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====== PHY 123 Lab 1 - Error and Uncertainty and the Simple Pendulum ====== | ====== PHY 123 Lab 1 - Error and Uncertainty and the Simple Pendulum ====== | ||

- | I hope that this material works on everyone's machines. If you have a problem, please try the following troubleshooting steps. | ||

- | -Use a decent browser, try either [[http://www.mozilla.com/firefox/|Mozilla Firefox]] or [[http://www.google.com/chrome|Google Chrome]]. | + | **Important: You need to print out and bring to lab the 1 page worksheet you find by clicking [[http://mini.physics.sunysb.edu/~mdawber/phy123summerworksheet1.pdf|here]]** |

- | -Make sure that you have both [[http://www.java.com/en/|Java]] and [[http://get.adobe.com/flashplayer/|Flash]] installed. | + | |

- | -If you still have trouble then try the [[http://www.ic.sunysb.edu/Class/phy122ps/labs/dokuwiki/pdfs/phy123lab1.pdf|.pdf]] version (videos and plotting won't work, but you can read/print the rest). | + | |

+ | If you need the pdf version of this page you can get it [[http://mini.physics.sunysb.edu/~mdawber/phy123summerlab1.pdf|here]] | ||

+ | |||

+ | |||

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- | ==== Estimate of error in the length of the string ==== | + | ==== Estimating the main errors in the experiment ==== |

- | Suppose that we had measured the string 5 times and found the following 5 values for the length of the string, L. | + | |

+ | === Error in the length of the string === | ||

+ | Suppose that we measure the length of the string 5 times and find the following 5 values for the length of the string, L. | ||

^Measurement No.^ L [cm]^ | ^Measurement No.^ L [cm]^ | ||

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- | ==== Error in the period ==== | + | ===Error in the period === |

If we measure the time for 10 oscillations we can find the time for one oscillation simply by dividing by 10. Now we need to make an estimate of the error. | If we measure the time for 10 oscillations we can find the time for one oscillation simply by dividing by 10. Now we need to make an estimate of the error. | ||

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$\Delta T = \Large \frac{\Delta (10T)}{10}=\frac{0.28}{10}= \normalsize 0.028$ s | $\Delta T = \Large \frac{\Delta (10T)}{10}=\frac{0.28}{10}= \normalsize 0.028$ s | ||

- | So you can see it was a good idea to measure several periods instead of one, we get a much more accurate result. Maybe you'd like to think about why we don't measure 100 oscillations (and because you'd get bored is only part of the answer!). | + | So you can see it was a good idea to measure several periods instead of one, we get a much more accurate result. Maybe you'd like to think about why we don't measure 100 oscillations (and because you'd get bored is only part of the answer!).. |

How accurately do you think you can press the button, is an 0.2 seconds and overestimate or underestimate for your reaction time? If you think that the accuracy of your button press is different to 0.2 seconds you should work out what you think $\Delta T$ is if you make a measurement of 10 oscillations. | How accurately do you think you can press the button, is an 0.2 seconds and overestimate or underestimate for your reaction time? If you think that the accuracy of your button press is different to 0.2 seconds you should work out what you think $\Delta T$ is if you make a measurement of 10 oscillations. | ||

+ | ==== The effect of angle on the period of oscillation ==== | ||

- | ==== Making a plot of our data ==== | + | We mentioned above that the equation we want to test is only valid for small angles. The first measurement we will make is to measure the period of oscillation for 3 angles of release. |

- | Now we have some idea of the uncertainty in our measurements we can look at some data and try to see if it matches the formula we expect. What we would do is to, for a fixed angle $\theta$ change the length of the string and find the oscillation period.Take a look at the following data set which was taken by one of our TAs: | + | |

- | ^L[cm ]^ΔL [cm]^ 10T[s]^T[s]^ΔT[s]^T<sup>2</sup>[s<sup>2</sup>]^ΔT<sup>2</sup>[s<sup>2</sup>]^ | + | Choose three angles at ~15<sup>o</sup>,~30<sup>o</sup> and ~80<sup>o</sup> and measure the time for 10 periods for each angle. Enter your data into the table on your worksheet. You measure the time using the computer as a simple timer: Double click the Timer Program Icon. The Timer Program comes up. It has a “Start” “Stop” toggle button. |

- | |10.6|0.1|6.2|0.62|0.028|0.38|0.03| | + | |

- | |21.9|0.1|9.1|0.91|0.028|0.82|0.05| | + | |

- | |33.2|0.1|11.6|1.16|0.028|1.34|0.06| | + | |

- | |40.5|0.1|12.8|1.28|0.028|1.65|0.07| | + | |

- | |48.4|0.1|14.0|1.40|0.028|1.95|0.08| | + | |

- | |61.6|0.1|15.8|1.48|0.028|2.48|0.09| | + | |

- | |73.1|0.1|17.4|1.74|0.028|3.01|0.10| | + | |

- | |81.4|0.1|18.1|1.81|0.028|3.27|0.11| | + | |

- | |89.6|0.1|19.4|1.91|0.082|3.75|0.08| | + | |

- | You should understand from what we discussed above how we got the first 5 columns. The rest of the table shows the necessary transformation of the data into the quantities we need to plot. You might be wondering why we have calculated $T^2$. Recall that we said earlier that we expect that $T=2 \pi \Large \sqrt{\frac{L}{g}}$. We can rearrange this as $L=\Large\frac{g}{(2\pi)^2}\normalsize T^2$, which means that we should get a straight line if we plot $L$ against $T^2$, and of course we need to know what the error in $T^2$, $\Delta T^2$, is so that we can draw error bars on the graph. | + | Taking into account your estimate for the error in T above are the periods the same for all 3 angles. Note that experimentally two measurements are considered to be the same if their is an overlap of their range taking into account the error of their measurement. e.g 15+/-1 is equal within error to 16+/-1, 13+/-1 is not. |

- | Let's try using the plotting tool we will be using in this course to plot this data. It's built right in to the webpage, but when you enter your data and click "submit" it will make the graph in a new tab. This makes it easy to change something and get another graph if you made a mistake. You should enter the $T^2$ values as your x values and your $L$ values as your y values. According to the equation we are testing when $T^2=0$, $L=0$ so you should check the box which asks you if the fit goes through (0,0). Enter the appropriate errors in the +/- boxes and choose "errors in x and y". Click "submit" when you are done. | + | |

+ | ==== Taking data ==== | ||

+ | | ||

+ | Now we have some idea about the errors involved in our experiment we should take some data. You will need to measure the period of oscillation of the pendulum for different values of the string length L. You will want to have the same angle $\theta$ for each L. So the procedure you should follow is to set your string length to a given L, which you should write in the table on your worksheet, and then pull it back to angle of 15<sup>o</sup> which you can measure with the protractor. After releasing the pendulum you should measure, using the computer stopwatch, the time for 10 oscillations and record that on the worksheet as well. Do this for 10 different lengths of the string L. Once you have your data for $10T$ you can then work out what $T$ is for each $L$ simply by dividing by 10. You should also enter in to your table values for $\Delta L$ and $\Delta T$ based on the estimates you made earlier. | ||

+ | | ||

+ | ==== Making a plot of our data ==== | ||

+ | | ||

+ | Recall that we said earlier that we expect that $T=2 \pi \Large \sqrt{\frac{L}{g}}$. We can rearrange this as $L=\Large\frac{g}{(2\pi)^2}\normalsize T^2$, which means that we should get a straight line if we plot $L$ against $T^2$. To test this we need to work out what $T^2$ is for each value of $L$ and of course we need to know what the error in $T^2$, $\Delta T^2$, is so that we can draw error bars on the graph. So you need to complete the last to columns of your table. Finding $T^2$ is easy enough. To obtain $\Delta T^2$ we will need to propagate the error in $T$, $\Delta T$ using some of the equations above. Equation (1.8) tells us how to propagate the error of a quantity in to the error of the power of that quantity. In this case the **relative** error in $T^2$, which is $\frac{\Delta T^2}{T^2}$ is the same as twice the **relative** error in $T$, which is $\frac{\Delta T}{T}$. You can thus find that $\Delta T^2=T\Delta(T)$. Make sure you can understand how to get this equation, your TA may ask you to show how it is obtained! | ||

+ | | ||

+ | Once you have completed your table we can use the plotting tool. It's built right in to the webpage, but when you enter your data and click "submit" it will make the graph in a new tab. This makes it easy to change something and get another graph if you made a mistake. You should enter the $T^2$ values as your x values and your $L$ values as your y values. According to the equation we are testing when $T^2=0$, $L=0$ so you should check the box which asks you if the fit goes through (0,0). Enter the appropriate errors in the +/- boxes and choose "errors in x and y". Click "submit" when you are done. | ||

<html> | <html> | ||

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</html> | </html> | ||

- | If you entered everything right then on your new tab you should see something that looks like this: | + | If you entered everything right then on your new tab you should see a plot that looks something like this: |

{{lab1plotfig.png}} | {{lab1plotfig.png}} | ||

- | The data is clearly quite linear when plotted like this, so it gives us an indication that our formula at the least has the right form. (Maybe you would like to try plotting $L$ directly against $T$ and see what looks like). Notice that you can't see the y error bars because they are very small. The program has fitted the data using a least-squares fitting approach. This means that it has calculated for each data point the square of the difference between the data point and the line. It then adds up all these "squares" and uses this number to determine how good the fit is. The computer tries to find the line that gives the smallest sum of squares and calls this the line of best fit. It's drawn this on the graph and called it "y=a*x". It's also given you the value of a and its estimate for the uncertainty in a. The value the program gives for the error in a is often fairly small, it relies mostly on the scatter in a and only uses the errors you enter to weight the points differently in its fit. | + | The data is clearly quite linear when plotted like this, so it gives us an indication that our formula at the least has the right form. Notice that you can't see the y error bars because they are very small. The program has fitted the data using a least-squares fitting approach. This means that it has calculated for each data point the square of the difference between the data point and the line. It then adds up all these "squares" and uses this number to determine how good the fit is. The computer tries to find the line that gives the smallest sum of squares and calls this the line of best fit. It's drawn this on the graph and called it "y=a*x". It's also given you the value of a and its estimate for the uncertainty in a. The value the program gives for the error in a is often fairly small, it relies mostly on the scatter in a and only uses the errors you enter to weight the points differently in its fit. |

Another technique you can use to estimate the error in the slope is to draw "max-min lines". Here we draw in two lines, one that has the maximum slope that seems reasonable, the "max" line, and another with the smallest slope that seems reasonable, the "min" line. Normally we do these exercises on paper, but you can probably do it simply by holding a clear plastic ruler up to the screen to decide where you think the max-min lines should be (please DON'T draw on the screen!!). A line is reasonable if it just passes within //most// of the error bars. You then just take two convenient points on the line, and find the change in y over the change in x to calculate the slope. You can then work out the slope of both lines to give yourself an estimate of the error in the slope. In the example below you can calculate that the "max" line has a slope of about 90/3.6=25 cm/s<sup>2</sup>, and the "min" line has slope of about 90/3.8=23.7 cm/s<sup>2</sup>, therefore if you used this method you would conclude that the value of the slope is 24.4+/-0.7 cm/s<sup>2</sup>, as compared to the computers estimate of 24.41+/-0.16 cm/s<sup>2</sup>. Note that I drew the lines through (0,0) which we can consider as an error free point, i.e. the fits **must** go through this point. | Another technique you can use to estimate the error in the slope is to draw "max-min lines". Here we draw in two lines, one that has the maximum slope that seems reasonable, the "max" line, and another with the smallest slope that seems reasonable, the "min" line. Normally we do these exercises on paper, but you can probably do it simply by holding a clear plastic ruler up to the screen to decide where you think the max-min lines should be (please DON'T draw on the screen!!). A line is reasonable if it just passes within //most// of the error bars. You then just take two convenient points on the line, and find the change in y over the change in x to calculate the slope. You can then work out the slope of both lines to give yourself an estimate of the error in the slope. In the example below you can calculate that the "max" line has a slope of about 90/3.6=25 cm/s<sup>2</sup>, and the "min" line has slope of about 90/3.8=23.7 cm/s<sup>2</sup>, therefore if you used this method you would conclude that the value of the slope is 24.4+/-0.7 cm/s<sup>2</sup>, as compared to the computers estimate of 24.41+/-0.16 cm/s<sup>2</sup>. Note that I drew the lines through (0,0) which we can consider as an error free point, i.e. the fits **must** go through this point. | ||

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**g=9.64+/-0.28 m/s<sup>2</sup>** | **g=9.64+/-0.28 m/s<sup>2</sup>** | ||

- | The accepted value for g is **9.81 m/s<sup>2</sup>**, which falls within the range we found using the max/min method and so we can say, based on that estimate, that our experiment is **consistent** with the equation we tried to fit the data with. If we used the computers error estimate then we would conclude that the data was **inconsistent** with the textbook value. This demonstrates why we need to be careful about the methods we use to estimate error, depending on the data one method may be better than the other. Generally it is safer to take the larger of the two error estimates, but these kinds of judgments are the kinds of things it will be useful to discuss with your TA when you are doing experiments and analyzing your data. Another thing to bear in mind is that we were quite careful here about trying to eliminate random error, but if systematic error was present then our methods would not have done very much to help. This is always something we should bear in mind when comparing values we measure in the lab to textbook values. We also need to think about simplifying assumptions we make. For example, we assumed that the pendulum did not slow down at all over the 10 swings we measured, and there are also approximations in the derivation of the equation we test here. Bearing these things in mind, an important point to make is that in general we should not necessarily be surprised if something we measure in the lab does not match exactly with what we might expect. When things don't seem to work we should think about why they don't, but, most importantly of all, we must **never** modify our data to make it match our expectations! | + | The accepted value for g is **9.81 m/s<sup>2</sup>**, which falls within the range we found using the max/min method and so we can say, based on that estimate, that our example experiment was **consistent** with the equation we tried to fit the data with. If we used the computers error estimate then we would conclude that the data was **inconsistent** with the textbook value. This demonstrates why we need to be careful about the methods we use to estimate error, depending on the data one method may be better than the other. Generally it is safer to take the larger of the two error estimates, but these kinds of judgments are the kinds of things it will be useful to discuss with your TA when you are doing experiments and analyzing your data. Another thing to bear in mind is that we were quite careful here about trying to eliminate random error, but if systematic error was present then our methods would not have done very much to help. This is always something we should bear in mind when comparing values we measure in the lab to textbook values. We also need to think about simplifying assumptions we make. For example, we assumed that the pendulum did not slow down at all over the 10 swings we measured, and there are also approximations in the derivation of the equation we test here. Bearing these things in mind, an important point to make is that in general we should not necessarily be surprised if something we measure in the lab does not match exactly with what we might expect. When things don't seem to work we should think about why they don't, but, most importantly of all, we must **never** modify our data to make it match our expectations! |

+ | | ||

+ | You should now work out the value of g from the plot of your own data. If it is not consistent with the accepted value you should think about why this might be. Discuss these issues with your TA! | ||

+ | | ||

+ | Once you have your value of g we should also make a plot of $L$ directly against $T$. | ||

+ | | ||

+ | <html> | ||

+ | <form method="post" action="http://mini.physics.sunysb.edu/~mdawber/plot/basicplot2.php" target="_blank"> | ||

+ | x axis label (include units): <input type="text" name="xaxis" size="60"/><br> | ||

+ | y axis label (include units): <input type="text" name="yaxis" size="60"/><br> | ||

+ | Check this box if the fit should go through (0,0). <input type="checkbox" name="zero" value="y" /><br> | ||

+ | (Don't include (0,0) in your list of points below, it will mess up the fit.)<br> | ||

+ | What kind of errors are you entering below?<SELECT name="errortype"> | ||

+ | <OPTION value="none">None</OPTION> | ||

+ | <OPTION value="x">Errors in x</OPTION> | ||

+ | <OPTION value="y">Errors in y</OPTION> | ||

+ | <OPTION value="xy">Errors in x and y</OPTION> | ||

+ | </SELECT> | ||

+ | <br> | ||

+ | | ||

+ | x1: <input type="text" name="x1" size="10"/>+/-<input type="text" name="dx1" size="10"/>    y1: <input type="text" name="y1" size="10"/>+/-<input type="text" name="dy1" size="10"/><br/> | ||

+ | x2: <input type="text" name="x2" size="10"/>+/-<input type="text" name="dx2" size="10"/>    y2: <input type="text" name="y2" size="10"/>+/-<input type="text" name="dy2" size="10"/><br/> | ||

+ | x3: <input type="text" name="x3" size="10"/>+/-<input type="text" name="dx3" size="10"/>    y3: <input type="text" name="y3" size="10"/>+/-<input type="text" name="dy3" size="10"/><br/> | ||

+ | x4: <input type="text" name="x4" size="10"/>+/-<input type="text" name="dx4" size="10"/>    y4: <input type="text" name="y4" size="10"/>+/-<input type="text" name="dy4" size="10"/><br/> | ||

+ | x5: <input type="text" name="x5" size="10"/>+/-<input type="text" name="dx5" size="10"/>    y5: <input type="text" name="y5" size="10"/>+/-<input type="text" name="dy5" size="10"/><br/> | ||

+ | x6: <input type="text" name="x6" size="10"/>+/-<input type="text" name="dx6" size="10"/>    y6: <input type="text" name="y6" size="10"/>+/-<input type="text" name="dy6" size="10"/><br/> | ||

+ | x7: <input type="text" name="x7" size="10"/>+/-<input type="text" name="dx7" size="10"/>    y7: <input type="text" name="y7" size="10"/>+/-<input type="text" name="dy7" size="10"/><br/> | ||

+ | x8: <input type="text" name="x8" size="10"/>+/-<input type="text" name="dx8" size="10"/>    y8: <input type="text" name="y8" size="10"/>+/-<input type="text" name="dy8" size="10"/><br/> | ||

+ | x9: <input type="text" name="x9" size="10"/>+/-<input type="text" name="dx9" size="10"/>    y9: <input type="text" name="y9" size="10"/>+/-<input type="text" name="dy9" size="10"/><br/> | ||

+ | <input type="submit" value="submit" name="submit" /> | ||

+ | </form> | ||

+ | </html> | ||

+ | | ||

+ | | ||

+ | \\ | ||

+ | | ||

+ | Was this plot linear? Do you think that based on the difference between the graphs you are able to confirm what the correct equation is that relate the period of oscillation to the length of the string? | ||

+ | ==== Finishing up ==== | ||

+ | Once you have finished your experiment you need to discuss your results with your TA. Don't leave until they say you can as you only get credit for the lab if the TA thinks you did a good enough job on the experiment and have mastered the key points that the experiment covers! |