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Acceleration experimented with the velocity and time of a falling object to produce a reasonable calculation for the acceleration due to gravity of Earth. The experiment was divided into parts, each with a specific procedure to determine the acceleration due to gravity. Both parts, however, required a basic photogate, an instrument that maintained a constant light beam between two sides. The purpose of this instrument is to electronically measure the velocity and time it takes for a falling object, in this case a ruler with striped tape bands, to fall a certain distance. The falling object, a clear ruler with masking tape wrapped a regular interval (d). The tape marks at regular interval (d) are used to plot points for a distance vs time, velocity vs time, and acceleration vs time graphs electronically drawn by a computer program, Exp2_xva_t. Errors in calculation were considered.
Measuring the length of one interval on the ruler (d): One interval measure (d) on the ruler is determined by the distance from the beginning of one segment of clear ruler to the end of the proceeding segment of masking tape. However, there are more than a singular quantity of these intervals and thus it would be inefficient (a large error) to simply record the measure of one interval to represent the population of intervals. As a result, the total amount of intervals (N) was measured and the total distance (D) with given absolute error of (± 0.1 cm) spanned by N intervals was measured as well. To calculate interval measure (d), the total distance (D) was divided by the total amount of intervals (N) to produce an average measure for the interval measure to represent d. The error of this average interval measure was calculated by dividing the error of the total distance (± 0.1 cm) by the total amount of intervals (N).
Part 1 – Calculating g through slope analysis of Velocity vs Time Graph: To calculate the slope of a velocity versus time graph, the velocity and time of a free-falling object must first be measured and the points plotted on a graph. The free-falling object used is the clear ruler with masking tape covering regular intervals described in the paragraph above. The instrument used to record the velocity and time was a photogate. As described in the introduction, it is an instrument that has a constant light beam that is shot from one end to another end on the photogate. Light is allowed to passed through during the clear parts of the ruler but the beam will be severed once in contact with the masking tape, thus allowing accurate and precise measurement of velocity and time. All data measured by the photogate is then displayed on a computer through a program called Exp2_xva_t. To begin measurements, the program Exp2_xva_t was set up and the photogate was powered on and positioned accordingly. The ruler with the masking tape intervals was held vertically right above the light beam of the photogate. Once prompted by a partner to drop the ruler, the ruler was released with no acting force on it to simulate free-fall. The data observed by the photogate was verified to see if the acceleration vs time graph produced by the computer program was relatively constant. If the acceleration vs time graph was relatively constant, then the data observed was recorded. This method was repeated 5 consecutive times and the data for each was recorded. The error in the velocity was calculated and a velocity versus time graph was constructed with error in mind. The slope of the linear function was computed and the error of the linear function was shown. All findings were recorded.
Part 2 – Calculating g through slope analysis of Velocity vs Time graph computed by Exp2_xva_t: In order to calculate the slope this time, the same procedure is performed up to the point of recording data. The ruler and photogate are dropped and positioned the same way respectively. The one difference is that this part no longer requires the recording of variables time and velocity. Instead, the slope of the velocity vs time graph analyzed by the computer programs is used. So there will be a recording of g (acceleration due to gravity) for every trial. The error of the velocity is then computed from the 5 recordings of g from this part of the experiment.
DD = Given by TA, Yue Wang
= ± 0.1 cm
Dd = DD / N
= ± 0.1 cm / 8
= ± 0.0125 cm
Dv = Dd / (tX – tX-1)g1
Ex: Dv for measurement #3
= (± 0.0125 cm ) / (3.596s – 3.570s)
= ± 0.481 cm/s^2
Dg (part 2) =
= 0.061 m/s^2
In conclusion, although the experiment demonstrated that measuring the velocity and time individually to calculate the acceleration due to gravity had an error that included the actual magnitude of g (9.81 m/s2) was a better procedure to determine g. In reality, the second part procedure is actually a better measurement of the magnitude. The reason being that the second part procedure propagated less errors in its calculation. The only error propagated in the second procedure was the error in interval length (Dd). Meanwhile for the first procedure there were two errors that were propagated to calculated g; the error in velocity (Dv) and likewise the error in interval length (Dd). This increase in error propagation can be seen in the magnitude of the error for gravity (Dg) for both parts 1 and 2. The magnitude of Dg in part 1 was calculated to be 0.06645 m/s2 which is larger than that of the Dg calculated in part 2; 0.061 m/s2.
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