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Testing Newton's Second Law--Can We Neglect Friction?
Earlier we looked at the modified Atwood's machine, shown to the right, to get some qualitative feel for Newton's Second Law. We analyzed this for the case of no friction, an ideal string, and an ideal pulley. Real systems have friction, and we must decide experimentally whether friction plays a significant role.
Theory:
Find the acceleration of the system including a constant frictional force, f = µ m1g. Be prepared to turn in your analysis at the start of the period. [Start from free body diagrams and Newton's Second Law. Do not try to patch up an equation obtained for no friction!]
The result you should obtain is
.
Experiment:
Your task is to check the validity of the analysis, and to determine whether friction must be included or not. Based on your measurements you will either be able to say that friction is unimportant within the uncertainties of your experiment, or that it is important and find the coefficient of friction. Along the way you should learn something about interpreting graphs.
Look at the expression for acceleration, it contains some easily varied quantities, m1 and m2, and other quantities that are hard to vary, µ and g. You want the simplest experimental design that will test this equation. First consider the case when m1 >> m2, and rewrite the equation using this approximation. Your graph will have acceleration on the vertical axis, but you need to decide what to plot on the horizontal axis.
Possible approaches are
(a) Keep m1 fixed and vary m2, [e.g. m1 = 500 g, m2 = 10, 20, 30, É, 80, 90 g]
(b) Keep m2 fixed and vary m1, [e.g. m2 = 100 g, m1 = 500, 700, 100, 1500, 2000 g]
(c) Keep (m1 + m2) fixed and vary m2 [e.g. m1 + m2 = 500 g, m2 = 10, 20, É, 90 g]
Discuss in your groups the advantages of doing each approach and decide on one.
Modeling: You can use the theory to predict what you should see. Use a spreadsheet like Excel to calculate the acceleration and make plots. Use the suggested values indicated above with a small coefficient of friction like µ = 0.01. What graphs give a true straight line? What are the meanings of the slope and intercept?
Suggested range of values:
Keep m1 greater than 500 g and m2 less than 50 g. As you have seen with earlier labs, you need to consider how many different masses to use and how to get standard deviations on the data collected.
Graphing: Plot your data in such a way as to get a straight line. Draw the best-fit line and lines showing the possible range of slopes. What does the slope of this line tell you (compare with theory)? What does the intercept tell you?
You will be graded on the quality of your experimental plan, your data, your graphs, and the conclusions you draw from the data.
How do I show the range of possible values?
Here is a graph having nothing to do with the data that you will collect, showing data fit with a trendline and including hand drawn lines that indicate the possible range of values. The dashed lines have different slopes and pivot around the center of the data. Data points are within 2 standard deviations (twice the size of an error bar) from the lines. For a brief reference to graphing download the Excel sheet
And for more detail see