MEASUREMENTS ACCELERATION OF GRAVITY Purpose: A. To illustrate the uncertainty of a measurement in the laboratory. The measurement is that of time. The data obtained from these measurements will be used to illustrate the methods of estimating the best values of the measured quantity and its reliability. B. To employ the results of Part A to determine the acceleration due to gravity g. Part A Theory: It is not possible to perform an absolute measurement. There are three causes of uncertainty or error of measurements: human, instrumental, and statistical. The human uncertainty depends on the skills of the person performing the measurement. The instrumental uncertainty usually equals the smallest value which can be measured with the instrument as described by the scale of the instrument. For greater accuracy, a more precise instrument could be used. The statistical uncertainty includes all other uncontrolled factors. Apparatus: Electronic free-fall timer equipment, two-meter stick, support stand, steel balls of different masses. Figure In the first part of the experiment, we will measure the time of fall (t) of a metal ball for a specified distance (d) by using the electronic free-fall timer. In this apparatus, a metal ball is clamped into the release mechanism and held directly over a receptor pad, as shown in the Figure. The ball is released by loosening the thumbscrew, which starts the digital timer. The impact of the ball on the receptor pad stops the timer, which then displays the time of fall to three decimal places (one thousandth of a second). The instructor will assign a different distance d = 40, 60, 80, 100, 130, 160, 190 cm to each group. Procedure: Start with the larger, heavier ball. 1. Place it in the release mechanism. The release plate is loaded by pressing inward on the metal pin, thereby bending the plate and compressing the spring. The
release plate is held in place by tightening the thumbscrew. The ball should be held in place between the contact screw and the hole in the release plate. The spring pressure should be just sufficient to hold the ball in place too much pressure will delay the release of the ball from the mechanism.. Make sure the receptor plate is aligned directly under the ball in the box provided. For the timer to operate properly, the point of impact must be close to the center of the pad. 3. Carefully adjust the distance d assigned by the instructor with the two-meter ruler. The distance d (see Figure) is from the bottom of the ball to the top of the pad. 4. Reset the timer to zero. 5. Activate the timer by gently loosening the thumbscrew so that the steel ball falls. 6. For the distance d assigned to your group by the instructor, perform the measurement of time t of fall in seconds (s) for twenty trials. Write these values down in Table 1. Calculations: 1. Determine the Average time t avg in seconds (s) of the twenty trials. Definition: If you have N measurements, x 1, x, x N, then the average is defined as x avg = x 1 + x + N... + x N. (1). Determine the Standard Deviation σ in seconds (s) of these trials. Definition: The standard deviation is defined as N ( xi xavg ) i= 1 σ =. () N 1 3. The experimental value of the free-fall time t is then
t tavg ± σ (s) =. (3) (This is of the form Best Estimate ± Uncertainty.) 4. Complete Table which lists for each time value t obtained in Table 1, the number of trials for which that time value occurs. 5. Plot a graph of Table, i.e., the number of trials for which each time value occurs as a function of the different time values obtained. This ought to look like a Gaussian distribution. Theory: Part B It is not too difficult to observe that a falling body falls faster and faster as it descends; it accelerates. Under certain conditions (mainly when the resistance of air is not strong) this acceleration is constant, and the same for all bodies heavy or light. The value of this acceleration of gravity is called g. One object of this experiment is to determine g. You will learn in lecture that a body released from rest and falling with acceleration g, will fall a distance d in time t, where From Eq. (4) we have that 1 d = g t, (4) d g =. (5) t Hence, by measuring the time t it takes an object to fall a distance d, we could determine g from Eq. (5). It is evident from Eq. (5) that the units of g must be those of Length/[Time]. In our experiment the units of g will be cm/s.
Procedure and Calculations: 6. Determine g from Eq. (5) using your value of the distance d and the average value of t as calculated by you from Eq. (3). 7. Repeat the measurement of time t using the lighter and smaller of the two steel balls for 5 trials only. As the difference in diameter of the two balls is small, you may assume that the distance d of fall is the same as before. Put these values of time in Table 3. Calculate the average time of these 5 trials. According to the law of falling bodies, the time of fall is independent of the weight of the object. Compare your result with that obtained previously for the heavier ball. Do your results agree with the law? The instructor will put on the board the values of d, t avg, and g obtained for the heavier ball by each group, as in Table 4. 8.Copy the results of t avg obtained by each group into Table 4. Complete the column for t avg. Determine g for each distance d. Do these results show that g is independent of the distance d? Determine the average g avg of the different values of g obtained by each group. Also determine the standard deviation σ. The experimental value of the acceleration due to gravity g expt is then g exp t = gavg ± σ. (6) (This is again of the form Best Estimate ± Uncertainty.) 9. The acceleration of gravity in New York City is 980 cm/s. Determine the percent error of the average value g avg. Definition: When the correct value of a property X exact is known, the percent error of the result obtained, X, is the error (X X exact ) divided by the correct value and multiplied by 100: ( X X exact ) error = 100. (7) X 0 0 exact 10. Plot the points of Table 4 with the values of d on the y-axis and t avg on the x-axis. Draw a straight line that best fits these points, and determine its slope. From Eq. (4) we see that since we are plotting d versus t, the slope of the line is ½ g. Thus, we can also obtain an estimate of the experimental value g expt from the graph as = slope ( cm s ). (8) g t exp
Calculate the percent error of the value of g expt obtained from the graph. 11. The manufacturer of the free-fall timer claims that the timer is accurate to 1%. Are your results consistent with the manufacturer s claim? 1. What do you think are the sources of error in your experiment?
Heavier Steel Ball Distance d = cm Trial # Table 1 Time t(s) 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 Average time t = avg (s) Standard deviation σ = (s) Experimental value of free-fall time from the distance d = cm is t = tavg ± σ = ( s) Acceleration due to gravity g = d t = cm s
Table Time t(s) # of Trials
Lighter Steel Ball Distance d = cm Table 3 Trial # Time t(s) 1 3 4 5 Average time t avg = ( s)
Heavier Steel Ball Table 4 Distance d (cm) Time t avg (s) (Time) t avg (s ) Acceleration g (cm/s ) 40 60 80 100 130 160 190 Average acceleration g avg = ( cm s ) Standard deviation ( ) σ = cm s Experimental value gexpt = gavg ± σ ( cm s ) Percent error = Slope of graph of ( d vs tavg ) = ( cm s ) Experimental value from graph g t = slope = ( cm s ) Percent error = exp