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Brown University Physics 50/70 Department of Physics Free Fall Velocity This experiment studies uniform acceleration in one dimension by systematic measurements of a falling body's position and instantaneous velocity. The positions of timing detectors are varied to generate precise values of these quantities from average quantities. The Basis, Plan, and Procedure sections that follow describe the experiment; the function and nomenclature of the timing equipment will be found in the Guide to Laboratory Measurements. 1. Basis of the Experiment It is shown in many texts that if an object is released from rest and allowed to fall, its instantaneous velocity at a distance S is given by v 2 2aS (1) where the accepted value of the acceleration due to gravity, a, is about 9.81 m / s 2 . The instantaneous velocity cannot be measured directly, because the body must move over a finite distance S in an interval of time t in order for us to measure a velocity. What we measure is an average velocity v S / t . There is a way, however, to relate a particular instantaneous velocity to a measured average velocity. The method is based on the fact that the instantaneous velocity v (t ) is linearly related to the elapsed time if the acceleration is constant: v(t ) at (2) U tU Here the zero of time is defined as the instant of release v(0) 0 . Suppose the body falls from an upper point (U) along its trajectory at time t U to a lower point (L) at time t L . The average velocity M tM v (tU , t L ) in this time interval between t U and t L is defined as the mean of the instantaneous velocities at the instants t U and t L , v (tU , t L ) v(tU ) v(t L ) 1 (3) L 2 tL v (tU , t L ) 1 v(tU ) v(t L ) 1 atU atL a tU tL (4) 2 2 2 Fig. 1 The “mid-time” instant t M in the fall from U to L is by definition, t M tU t L 1 (5) 2 Substituting this in the right hand side of Eq. (4), we have v (tU .t L ) atM (6) But at M , as shown in Eq. (2), is just the instantaneous velocity at time t M , so we have converted our measurement of average velocity over an interval to the instantaneous velocity at t M , the TIME MIDPOINT of that interval: v(t M ) v (tU , t L ) (7) We go through all this trouble because the average velocity is easy to measure. The average velocity is s(t tU ) s(t t L ) SUL v s / t (8) (tU t L ) TUL 1 2 1 2 s(tU ) v0tU (atU ) ( gtU ) (9) 2 2 1 2 1 2 s(t L ) v0t L (atL ) ( gtL ) (10) 2 2 v 0 is 0 because the cylinder is dropped from rest with no initial velocity (free fall ). S As a check, let us verify that equation 7 is in fact true. Does v (t M ) UL ? TUL g v(t M ) (tU t L ) (11) 2 1 2 1 2 ( gt ) ( gtU ) SUL 2 L 2 1 ( g (t L tU )) (12) TUL (t L tU ) 2 Equations 11 and 12 are thus equal and hence validate equation 7. Notice that the time midpoint t M is not the space midpoint, as shown in Fig. 1. Because of the acceleration, the body travels farther in the second half of the time interval than in the first half. But if we develop a way of locating M, the space point corresponding to the time midpoint t M , we can take the instantaneous velocity at M to be the measured average velocity over the UL interval, and use Eq. (1) to calculate the acceleration a. Solving Eq. (1) for a, we get for the acceleration v2 a (13) 2S ZM In our case, S ZM (see figure2) is the total distance traveled in the time interval between t = 0 and t = t M . It is not the distance S UL or S UM . So we can calculate g from equation 13 by substituting v S UL /(tU t L ) and measuring S ZM . But remember this will only be 1 1 true when the photobridges are set so that TUM tU t M TUL (tU t L ) . Electronic 2 2 timer1 will read TUM and electronic timer2 will read TUL 2. Plan of the Experiment We use photobridges across the path of the falling body to measure the time intervals we need. The apparatus consists of a rigid vertical rod adjacent to the body's trajectory, on which the photobridges, marked U, M, and L in Figure 2, are mounted. The body, latched magnetically at Z until released, defines an exact zero point in time, distance and velocity. Two electronic timers marked TUM and TUL in the figure, are set to operate in pulse mode. Not intended to be a wiring diagram (these are present in the laboratory) the figure indicates the logic flow of signals from the photocells to the timers. The pulse from the U cell as the body first cuts its beam is passed to both the UM and UL timers, starting both counters. When the body first cuts the M z beam, its photocell sends a second pulse to the TUM timer, which start causes the timer to stop, giving the TUM time of fall from U to M. The TUL U timer continues until the beam is stop cut to photocell L, at which time its pulse stops the timer with the time of fall from U to L. M TUL All the bridges are movable on the rod. Suppose we start with the U bridge high on the rod and the L stop L bridge mounted about a meter below it. Now let the M bridge be placed midway in space between the other two. When a drop is made, the UL timer will contain the total fall time through the bridges, Fig. 2 and the UM timer will show the fall time from U to the space midpoint. The latter, because of acceleration, will be larger than one-half the UL reading. But now, leaving the other bridges locked in their positions, we move the center photobridge upward, searching for the time midpoint. At the next drop, we can verify the constancy of the TUL reading, and check to see if we have reached half that value on the TUM timer. This step is repeated until the position corresponding to the time midpoint is found. Once the time midpoint is found, we can apply Equation (8), using the time readings and distance measurements as described on pages 1 and 2, to get a precise value of the acceleration. The measured average velocity over the UL interval equals the instantaneous velocity of the body at the instant it cut the M photobeam at the TIME MIDPOINT. The distance S is that from the rest position Z (not merely from the U bridge) to photobeam M. Distance measurements are critical. Note that measurements at the rest position always refer to the lower edge of the body, because that is the edge that activates the photobeam “switches”. The distance from the rest level to the upper photobeam can be made a one- time problem by choosing a good location for the upper bridge (one that allows easy access for placing the mass at the rest position) and locking it there for the entire experiment. All measurements to or between photobeams are best made by using the well-defined metal frame of the photobridge itself. The distance between photobeams, for example, is exactly the distance between corresponding edges on their photobridges. Where the beam itself must be located, as in the case of the upper beam relative to the rest position, use the “offset” of the beam from the edge of the photobridge that you are using. This can be obtained (again a one-time problem) by measuring the vertical underline{width} of any bridge with a caliper; the offset is just one-half this width. 3. Procedure and Data KEEP A RECORD OF YOUR PROCEDURE THROUGHOUT THE EXPERIMENT. Align the apparatus so that the beams are cut reliably over the entire drop length. Small shifts of the mounting board on the floor, and small rotations of the bridges, may be needed. Be sure that there is a box at the base to catch the body. Set the top bridge position high, but allow ready access to the launch position. Make several drops to check for good alignment, for repeatability at fixed bridge positions, and to decide on a good range of positions for the lowest bridge. Note that the highest position of the lowest bridge should not be such as to give small (two-digit or very low three digit) time readings, since any digital reading can inherently be in error by one in the lowest digit. Measure carefully the constant distances discussed above and record them. In your notebook set up a Table in which to enter your data in a clear, understandable way. Always record the numbers as you measure them - leave calculations, even simple ones, for later. Include units for all numbers. Choose a lowest setting of the lower photobridge, and hunt with the middle bridge for the half-time (TIME MIDPOINT) position. Once located and verified, record all distances and the timer readings. These will be used to calculate the acceleration a from equation (8) as described in the Basis section. Record all the UL times for a fixed L position: The variation in this number reflects the reproducibility of the measurements with this apparatus. For at least four more (higher) positions of the lower photo bridge, repeat and record the procedure and data as you did in the preceding steps. 4. Calculations For each setting of the lowest bridge, calculate the acceleration by determining v (U , L) and S from your measurements, and then using Eq (8). Expect some variation among your values of a. 5. Results The best value obtained from a series of N measurements of a quantity is the mean value, simply the arithmetic average of the individual measurements. Using all N independent acceleration determinations a i , (where N is at least 5), calculate a “best value” for your experiment as the mean, or average of the individual values a i , N 1 a N a , i 1 i No experimental result is complete or meaningful without an estimate of the experimental uncertainty. A good measure of the uncertainty in the mean is the standard deviation of the mean, S.D., which is obtained from the mean square deviation (MS) of your measured values from their mean: 1 N MS (ai a ) 2 , N 1 i 1 and MS S .D. , N where the a i are your individual determinations of a. A final best value with its uncertainty is then a S.D. 6. Discussion and Conclusion Compare your measured value to the accepted value of the acceleration of gravity, and discuss the result, taking into account your experimental uncertainty and the reproducibility of measurements with the apparatus. Try to include a discussion of sources of experimental uncertainties. Note: In your report you are not expected to repeat the plan of the experiment as given in the handout, but to say briefly what you actually did - e.g. how many drops you made to find the time midpoint and what the time was for each drop, any problems you encountered . REFERENCES [1] Laboratory Measurements (Physics 0030); and Young and Freedman; [2] University Physics (9th Ed, Extended Version), Chapter 2.