The Acceleration Due to Gravity: Free Fall Name

Transcription

1 The Acceleration Due to Gravity: Free Fall Name I. Discussion Partner Early in the 17th century the very important discovery was made that, when the effects of air resistance are eliminated, all bodies, regardless of their weight, experience the same acceleration when they fall freely near the earth s surface. This may easily be shown to be a consequence of the fact that a freely falling body is acted upon by a constant unbalanced force, its weight. Thus consider a body of mass m at the earth s surface. The weight of the body is the force with which the body is attracted by the earth. The magnitude of this force is given by Newton s law of universal gravitation. F = G Mm R 2, (1) where m is the body s mass, M is the earth s mass, R is the earth s radius, and G is the gravitational constant. When a body is falling freely, the gravitational force is the only force acting on it, and the acceleration which the body experiences is given by Newton s second law of motion, F=m a (2) When the expression for the gravitational force, from (1) above, is substituted in eq. (2), the result is GMm = ma, R 2 or GM a =. R 2 (3) Since G, M, and R are all constants, eq. (3) indicates that the acceleration due to gravity is a constant, independent of the mass of the particular body experiencing it. The value of this constant acceleration, usually symbolized by g, is about 32.2 ft/sec 2, or about 980 cm/sec 2. Of course, the acceleration of a body falling through the earth s atmosphere may be reduced seriously by the retarding force exerted on it by the atmosphere. For example, if the retarding force equals 1/10 of the body s weight, the acceleration is reduced to 9/10 of the value for a freely falling body. In addition, the magnitude of the retarding force depends on the shape of the body and on its instantaneous speed. Also, inhomogeneities in the earth s composition, and irregularities in its shape cause the acceleration due to gravity to vary from one location to another. Variations in height of acceleration by changing the value of R in the equation, but this change is usually small for ordinary changes in height. Even at a height of two miles above the earth s surface, for instance, the acceleration of a falling body is only 0.1% lower than its value at the surface. In the laboratory, any measurements of the motion of a falling body are made at a given location on the earth s surface, and the height above the surface from which the body falls is, at most, a few meters. Only moderate speeds are attained, so effects due to air resistance are minimal. Usually, then, it may be assumed that this limited type of free fall involves no serious departures from linear motion having constant acceleration. If the downward direction is taken as positive, and if the acceleration is symbolized by g, the equation of motion for a freely falling body is

2 v = v 0 + gt. (4) Equation (4) is represented by a straight line, when v is plotted as a function of t, as is shown in Fig. 1. v2 v = v0 + gt v1 v0 t1 t2 time Fig. 1 The slope of the line is the acceleration, g, which is given, for any time interval, by g = v 2 v 1 t 2 t 1 = Δ v Δ t. (5) The intercept v 0 is the initial speed, when timing was begun, and the intercept t is the time required for the body, starting from rest, and accelerating at the same rate, to attain the speed v. Also, the average speed, for any time interval, is simply an arithmetic average, given by v ave = v 1 + v 2 2 (6) and this value of average speed occurs at the mid-point of the interval for which it is calculated. As always, the average speed and the distance traveled in a given time interval satisfy the relationship, v ave = s 2 s 1 t 2 t 1 = Δ s Δ t. In view of equation (5) above, it is clear that the acceleration due to gravity, at a given location, may be obtained by determining the slope of a v vs t graph for a freely falling body.

3 II. Collection of Data Manual timing of a freely falling body over the distances normally encountered in the laboratory is not practical, because of the small time intervals that occur. For a body to fall from rest, for example, through a distance of 4 meters (the height of a high ceiling) requires only 9/10 second. A stop-watch can barely be started and stopped within so short a time. If the height through which a body falls cannot be greatly increased, some more refined method of timing must be used. (Of course, increasing the height would have the adverse effect of increasing the speed, which would then increase the effect of air resistance.) In this experiment a body is allowed to fall between two parallel vertical wires. Once every 1/60 second, for a very brief interval, a high electrical potential difference is established between the wires. Encircling the body is a metal ring through which a spark may jump from one wire to the other each time the potential difference occurs. A wax-coated or heat sensitive paper tape is placed between the falling body and one of the wires. The sparks, in passing through this paper, each leave a small visible spot on the surface. A permanent record is thus created of the positions of the falling body at intervals of 1/60 second. The apparatus is shown schematically in Fig. 2. electromagnet switch DC power supply falling body wire wire paper tape spark timer (high voltage) Fig. 2

4 As was pointed out, with reference to the graph of Fig. 1, the average speed actually occurs at the midpoint of the time interval for which it is calculated. Thus a graph of instantaneous speed vs. time may be made by plotting each calculated value of average speed at the mid-point of the time interval to which it pertains. 0 S0 ΔS1 1 S1 ΔS2 2 S2 ΔS3 3 S3 4 S4 Fig. 3 A typical record of the falling body is shown in Fig. 3. Every other dot may conveniently be used as a time indicator. Then each position of the body is determined at intervals of 1/30 second, instead of 1/60 second. The particular dots selected for use should be circled and numbered, so as to reduce confusion. At the beginning of the motion the dots are close together, and labelling is begun at some convenient point marked zero. It is more accurate to choose a sequence of dots farther apart. This means that the portion of the motion which is analyzed possesses an initial speed, v 0, the speed with which the body passes the arbitrary zero mark. (The actual starting position, where the body was at rest, is above the beginning of the tape and is not recorded.)

5 The distances S 0, S 1, S 2, etc. are measured directly from the tape, and may be recorded in the table. In measuring the distances S 0, S 1, S 2, etc., place the meter stick on edge on the tape as close to the dots as possible, and read all positions without moving the meter stick from its original placement. This will help reduce uncertainty in the measurements. Attempt to measure the positions of the dots to a fraction of a mm and record the positions of about ten dots. Then the intervals ΔS 1, ΔS 2, etc. should be calculated by subtracting the previous position from the current one. Finally, based on the time interval of 1/30 second, the average speed for interval may be calculated, and entered in the table. If eleven positions are recorded this will yield ten average velocities to plot. III. Results and Conclusions You will be plotting two graphs and using them to determine the freefall acceleration and to examine features of the motion. Graph 1 velocity vs. time a. Make a graph of v vs. t by plotting the average velocity in the middle of the time interval. For example, the average velocity computed during the first 1/30 of a second is plotted at the time 1/60 (.01667) seconds, the next point at 3/60 (.05000) seconds, and so on. The available graphing software is called Graphical Analysis. Under the Analyze menu heading select Linear Fit and draw the best straight line that fits this data. Remember to include the uncertainty (standard deviation) of the slope and intercept. Print and include this graph. On this graph, the slope is the free-fall acceleration and the y intercept is the initial velocity. b. Report the freefall acceleration along with its uncertainty. Be sure to use correct units and the appropriate number of significant digits. g = % difference from 980 cm/s 2 = c. Report also the initial velocity- the speed of the falling object when it passed your zero point S 0.. v 0 = Graph 2 position vs. time a. Make a graph of s vs t. Join the data points with connecting lines. Motion with constant acceleration predicts that the position varies with time to the second power. In other words, the plot should be some portion of a parabolic curve. Is your s vs. t graph curved or straight? b. Use the s vs. t graph and determine the speed at the time t = 5/30 sec. Remember that the slope of a tangent line on this graph gives the instantaneous speed at that time. Draw a tangent line and find the slope or use the Tangent feature under the Analyze menu. Compare this calculated speed with the value read or calculated from the v vs. t graph at t=5/30 sec. v(5/30 s) from s vs. t graph = (use tangent feature) v(5/30 s) from v vs. t graph = (use v= v 0 + g t)

6 Data and Calculation Table for Free-fall Experiment Name Date Graph 2 (s vs. t) Graph 1 (v vs. t) Position Time Position Si ΔSi V ave Time # (sec) (cm) (cm) (cm/sec) (sec) 0 0 (middle of interval) /60 1 1/ /60 2 2/ /60 3 3/ /60 4 4/ /60 5 5/ /60 6 6/ /60 7 7/ /60 8 8/ /60 9 9/ / / by J.R. O Donnell and B.L. Hurst (2012)