examine the data qualitatively. First Launch After you have levelled your air track to your satisfaction place the spacer under the front footpad of the track. The track is now inclined by some angle. Now release (don’t push) the glider from the top end of the track. Make sure the glider’s reflector is facing the motion detector. It is suggested that you record data between two bumps of the glider with the lower end of the track. When you are ready, click the Collect button. Repeat a few times to get graphs to your liking. After 10 seconds you should see screens resembling Figure 10. You should have 6 graphs of information altogether: Position, Velocity, Acceleration, Kinetic Energy, Potential Energy and Total Energy (the last three you have to select using Option 1 below). Before you get down to serious business you may wish to investigate one or more of the following options. However, if you wish to bypass the options you can go directly to the section First Observations. Option 1. Changing the Graph in an Area Notice in Figure 10 the label â€Å"r (m)†. You can L2-10 Linear Motion L2 change what is displayed in a graph area. To do this move the mouse to place the pointer over the label and hold down the mouse button. A list of Columns to Plot will then pop down. Select the graph you wish to be displayed in the area from this list. Option 2. Number of Panes You may wish to display one graph at a time or as many as four graphs. To change the number of panes select View Graph Layout†¦. Then click on the number of panes selection you wish. Graph produced by LoggerPro, copied to the clipboard and pasted directly into Microsoft Word. Figure 10. A Position vs time graph. This is one of the six screens of information you have collected at this stage of the experiment. Option 3. Saving If you wish to save your experiment choose File Save As†¦. Make sure you save your file in your account space on the campus server or in the â€Å"Student Temp Save† folder on the local hard drive. Option 4. Preparing for Printing Eventually, you will want to print your graphs on the laserwriter printers in the physics lab as proof of your work. To prepare for printing, choose File Page Setup. Fill in your name and any comment you wish to have appear beneath your graphs on the printed page and then click OK. Your partner may change this later to print graphs for his or herself. Option 5. Printing Before you attempt to print anything on the laserwriter printers in the physics lab, confirm that your output will go to the printer you expect. Select Chooser LaserWriter 8, then confirm that the correct printer (Physics LaserWriterI or PhysicsLaserWriter2) is selected. If in doubt about a printing issue ask your TA. To actually print a graph choose File Print Graphs. First Observations Before you apply mathematics to your data in any experiment it is often useful to obtain an overall qualitative â€Å"impression† of your data to determine if it is â€Å"reasonable†. To this end consider these questions: Questions: ? With reference to eqs[10] and [11] do you have a graph which is quadratic in time? Linear in time? Which graph describes a constant (between bumps)? ? Your position versus clock time graph resembles the path of a projectile in two dimensional space. Why? ? Explain in your own words why the velocity and acceleration graphs have the form you observe. ? From your velocity versus clock time graph you should be able to find a clock time at which the velocity of the glider was zero. What L2-11 L2 Linear Motion was the acceleration of the glider at this clock time? If the velocity of a body is zero can its acceleration be non-zero? Explain. With LoggerPro’s tools you can call up instantaneous values and tangents. To see instantaneous values choose Analyze Examine. One of the graphs will resemble Figure 11. If you now choose Analyze Tangent the tangent or slope at the position of the cursor will be shown on the graph. By means of the tangent function can you demonstrate to your satisfaction the validity of eq[4] and its counterpart for acceleration? A few tests are sufficient because you will continue this study in Exercise 3. When you wish to turn Analyze OFF choose Analyze Examine a second time. A better test of the validity of eqs[10] and [11] is a polynomial fit. This you will do in the next exercise. Graph produced by LoggerPro, copied to the clipboard and pasted directly into Microsoft Word. Figure 11. One of the six graph screens with Examine turned on. As you move the cursor with the mouse the position and time are updated at the top left corner of the screen. Exercise 2. Fitting Functions to Data The objective of this exercise is to test the validity of eqs[10] and [11] by seeing if they can be fitted to the appropriate set of your data, position, velocity or acceleration. The result of the fitting will yield the acceleration â€Å"a†. This value you can then compare with the â€Å"expected value† of Exercises 0 and 1. Fitting To fit a function to your data first select (click on) the appropriate graph. Then choose Analyze Examine. Place the cursor at the beginning of the range you wish to fit, hold down the mouse button and drag over the range you wish fitted. Make sure the range you select does not include a bump (after all, you are interested in linear motion and a bump is defnitely non-linear). If you wish to fit a straight line choose Analyze Linear Fit. If you wish a different kind of fit (a quadratic for example) choose Analyze Curvefit. The result of a linear fit might appear as shown in Figure 12. For this process to succeed you must pay close attention to the function to be fitted and the terms to be included. For example, eq[11] is what kind of function of time? Linear? Quadratic? What is the slope? What is the intercept? If you wish more information on the linefit (more digits etc) double click on the bar of the information window. The result will resemble Figure 13. L2-12 Linear Motion L2 Fit Results Find the acceleration â€Å"a† from the results of the fit of all three graphs (position, velocity and acceleration). How do the a alues compare with each other and with the expected value? Comment on the quality of the fits. If the fits are good it can be said that the equations â€Å"describe† the data. Graph produced by LoggerPro, copied to the clipboard and pasted directly into Microsoft Word. Figure 12. A range of data on a velocity vs clock time graph has been selected (shown by the square brack ets) and fitted to a straight line. To request more information about a fit double click on the bar of the fit information window (Figure 13). Graph produced by LoggerPro, copied to the clipboard and pasted directly into Microsoft Word. Figure 13. The graph of Figure 12 showing more information about the linefit. The slope is now seen to be -0. 0897  ± 0. 0001. Cor is the correlation coefficient. Exercise 3. Differentiation and Integration The objective of this exercise is to investigate how the variables of motion are related by differentiation and integration. Differentiation (Slope Studies) Identify the sections on the position versus clock time graph where the glider is moving towards and away from the origin. Explain in your own words why the position increases or decreases and when. Is the value obtained for the slope of the position versus clock time graph at some arbitrary clock time consistent in magnitude and sign with the value given for the velocity? Is the value obtained for the slope of the velocity versus clock time graph consistent in magnitude and sign with the value given for the acceleration? A few tests will suffice. L2-13 L2 Linear Motion Integration (Area Studies) Comparing two variables of motion related by differentiation—when one is the slope of the clock time graph of the other—is easy. But what if the relation is integration? To prepare ourselves we put eqs[9] and [10] in this form: (t 2 ) – r(t1 ) = ? t v(t)dt , 1 t2 †¦[18] where t1 and t2 are, of course, the clock time limits. As preparation for the following studies describe eq[18] in your own words. What does its left hand side mean? What does its right hand side mean? You might wish to include in your word description the phrases â€Å"difference between† and â€Å"area under the curve of†¦ † or other equivalent phrases. Pay attention to the clock time limits. If you understand this expression enough to explain it in your own words you will have no trouble with the following activities. Calculating the area under a curve is easy with the software supplied. LoggerPro will compute the area under a region of a curve you select, i. e. , between two limits. You must first select the region by dragging with the mouse as was illustrated in Figure 12. Pick a region of the velocity versus clock time graph between two bumps (which is linear) and drag over the region to select it. Next choose Analyze Integrate. Your result might resemble Figure 14. Do your best to confirm eq[9]. Repeat in a similar fashion for the relationship between the velocity and acceleration. Graph produced by LoggerPro, copied to the clipboard and pasted directly into Microsoft Word. Figure 14. A velocity vs clock time graph showing an integration. The area under the curve for the time limits chosen is 0. 222 m/s/s. Other Features and Other Questions There are other features of the graphs whose interpretation involves interesting physics. For example, explain in your own words and with sketches why the features shown in Figures 15a and 15b appear on the velocity and acceleration graphs. Why does the acceleration go from zero to a maximum positive value then drop to zero again? Explain why both â€Å"positive† and â€Å"negative† bumps appear on the acceleration graphs. Does the magnitude of the velocity have the same value after, as before, a bump? What does this say about the kind of collision at the bump? L2-14 Linear Motion L2 Figure 15. Interesting structures in the velocity and acceleration graphs. Exercise 4. Energy The objective of this exercise is to examine as many aspects of energy as time permits. Viewing the Energy Graphs To view the energy graphs follow the procedure of Option 1 of Exercise 1. Your total energy graph may or may not resemble Figure 16. Observations and Questions Explain in your own words the characteristics of the kinetic energy graph. That is, answer or otherwise account for the following (interrelated) questions: Conservation of Energy Questions: ? Is mechanical energy conserved in this experiment? If not, why not? ? Why are the potential energy and kinetic energy graphs curves and not straight lines? Energy Loss Of course, friction is present here and does account (at least partially) for the decline in total energy. Explain how you might calculate the total energy lost per second? Other Questions What might be another source of energy loss besides friction? How would you go about quantifying this loss? ? ? ? What accounts for the â€Å"dips† in the kinetic energy graph? The dips in the kinetic energy graph show that the kinetic energy is zero for a brief instant. How is it possible for the glider to lose all of its kinetic energy and then regain it again? Describe all instances of work being done and by what agent. ? ? Graph produced by LoggerPro, copied to the clipboard and pasted directly into Microsoft Word. Figure 16. A total energy vs clock time graph. The energy is nearly a constant here at just under 0. 05 J. It would still be useful to find the energy lost per second from a linefit. L2-15 L2 Linear Motion Physics Demonstrations on LaserDisc There are many physics demonstrations on laserdisc dealing with motion in one dimension. Here are just a few of them: from Chapter 2 Linear Kinematics Demo 01-08 Constant Velocity Demo 01-11 Constant Acceleration from Chapter 3 Linear Dynamics Demo 03-01 Air Track Friction Stuart Quick 2003 End Notes for Linear Motion 1 2 If you can’t follow the calculus here go directly to the main equations [10] and [11]. By point mass is meant an object whose internal structure is to be neglected. Or in other words, a point mass is an idealized object of mass m of infinitely small size. Some of the work done on an object of finite size could conceivably go into increasing the object’s internal energy, something we wish to avoid here. In this guidesheet the word object will mean point mass. 3 If you wish to bypass the calculus go directly to the main results, eqs [15] and [16]. 4 5 This change is effected by putting a = dv and dr = dr dt. This is similar to how a bat sees in the dark using its own kind of ultrasound. It is also how a Polaroid autofocus camera determines the distance to an object to focus properly. Similar principles apply to ultrasound devices used in medicine and to various range-finding devices. dt dt L2-16