LAB 2 - Free Fall Determination of Acceleration due to Gravity
Tony Wu, Leslie Zhao, Isaiah Hernandez
August 31, 2016
Lab Goal: To examine the validity of the statement: In the absence of all other external forces except gravity, a falling body will accelerate at 9.8 m/s². The goal is to estimate the acceleration due to gravity and compare it to the accepted value.
Theory/Introduction: Using a well-known apparatus, we study the basic laws of motion in order to determine the acceleration due to gravity (g) of a freely falling body. We predict that the acceleration due to gravity should be lower than 9.8 m/s² due to drag and/or wind resistance. The apparatus we used releases an electromagnet. The fall is precisely recorded by a spark generator. The marks made at intervals on the spark-sensitive tape attached to the column give students a permanent record of the fall. This data is then used to make a distance vs. time graph and a velocity vs. time graph. Finally, the acceleration is calculated from velocity graph.
Apparatus: The apparatus is 1.86 m tall. It holds a 1.5 m length of spark-sensitive tape. To set up the apparatus, pull a piece of paper tape tight between the vertical wire and the vertical post of the device. Clip it with a weight to keep the paper "tight". The apparatus is now ready.
Experimental Procedure: We turned the dial hooked up to the electromagnet up a bit, then hung the wooden cylinder with the metal ring around it. After turning on the power to the sparker, the spark leaves a dot on the paper. We then turned off the electromagnet to initiate the fall of the object. After the object has fallen, we turn off the power to the sparker thing and tear off the paper strip. The marks on the paper strip correspond to an interval of 1/60th of a second. Now we take a two-meter stick and mark the position of each dot as measured from the first dot.
After we have all the dots marked and measured, we input the data into Excel. In column A we have the time intervals and in column B we have the distance between each mark. In column C we have the change in distance (delta x). In column D we have the Mid-Interval Time, this gives the time for the middle of each 1/60ths interval. In Column E we have Mid-Interval Speed.
We then graphed all this data into a Marked Scatter chart. We added a Linear Fit Trendline and also displayed an R-squared value on the chart. The closer the value of R-squared to 1 is, the better the fit of the trendline. The Time vs. Distance graph is shown with a Polynomial Fit of order 2.
Data and Calculations:
The linear fit trendline equation, given slope-intercept form, best approximates the data recorded. The coefficient in front of the variable gives us the slope of the line which is the estimated value for the constant acceleration due to gravity. The value we got was 9.63 m/s² which is lower than our expected value of 9.8 m/s². We are also able to get the acceleration due to gravity from the Time vs. Distance graph. We take the coefficient in front of the x² and multiply it by 2, 9.59 m/s². This is also below our expected value of 9.8 m/s². There are a number of reasons why the estimated values are different from the expected values, systematic errors such as air resistance and/or random errors such as user errors during the measuring of each spark mark on the tape. Below are the values that each group got for the acceleration:
Group #1: 9.58 m/s²
Group #2: 9.63 m/s²
Group #3: 8.93 m/s²
Group #4: 9.57 m/s²
Group #5: 9.25 m/s²
Group #6: 9.55 m/s²
Group #7: 9.50 m/s²
Group #8: 9.85 m/s²
After we have all the dots marked and measured, we input the data into Excel. In column A we have the time intervals and in column B we have the distance between each mark. In column C we have the change in distance (delta x). In column D we have the Mid-Interval Time, this gives the time for the middle of each 1/60ths interval. In Column E we have Mid-Interval Speed.
We then graphed all this data into a Marked Scatter chart. We added a Linear Fit Trendline and also displayed an R-squared value on the chart. The closer the value of R-squared to 1 is, the better the fit of the trendline. The Time vs. Distance graph is shown with a Polynomial Fit of order 2.
Data and Calculations:
Graph 1: Mid-Interval Time vs. Mid-Interval Speed Graph 2: Time vs. Distance
The linear fit trendline equation, given slope-intercept form, best approximates the data recorded. The coefficient in front of the variable gives us the slope of the line which is the estimated value for the constant acceleration due to gravity. The value we got was 9.63 m/s² which is lower than our expected value of 9.8 m/s². We are also able to get the acceleration due to gravity from the Time vs. Distance graph. We take the coefficient in front of the x² and multiply it by 2, 9.59 m/s². This is also below our expected value of 9.8 m/s². There are a number of reasons why the estimated values are different from the expected values, systematic errors such as air resistance and/or random errors such as user errors during the measuring of each spark mark on the tape. Below are the values that each group got for the acceleration:
Group #1: 9.58 m/s²
Group #2: 9.63 m/s²
Group #3: 8.93 m/s²
Group #4: 9.57 m/s²
Group #5: 9.25 m/s²
Group #6: 9.55 m/s²
Group #7: 9.50 m/s²
Group #8: 9.85 m/s²
Conclusions:
In this lab, we assumed that air resistance is negligible. Is it reasonable for us to make that assumption? Every group but one had a lower estimated acceleration than the expected value. This is likely due to the air resistance slowing down the freefall object. As mentioned before, user error could also affect the data gathered. We all measured the mark of each spark using a meter stick and eyeballing each measurement. We can quantify how accurate our estimated value is by computing the absolute and relative difference to the expected value.
Absolute Difference - Experimental Value minus Accepted Value: 9.63 m/s² - 9.8 m/s² = -0.17 m/s²
Relative Difference - [(Experimental Value - Accepted Value)/Accepted Value] * 100%
In this lab, we assumed that air resistance is negligible. Is it reasonable for us to make that assumption? Every group but one had a lower estimated acceleration than the expected value. This is likely due to the air resistance slowing down the freefall object. As mentioned before, user error could also affect the data gathered. We all measured the mark of each spark using a meter stick and eyeballing each measurement. We can quantify how accurate our estimated value is by computing the absolute and relative difference to the expected value.
Absolute Difference - Experimental Value minus Accepted Value: 9.63 m/s² - 9.8 m/s² = -0.17 m/s²
Relative Difference - [(Experimental Value - Accepted Value)/Accepted Value] * 100%
[(9.63 m/s² - 9.8 m/s²) / 9.8m/s² ] * 100% = -1.73%
Our experimental value for the acceleration due to gravity is only 1.73% lower than the accepted value.
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