Falling in Air. "Facts do not cease to exist because they are ignored." A. Huxley
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1 Falling in Air "Facts do not cease to exist because they are ignored." A. Huxley OBJECIVES o learn another technique for measuring motion and to study an example of motion along a line under the influence of various forces. HEORY It is well known that light-weight objects like leaves fall less rapidly in air than dense objects such as rocks. his is attributed to "air resistance" or "air drag", and usually ignored in the idealized world of physics textbooks. Since it is a complex but important phenomenon, we will examine some of the effects of drag experimentally. Like friction forces between solid objects, drag forces on objects moving in a fluid tend to slow and eventually stop the motion. In simple situations, the force might be proportional to some power of the speed, and directed opposite to the velocity of the object. One possible expression for a drag force is F d = C d Av p (1) where C d is called the coefficient of drag, A is the cross-sectional area of the object normal to the velocity, and v is the speed of travel. For low-speed motion through viscous fluids, like a spoon sinking in honey, the exponent p will be near 1. For macroscopic objects moving through air at speeds typical of thrown balls, runners, cyclists or automobiles, p will be about. If the speed varies significantly, there may be a transition between the different regimes. he velocity dependence of the drag force makes it difficult to solve the equations of motion in any but the simplest cases. Here, we will consider only an object of mass m falling vertically from rest, not the more interesting trajectory problems that arise in sports. he equation of motion is then ma = mg! F d () Before solving this in detail, we note a qualitative feature: As the speed increases the drag force increases, until it is the same magnitude as the gravitational force and there is no further
2 acceleration. his contrasts sharply with the text book case of free fall, where the acceleration continues indefinitely. he steady final velocity, called the terminal velocity, can be calculated by finding the speed at which the gravitational force mg is equal to the drag force given by Eq. 1. he result is v p = mg C d A (3) Evidently the terminal velocity increases as the ratio of mass to area increases, so small heavy objects have a higher terminal velocity than large light ones, as you might expect. his also accounts for the observation that typical objects dropped from reasonable heights do not seem to reach a steady velocity. (Amusing fact: he terminal velocity for small animals, like squirrels, is low enough that they will almost always survive the encounter with the ground, no matter how far they fall. Large animals, like people, do not fare so well.) he complete equation of motion for an object falling vertically with p = is ma = mg! C d Av (4) his can be simplified slightly by writing it in terms of v dv dt = g " 1! v % # $ v &' (5) Eq. 5 can be solved by the usual methods for second order differential equations, to yield the speed and position as functions of time " v = v tanh gt + tanh!1 v 0 % # $ v & ' v (6) x = v g '! gt $ ln cosh# & + v 0 sinh gt * ), + x 0 (7) ( " v % v v + where v 0 and x 0 are the initial conditions, and v is given by Eq. 3 with p =. he properly skeptical reader can verify this result by differentiating twice to recover Eq. 5. Actually, we will only be concerned with the position of an object dropped from rest, so Eq. 7 becomes PHYS 111 Falling in Air
3 x = v g '! gt $ * ln) cosh# &, + x 0 (8) ( " v % + he predicted speed and position are plotted in Fig. 1. Note that the speed becomes constant, corresponding to the straight portion of the x-t graph. A similar derivation for the p=1 case leads to dv dt = g(1! v /v ) (9) v = v 1! (1! v 0 /v )e!gt / v [ ] (10) x = v t! v g (1! v /v )1! 0 e!gt / v [ ]+ x 0 (11) For an object dropped from rest, this becomes x = v t! v g 1! e!gt / v [ ]+ x 0 (1) 3 3 v (m/s) x (m) t (s) t (s) Fig. 1 Calculated speed and position vs time for an object with v = 3.3 m/s and p =. PHYS 111 Falling in Air 3
4 he resulting motion is qualitatively similar to Fig. 1, although it may be possible to distinguish the cases by careful comparison with data. EXPERIMENAL PROCEDURE o find out if the motion is adequately described by Eq. 8 or 1, you need to measure x(t) for some falling objects. his can be done with LoggerPro and a video camera, following the directions in Section V. If the data and calculation can be made to agree by proper choice of p and v, we can have some confidence in our description. As a further test, we can vary the mass of the falling object to see if v is proportional to the mass or the square root of the mass, as expected from Eq Physical arrangement he falling object is a styrofoam cup, suspended from three threads. It can be dropped from a clamp situated about 4 m above the floor. When you are ready, use the clamp to grab the suspension threads at the knot where they are tied together. Be sure the cup hangs freely, and that the heavy release cord for the clamp is out of the way. o release the cup, pull gently on the cord from the ground after starting the video recording. Some cardboard disks are available to increase the weight of the falling cup. he camera should already be set up on a support across the room from the drop area. Be sure the power is on (plugged in) and the round switch on the back is set to MOVIE. You can check the orientation and field of view when you open the preview screen of the capture program. Consult with the instructor if it appears to be necessary to move or adjust the camera, and please leave it undisturbed for the next group of students. In order to calibrate the pictures we need to include a known length in the video. A white stick with black lines at 0.5 m intervals is available. Place it near the path of the falling cup, at the same distance from the camera. he camera has been set for a fast shutter speed (1/500 th s) in order to produce a sharp image of rapid motion. hree lamps, mounted on a tall stand, provide the additional light needed for a good exposure. Be sure the lights are turned on when you are taking data, and turn them off when you stop taking pictures.. Picture data Start LoggerPro from Falling.cmbl, since it has Eqs. 8 and 1 loaded into the fitting menu. Capture a movie of the falling cup and then go to the analysis window. Mark as many points as possible during the fall, and calibrate the distance scale. For fitting, it is convenient to put the origin at the starting point of the motion, so that x 0 = 0. PHYS 111 Falling in Air 4
5 Select the portion of the x vs t graph corresponding to the period between release and impact on the floor. Use Curve Fit to find the best parameters in both Eq. 8 and 1 to describe your x(t) data. Note that an additional parameter t 0 has been included, to account for the fact that the camera clock started before the motion began. hese are difficult expressions to fit, so you will have to manually modify the parameters to get very close to the correct values before starting the automatic fit. Even so, you need to check that the final values of x 0 and t 0 are reasonable. If the cup actually reaches terminal velocity you can also compare a linear fit to that part of the motion to the v parameter found from 8 or 1. Once you are satisfied with the validity of the parameters, record them for later analysis. You will probably notice that neither Eq. 8 nor 1 fits the data perfectly. o quantify the precision of the fit, the program calculates the RMS (root mean square) error, defined as # & RMSE = %"(y i! y i, fit ) ( $ ' i 1/ (13) If both fits cover the same range of data, the one with the smaller RMSE is a more precise description. 3. Measurement Program Obtain x(t) and fit to find v for both models for several different masses covering the available range. Note that for large enough mass the motion will become indistinguishable from the quadratic x(t) of free fall. Prepare a plot and fit to determine if v is more nearly proportional to either m or m as appropriate for each model. Check the individual fits to see if the p = 1 or model is a better description of the x(t) data. If you do find differences, do they occur under particular circumstances, for example when v is large or small? PHYS 111 Falling in Air 5
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