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Unformatted text preview: 1 Physics 195 Mesa College Crivello Graphing on log-log Paper Suppose you were presented with the set of data shown below. A graph of x vs. t is also shown, and you can see it's a smooth curve. But other than that, it's not very informative. Suppose, however, in addition, there were reasons to believe that this data obeyed a power-law, x = kt n , meaning that the function could be of the form x = 8t 3 , where k = 8 and n = 3. How could you determine if this were true and, if it were, find the constants k and n? Perhaps this function is x = t 2 , or x = 5t 4 . Actually, it is probably impossible to determine the exponent ‘ n ’ and constant ‘ k’ by looking at this graph. x vs. t 5000 10000 15000 20000 25000 30000 35000 20 40 60 80 100 120 t x Your data Data table (1) Here is a slick technique to solve this dilemma. Let’s take the ‘log’ of both sides of our function: Eq. (1) log ( x ) = log ( k t n ) Recall: log AB = log A + log B and log A n = n log A So equation (1) becomes: log x = n log t + log k eq. (2) But this has the form: y = m x + b , a straight line! This means that we can just take the log of each data point and plot it on regular graph paper: log x vs. log t on regular paper 1 2 3 4 5 0.5 1 1.5 2 2.5 log t log x *** Note that the axes are log x and log t !!*** log of the data log t log x 00.477121 0.301031.079181 0.4771211.431364 0.602061.681241 0.698971.875061 0.7781512.033424 0.8450982.167317 0.903092.283301 0.9542432.385606 12.477121 1.301033.079181 1.698973.875061 1.903094.283301 24.477121 Data table (2) How to find ‘n’ : Now we have a straight line whose slope is the exponent ‘ n ’ and whose x-intercept is ‘log k ’. You can use this data table to show that the slope is ‘2’. Thus n = 2. Notice that since logs have no units, then the slope has no units. Rev. 2/15/10 t (s) x ( m) 1 3 2 12 3 27 4 48 5 75 6 108 7 147 8 192 9 243 10 300 20 1200 50 7500 80 19200 100 30000 2 How to find ‘k’: The constant ‘ k ’ is a little bit trickier. Just as you would find the y-intercept in y = mx + b by setting x = 0, you would find k by setting n log t equal to zero in equation (2). Eq. (2) log x = n log t + log k So equation (2) becomes log x = n log t + log k , or log x = 0 + log k , thus log x = log k. Now look on the second graph to see where the line crosses the log x axis. This occurs when log x = 0.477121. (* Remember, that value is not x , it’s log x .) So log x = log k , and we have 0.477121 = log k . Solving for k yields k = 3. So you have now found the constants k and n for the function x = kt n . You can now state that the data of x vs. t can be described by the function x = 3t 2 . The easy way… Taking the log of all the data and re-plotting it is tedious and time consuming. Fortunately there is an easier way! Instead of using your calculator to take the log of each data point, we can use special graph paper called logarithmic graph paper. Since the log of both variables x and t are needed, we can use log-...
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- Spring '11
- Physics, Logarithmic scale, Tn