Physics 6L, Summer 2008 Lab #2: Dynamics and Newton's Second Law

Transcription

1 Physics 6L, Summer 2008 Lab #2: Dynamics and Newton's Second Law Introduction: In Lab #1, you explored several different ways of measuring an object's velocity and acceleration. Today, we are going to build on those techniques, but will focus more strongly on the physical reasons why an object would accelerate. In other words, we will be exploring the consequences of Newton's Second Law, which says that an object accelerates only when one or more forces are pushing on it. The forces which we'll be dealing with today include, among others, gravity, friction (which we will try to eliminate as much as possible), and the force exerted by an ideal Hooke's Law spring. Apparatus: The main piece of apparatus in this week's lab is the two-meter-long air track shown in Fig. 1. When you attach an air hose to one end of ht track, the air is forces out of small holes along the length of the track. These air streams are strong enough to support a "glider" which can then move along the track with almost no friction. A length scale is marked along one side of the track, and can be used to measure the glider's position. One of the track's two "legs" has a screw which can be used to level the track. It is easiest to level the track with the air turned on and the glider on the track; then you can just adjust the screw until the glider does not accelerate in either direction along the track. You can track the glider's motion with our other important piece of equipment: a photogate timer, shown in Fig. 2. The photogate uses a beam of infrared light to detect any solid object -- such as the metal "flags" affixed to the top of your glider -- which passes through the beam. The timer has a number of controls, but you don't need to mess with them much for today's lab (unless you want to figure out what all the different settings do, and design some variant experiments of your own.) In Experiment #1, below, we describe how you will use the photogate to measure the speed of the glider as it passes through.

2 (Important: the photogate's power cord plugs into a jack on the right side of its base. There is no real reason to unplug this jack...but if you decide to (to help rearrange cords and keep them from getting tangled or something), please make sure the power cord is unplugged from the wall before inserting or removing the jack from the photogate base. Forgetting this rule has burned out some power supplies in the past.) Experiment #1: Measuring the Glider's Velocity We begin by studying the motion of the glider on a level, (nearly) frictionless air track. With the air turned on, level the track as described in the Apparatus section. Push the glider with moderate speed, and verify that as it moves along the track, friction does not appear to cause it to slow down significantly. (If friction does seem significant, check over your apparatus or talk with your lab instructor; you might have a badly connected air hose or other glitch.) Your next job is to calculate the glider's instantaneous velocity, by measuring the amount of time required for one of its flags to move through the photogate beam. (What you are really measuring is the average velocity during the interval when the flag is blocking the beam, but since this is a small interval compared to the length of the track, we are treating it as a good approximation to the instantaneous velocity.)

3 Measuring Velocity (Take these instructions as initial ideas to build on, not as gospel. If you experiment with the equipment available, you may find other ways to make this and other important measurements for this lab. Your lab instructor will be more interested in seeing that you understand the underlying concepts and can improvise a bit, than in watching you follow a set procedure "because it says so in the manual".) 1) Measure the width w of the glider's leading flag (the one which will pass through the photogate first.) You can ignore the other (trailing) flag, unless you want to use it to devise some variant procedures for any part of the lab. 2) Set one of your photogates to GATE mode, and turn on the MEMORY toggle switch. (You should see a red LED come on when you set the MEMORY toggle.) Set the resolution switch to 0.1 ms. (NOTE that the timer display reads in seconds, not milliseconds; the resolution setting just determines the maximum precision of the timer.) 3) Make sure the gate is working and the settings are right: first hit the RESET button, to reset the timer to zero seconds. Then block the beam briefly with your finger. You should see the timer display counting the amount of time the beam is blocked; when you move your finger out of the beam, the timer should stop, and should then be impossible to activate again until you RESET it once more. You should also notice that an LED above the detector lights up whenever the beam is blocked. 4) Place the photogate at any point along the track where you would like to make a velocity measurement. Make sure the glider can pass smoothly through the photogate without hitting any stray wires or other obstructions, and that the leading flag blocks the photogate beam as it passes through. (You should see exactly the same behavior as when you passed your finger through the beam in step 3.) 5) RESET the timer, then give the glider a gentle push and let it coast through the photogate. Record the time t measured on the photogate display; if all has gone well, this is the time that it took for the leading flag to move through the beam. 6) Now you can calculate the glider's velocity at the moment when the flag passed the beam: (eq 1) v = w / t (careful with units!) Repeat this velocity measurement several times, pushing the glider at a different speed each time. Did the runs in which the glider seemed to move faster, in fact give a higher velocity measurement? Try placing two photogates at different points along the track. If the glider truly moves without friction, and the track is perfectly level, how should the glider's velocity compare at these two

4 points? Record a prediction of what results you expect, then try it out, measuring the velocities at both gates. Do your results match your prediction? What, if anything, do your results tell you about possible systematic and/or random errors that might affect the rest of the lab? Experiment #2: Galileo's Inclined Plane experiment As we showed in lecture last Thursday (and as Wolfson derives in Chapter 5), if you tilt the track at an angle θ from the horizontal, then (if we continue to neglect friction), the glider should accelerate downhill with acceleration (eq 2) a = g sin θ where g is the local gravitational field, which near the Earth's surface should be approximately g = 9.81 m/s2. So let's give it a try. Place a wooden block or some other convenient object under one of the "feet" of the airtrack. Measure the height h by which you have elevated this end, and also measure the length L of the track, from one "foot" to the other. Using a bit of basic trigonometry, you should be able to calculate the angle θ that the track makes with the horizontal. Do so. (Why was it important to measure L from one "foot" to the other, instead of measuring the full length of the track? You should be able to answer this by thinking about the right triangle you used for your trig calculation.) You will get cleaner results if you stick to angles θ less than about 5 degrees or so; at higher angles, the glider will accelerate to faster speeds, and friction and air resistance will become more important. Calculate the glider's expected acceleration using equation (2). Now we want to measure its actual acceleration, and compare it to your expected value. There are several ways in which you could do this. We describe one method below...but again, make sure you understand why these

5 method works, and are not simply following directions. If you want to devise your own methods, that is better yet. Measuring Acceleration: Method 1 (using single photogate) 1) Choose a point on the track at which you will release the glider from rest. 2) Choose a point, somewhere well below the release point, where you will measure the glider's velocity. Place a photogate across the track, just like you did earlier. (You may need to adjust the height and angle of the photogate, now that you have tilted the track.) 3) Measure the distance Δx that the glider will travel, from its release point until its leading flag trips the photogate. (You can use the track's built-in ruler, or grab a meter stick; whichever is easier.) Think about what you're doing here, and make sure the distance you measure really is the distance the glider will travel. 4) Release the glider from rest at your chosen point. Using the same velocity-measurement protocol you learned in Experiment 1, calculate the glider's velocity v as its leading flag passes the photogate. 5) The kinematics equations at constant acceleration should, by this point in the course, be very familiar to you. For the measurement you've just made, the equation (eq 3) v 2 = v aΔx is perfect. Since you released the glider from rest, you can take v0 = 0. Solving for a then yields: (eq 4) a = v 2 / (2Δx) (again, careful with units!) Calculate the glider's acceleration using this method. By using equations 3 and 4, you have implicitly assumed that the glider's acceleration remained constant for the entire trip. Do you think this is an accurate assumption? Why or why not? Try releasing the glider from several different points, sometimes far above the photogate, sometimes much closer. (Of course, you will have to re-measure Δx each time.) Before doing this, predict two things: How should the glider's final velocity change as you vary Δx? (That is, should a higher value of Δx yield a faster velocity, a slower one, or the same?) How should the glider's measured acceleration vary with Δx? Make several measurements with different values of Δx. Were your predictions accurate? How do your measured values for acceleration compare to your predictions from eq. 2? What can you deduce about the forces affecting the glider, and/or about other sources of uncertainty that might be present? If you've got enough time, try varying the track's angle θ. Predict how this should affect the glider's acceleration, and explain whether your predictions were accurate.

6 Experiment #3: Using Hooke's Law to measure a spring constant This experiment should be very quick. If it takes you much longer than 5 minutes, you may be overcomplicating things. Recall from lecture, or from Wolfson Chapter 4, that when a spring is stretched a distance x beyond its equilibrium point, it will pull back toward equilibrium with a force given, to an excellent approximation, by Hooke's Law: (eq 5) F = -k x where the spring constant k, measured in Newtons per meter, is independent of the "stretch" Δx. Two different springs, though, may well have different spring constants k. The negative sign on the right-hand side is essentially just a reminder that the force always points back toward equilibrium, and can be ignored when you just want to calculate the magnitude of the spring force. You've been given one or more springs. Choose one, and measure its spring constant k. The easiest way to do this is to let the spring hang vertically from a fixed support. (We've already set up such a support at your lab station, complete with a meter stick clamped in place to measure position.) Attach a hanger to the spring, and note the hanger's equilibrium position on the meter stick. If you now add mass m to the hanger, the spring will stretch to a new equilibrium position, an additional distance Δx below the one you noted before. At this new equilibrium, that the downward pull of gravity mg (from the mass you added) is balanced by the additional upward pull k Δx (due to the fact that the spring has stretched farther than before): Eq. (6) mg = k Δx You can measure m and Δx, and you know g, so you should be able to solve for k. Again, be careful with units. The most convenient units for k are Newtons per meter, so think carefully about what units to use for m, g, and Δx. Repeat this measurement for several different values of m. As you vary m, how should k change? Make a prediction, then do the measurement and see if your prediction was accurate. If you have time, you may want to make a quick graph of Δx vs. mg. If you draw a best-fit line through your data points, the slope of this line serves as a measurement of 1/k. (Why?)

7 Experiment #4: Dynamics of a spring-driven glider. Now we're going to try to measure the acceleration of a glider which is being pulled along the airtrack by a stretched spring. This will be the trickiest measurement of today's lab, because Hooke's Law tells us that the spring's force will change as the glide moves along the track...thus, the glider's acceleration is not even approximately constant over its entire trip. Before devising a way to measure non-constant acceleration, let's take care of a few other details. Attaching the spring, and re-balancing the glider Connect one loop of a mylar tape sptrip to the glider's flag as shown above. Connect a spring (whose spring constant k you already measured in Experiment #3) to the other end of the mylar tape, and run the tape over the horn-shaped air bearing at the end of the track. (Adjust the air flow in the bearing by turning the little screw at its base until the tape moves over the bearing with negligible friction.) With the spring hanging freely, adjust the angle of the airtrack until the glider can rest without accelerating -- that is, the downhill pull of gravity on the glider, and the uphill pull due to the spring's weight, cancel one another out. (To adjust the angle, don't mess with the set screws on the track's legs -- that will make life difficult for whoever uses the track next. Instead, just be creative in using objects to raise one end of the track.) Notice that there is a heavy weight on the floor near your desk. Attach the free end of the spring to this weight, to anchor it in place. You should now find that the glider has a well-defined equilibrium position on the track, and that when you pull the glider away from this equilibrium (thus stretching the spring), the spring causes the glider to accelerate back toward equilibrium. Make a note of this equilibrium position, using the track's built-in ruler -- you will need it for step 1 in the upcoming procedure instructions. Now, you want to measure the glider's acceleration at several different points on the track, different distances from equilibrium. This time, though, you cannot assume that the glider moves with constant acceleration for its whole trip. (Well, you can, but it will be a wildly inaccurate assumption.)

8 So instead, we will use two photogates, and measure the glider's velocity at two closely spaced points. We will still need to assume the acceleration is constant during the short interval between the gates...which is not entirely accurate, but far better than assuming it is constant for the entire length of the track. Measuring Acceleration: Method #2 (using two photogates) 1) Choose a point -- we'll call it point A -- where you want to measure the glider's acceleration. Measure the distance Δx from point A to the glider's equilibrium point. (You will be estimating the acceleration as the glider's leading flag passes point A...so measure from point A to the equilibrium position of the leading flag.) 2) Place two photogate timers equal distances to either side of point A, as shown below. Measure the distance z from the center of one photogate to the center of the other. You will have to assume that the glider's acceleration is constant over the interval between the photogates...this is not actually true, but the smaller the distance z the better an approximation it will be. On the other hand, there will be physical constraints on how small you can make z...use some imagination, and decide how to best handle this issue. 3) Make sure the glider can move freely through both photogates without striking any cables or other obstructions, and that the leading flag trips both beams correctly as it passes through. Prepare both photogates to take velocity measurements, just as you did in Experiments #1 and #2. 4) Release the glider from rest, somewhere below both photogates, and let it accelerate under the force of the spring, passing both gates. Using your usual methods, calculate the glider's speeds v 1 and v 2 as its leading flag passes through the two gates respectively.

9 5) Now we can adapt equation (3) to this experiment. That is, assuming that the acceleration a stayed roughly constant for the brief interval between photogates (though definitely not for the whole trip), we can say: Eq. (7) v 2 2 = v az (Unlike our earlier strategy used in equation (4), we cannot say v 1 = 0.) Solve this equation for a...first symbolically, and then plug in your measured values of v 1, v 2, and z, to calculate the glider's (sort-of) instantaneous acceleration at point A. If you haven't already done so, measure the glider's mass M. (Scales are provided on the back counter of the lab room.) Using M and the measured acceleration at point A, calculate (via Newton's Second Law) the force that the spring exerts on the glider at point A. (For this calculation, you don't need to worry about any forces other than the spring's Hooke's Law force. You already carefully tilted the air track so that the downhill component of the glider's weight cancels the weight of the spring.) Finally, now that you know the spring force F, measured at a known distance Δx from equilibrium, you can deduce the spring constant k. Does the value of k you've just measured agree (within reasonable limits of experimental uncertainty) with the value you measured for the same spring in Experiment #3? If you still have time, choose a new point A, either closer or further from the glider's equilibrium point, and repeat your acceleration measurements at this new point. You should definitely get different velocities and a different acceleration...but the value of k should (within the limits of experimental accuracy) stay the same. Remember this for your future work with springs: for an ideal Hooke's Law spring, k is a property of the spring itself, and does not depend on how far the spring is stretched. Pre-Lab Questions: 1) Suppose you are working on Experiment #3, the static Hooke's Law measurement. You add a 200-gram mass to a spring, and find that it stretches an additional 16 cm. Calculate the spring constant k. 2) Suppose that you are using the two-photogate method for calculating acceleration, as described in Experiment #4. The width of the glider's leading flag is w = 2.0 cm. You place the photoogates such that the distance between their centers is z = 10.0 cm. When the glider passes through the two gates, the photogate timers read t 1 = s, and t 2 = s. Calculate the velocities v 1 and v 2 as the glider passed each photogate, and calculate its

10 acceleration a.