Measurement Of Free-Fall Acceleration Essay, Research Paper

Measurement of Free-Fall Acceleration

Introduction

Galileo Galilei (1564-1642), the man first accredited with the correct notion of free-fall with uniform acceleration, stated that “if one were to remove entirely the resistance of the medium, all materials would descend with equal speed.” Today, this statement holds true for all objects in free-fall near the Earth s surface. The purpose of this experiment is to verify Galileo s assertion that acceleration is constant. In addition, the magnitude of acceleration will be calculated.

Theory

By definition, acceleration is the rate of change of velocity with respect to time. Instantaneous acceleration is the derivative of velocity with respect to time.

a(t) = dv / dt.

Average acceleration is the change in velocity during a time interval, Dt, divided by the length of that interval,

aave = Dv / Dt.

In this experiment, average acceleration of gravity will be determined by measuring the change in position of a falling object at regularly timed intervals. With this, average velocities for these intervals will be calculated. A graph of the average velocities versus time should give a straight line whose slope is the acceleration of gravity (g).

Apparatus

To determine the acceleration of gravity the Behr apparatus will be used. The device consists of two vertical conducting wires, a thin strip of paper held between them, and a metal-girdled weight designed to fall between the wires along the length of the paper strip. A spark timer transmits a high voltage electric pulse to the wires approximately 60 times a second. Every time a pulse is transmitted, two main sparks flow through the system. One spark passes from one wire to the metal girdle around the weight. The second spark causes a small burn in the paper, marking the location of the weight at that instant.

Procedure

Turn on the electromagnetic power supply and suspend the weight from the end of it. Confirm that the weight falls smoothly into the cup at the base of the apparatus when the electromagnet switch is turned off. Run this test run about three or four times before you continue. Next, draw a fresh strip of paper from the base of the device and clamp it in place. Turn on the electromagnet, and suspend the weight at the end of the magnet. Hold down the spark switch, and then immediately turn off the eleectromagnet power supply. The weight should fall down to the base of the apparatus, causing sparks to pass between the two wires and itself. Turn off the power to the spark timer and inspect the paper strip. A series of burns should be visible along the length of the paper. Remove the paper strip from the apparatus and immediately mark the spots with a pen or pencil to see them more clearly.

Data and Results

The following table shows the data calculated for the experiment. The spots found on the paper strip are shown as (n). The distance of the metal girdle along the strip is denoted by (x). Velocity is (v) and acceleration is (a). The estimated time (Dt) for this test was 60.2 x 0.7s-1.

Calculations of distance, velocity, and acceleration of metal girdle.

n x n (cm) xn+1 – x n (cm) xn+1 – x n / Dt = v n (cm/s) vn+1 – v n (cm/s) vn+1 – v n / Dt = a (cm/s2)

1 0.00

2 0.70 0.70 x .02 42.1 x 2

3 1.43 0.73 x .04 43.9 x 3 1.8 x 5 108 x 302

4 2.43 1.00 x .04 60.2 x 3 16.3 x 6 981 x 373

5 3.72 1.29 x .04 77.7 x 3 17.5 x 6 1054 x 373

6 5.27 1.55 x .04 93.3 x 3 15.6 x 6 939 x 372

7 7.07 1.80 x .04 108.4 x 4 15.1 x 7 909 x 432

8 9.16 2.09 x .04 125.8 x 4 17.4 x 8 1047 x 494

9 11.5 2.32 x .04 139.7 x 4 13.9 x 8 837 x 491

10 14.1 2.61 x .04 157.1 x 4 17.4 x 8 1047 x 494

11 17.0 2.90 x .04 174.6 x 4 17.5 x 8 1054 x 494

12 20.1 3.15 x .04 189.6 x 5 15.0 x 9 903 x 552

13 23.6 3.45 x .04 207.7 x 5 18.1 x 10 1090 x 615

14 27.2 3.65 x .04 219.7 x 5 12.0 x 10 722 x 610

15 31.2 3.98 x .04 239.6 x 5 19.9 x 10 1198 x 616

16 35.4 4.20 x .04 252.8 x 5 13.2 x 10 795 x 611

17 39.9 4.52 x .04 272.1 x 6 19.3 x 11 1162 x 676

18 44.7 4.72 x .04 284.1 x 6 12.0 x 12 722 x 731

19 49.7 5.00 x .04 301.0 x 6 16.9 x 12 1017 x 734

20 55.0 5.33 x .04 320.9 x 6 19.9 x 12 1198 x 736

21 60.6 5.60 x .04 337.1 x 6 16.2 x 12 975 x 734

22 66.5 5.87 x .04 353.4 x 7 16.3 x 13 981 x 794

23 72.5 6.07 x .04 365.4 x 7 12.0 x 14 722 x 851

24 78.9 6.35 x .04 382.3 x 7 16.9 x 14 1017 x 855

25 85.8 6.68 x .04 402.1 x 7 19.8 x 14 1192 x 857

26 92.7 6.93 x .04 417.2 x 7 15.1 x 14 909 x 853

27 99.9 7.15 x .04 430.4 x 7 13.2 x 14 795 x 852

28 107.4 7.46 x .04 449.1 x 8 18.7 x 15 1126 x 916

29 115.0 7.74 x .04 465.9 x 8 16.8 x 16 1011 x 975

30 123.1 8.01 x .04 482.2 x 8 16.3 x 16 981 x 975

31 131.1 8.20 x .04 493.6 x 8 11.4 x 16 686 x 971

32 139.9 8.55 x .04 515.0 x 8 21.4 x 16 1288 x 978

33 148.7 8.80 x .04 530.0 x 9 15.0 x 18 903 x 1034

aAVE = 9.47 x .69 m/s2 s = 9.47 x .78 m/s2 slope (m) of graph = 8.9

Conclusions

The average value of acceleration for each time interval is closer to the desired value of 9.8 m/s2 than the calculated slope of the velocity-time graph. The average of uncertainties for the calculated accelerations is a better as choice of uncertainty because it provides a narrower field of uncertainty than does standard deviation. In conclusion, the calculated value of 9.47 x .69 m/s2 for acceleration is acceptable.

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