Free Fall Lab- determination of g and statistics for analyzing data
Purpose:
This lab aims to derive a value of gravitational acceleration g based on an object's motion as it experiences free fall. This lab will also demonstrate the use of Excel graphs to perform statistical analysis of data.
Theory:
Near Earth's surface, in absence of all other forces, an object can be expected to accelerate at approximately 9.81 m/s^2 towards the Earth. Ideally, this would mean that analysis of the motion of an object in free fall based on measurements of position and time would produce a derived value of g near the accepted value of gravitational acceleration. Statistical analysis of multiple trials would then produce a range of values between which the actual value of g is likely to reside.
Apparatus:
- Spark tape (1)
- 2 Meter stick (1)
- Excel graphing program
Procedure:
A previously prepared spark tape will be provided. The spark tape will have points marked along it at time intervals of 1/60th of a second, marking the position of an object as it underwent free fall. Measure the distance of each point from the origin and enter each position into an Excel graph with the time that corresponds to the position. Create columns for mid-interval times of t + 1/120 seconds and mid-interval speeds calculated based on dv/dt. Create a velocity vs time graph using mid-interval times and insert a linear trend line. Create a position vs time graph and insert a second-order polynomial trend line. Show the equation and r-squared values for both trend lines. Collect data from other experimental groups and calculate values of standard deviation of g.
Data and Graphs:
Analysis:
The value of g may be found using either the position vs time graph or the velocity vs time graph. To find g using the position graph, find the derivative of the graph to produce a velocity graph. For a constant value of acceleration, the slope of the resulting velocity graph is equal to the value of acceleration. When directly using the velocity graph, simply find the slope to determine g. This method works with constant acceleration because mid-interval velocity is equal to the average velocity across the interval where if a is a constant c, the change in position x is equal to (1/2)ct^2, meaning that average velocity over a time interval is dx/dt or (1/2)ct. Mid-interval velocity would be equal to a*(t/2) which is the same as average velocity. The value of g in this trial was determined to be 988 cm/s^2 or 9.88 m/s^s, slightly greater than the accepted value of 9.81 m/s^2. When data from other groups was collected, this value of g was found to be much greater than the average of 9.63 m/s^2. One other data point also differed greatly from the average with a value of 9.46 m/s^s. This suggests that both of these groups likely experienced large random error caused by poor preparation of the spark tape as it was the most error prone data collecting procedure for this experiment. Other systematic errors due to inaccuracy in measuring the spark tape would likely not have produced such a large difference.
Conclusion:
The average value of g was determined to be 9.62 ∓ 0.100m/s^2, a difference of 0.19 m/s^2 or 1.94% from the accepted value of 9.81 m/s^2. The fact that most experimental groups determined g to be a value lower than expected suggests the presence of some systematic error in the spark tape apparatus. A possible source of error would be if the spark generator itself was not correctly timed at 60Hz. If it was firing faster than intended, our time intervals of 1/60th of a second would be too short and thus produce a lower value of acceleration as our measured change in position would not correspond to a time interval of 1/60th of a second. Other sources of error such as air resistance and friction with the wire are too insignificant to produce error of this magnitude. Another possibility is that the location of the classroom is well above the average radius of the Earth which would reduce the force of gravity on the falling object, causing it to accelerate at a slower rate. It may be necessary to perform further experimentation to determine the value of g using a method different from the one used in these trials as the presence of significant outlying points and low average g value raises concerns that these results may not be reliable.
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