PHY 133 Lab 1 - The Pendulum

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1 3/20/2017 PHY 133 Lab 1 The Pendulum [Stony Brook Physics Laboratory Manuals] Stony Brook Physics Laboratory Manuals PHY 133 Lab 1 - The Pendulum The purpose of this lab is to measure the period of a simple pendulum, and experimentally determine the relationship between the period T and the lenth L of a pendulum. You will also bein to learn how to find, investiate, and understand major sources of uncertainty and error in measured and calculated quantities. Equipment Pendulum: steel ball, cross bar, and strin Protractor (to measure anles) Computer or stopwatch (to be used as a timer) Ruler (to measure lenth) Introduction A simple pendulum consists of a mass m (ideally, concentrated to a sinle point) havin ravitational weiht (a vector) w = m, suspended from a fixed point by a strin (ideally, massless) of lenth L. (For your convenience, the fiure below also shows the two components of w, one parallel to the strin and one perpendicular to the strin.) If carefully set into motion, the mass swins alon an arc (labeled by S ) lyin in a plane that contains the ravity vector. θ(t) is the time-dependent anle that the taut strin makes relative to the vertical, defined by the direction of. If the mass is NOT carefully set into motion, it becomes a conical pendulum, which does not swin alon a vertical plane and is, therefore, a more complicated physical system. For the purposes of this experiment, you will need to ensure that your simple pendulum swins in a vertical plane. 1/5

2 The period of this motion is defined as the time T needed for the mass to swin back and forth once. We will see, later in this L course, that the approximate relation between the period T and lenth L of a simple pendulum is T = 2π, where the manitude of is 9.81 m/s 2. In the derivation of this equation, the assumption is made that the anle θ is small, so that sin(θ) θ (where θ is measured in radians). By measurin the period T of oscillation of the pendulum as a function of the lenth L of the strin, you can experimentally determine an estimate for the value of, the acceleration due to ravity. The different quantities that are important within this experiment are T, L, and θ. Note that to minimize random errors, you should measure L several times and the time it takes for 10 oscillations rather than just one oscillation. By takin an averae value, you will reduce the effects of makin a sinle unusually hih or low measurement. If the computer at your lab table has the proram SnapTimePro installed, accessible via the Desktop icon labeled Shortcut to SnapTimePro, then use it as your timer. If not, use a stopwatch. Estimatin the main uncertainties in the experiment Uncertainty in the lenth L of the strin You should make an estimate of the uncertainty in your measurement of the lenth L of the strin. Factors to take into account should include the scale on your measurement tool, and how accurately you can determine the center of the ball and the pivot point. One approach you can take is to first measure the lenth from the center of the ball to the center of the pivot directly. Call this quantity L. Then, to obtain the uncertainty σ L in this lenth, measure the distance from the pivot point to the top of the ball, and the distance from the pivot point to the bottom of the ball, and take the difference of these two values. Call this quantity σ L. Throuhout the experiment, since the ball's size does not chane even if you chane the strin lenth, you may use this value of as the uncertainty in all of your subsequent strin lenth measurements. Uncertainty in the period T If you try to time one oscillation of the pendulum, you will find that it is fairly difficult to et a sinle accurate value after multiple attempts. So, to reduce this uncertainty, you will measure the time for ten oscillations, then simply divide your value by 10 to obtain the time for one oscillation, called the period T of the pendulum. You can also propaate your uncertainty in the measurement of 10T to find the uncertainty in T. Bein by considerin the uncertainty in your measurement of the time for ten oscillations. How accurately do you think you can press the button to tell the computer when to start and stop the measurement? Let's say that you think you can press the button within 0.2 seconds of either the start or the stop of the measurement. Hence, your reaction time is 0.2 s. Since your reaction time affects both the start time and the stop time, these uncertainties are sources of random error, and they add in quadrature so that: σ 10T = (0.2) 2 + (0.2) 2 = 0.28 s Now, we find the uncertainty in T by dividin by 10: σ L σ 10T = = = s So, measurin several periods instead of one sinificantly reduces the overall uncertainty in the measurement, and we et a much more precise (small ) estimate. The only reason why we don't measure 100 oscillations is because there may be other sources of error that we may wish to reduce usin our time! To find your own personal value of as in the example above, do the followin. First, open the SnapTimePro proram on the Desktop computer, and chane the events in 1 sec settin to 100, then click Set. This allows the proram to measure time like a normal stopwatch. Now, you will find your own reaction time by startin the timer, and attemptin to stop it at exactly s. Take the difference between your stop time (for example, s) and s, and use this as your reaction time (which is 0.15 s in 2/5

3 this example). Usin the equations above and your own reaction time value, calculate your own value of. For all subsequent trials of measurin the time for 10T, you may use this uncertainty value since your reaction time should be rouhly the same! Procedure For this experiment, you will conduct two analyses: one of the effect of anle θ on period T, and one of the effect of strin lenth L on period T. The effect of anle θ on the period T of oscillation As mentioned above, the pendulum equation that we want to test is valid only for small anles of θ. For the first measurement, you will test this expectation by findin the period of oscillation at 3 different anles of release: θ = 15, 30, and 80. For a sinle lenth of the strin (L 50 cm ), use the SnapTimePro proram or a stopwatch to measure the time 10T for ten oscillations of the pendulum, startin from each of the three different startin anles. Use the protractor to precisely determine these anles of release. Also, remember that the anle θ is defined as the anle away from the vertical! Calculate the period T for each startin anle θ, and use the uncertainty from before as your uncertainty in each of these times. Considerin that two period measurements may actually be the same if their uncertainty ranes overlap, how does the startin anle θ affect the period T of the pendulum? Explain why your results make sense, based on our assumptions of the simple pendulum. The effect of lenth L on the period T of oscillation In order to test the pendulum equation above, we will need some data relatin strin lenth L to the period T of the pendulum. You will need to measure the period of oscillation of the pendulum for different values of the strin lenth L, ranin from ~10 cm to ~100 cm. One suestion is to take even increments, measurin the period at strin lenths of 10 cm, 20 cm, 30 cm, up to 100 cm. For each trial, you will want to keep everythin else the same (especially the anle θ at ~15 ), except for the strin lenth L. For each trial, set your strin lenth L, pull back the pendulum mass to the same anle (usin the protractor), and use SnapTimePro or a stopwatch to measure the time 10T for ten full oscillations. Do this for 10 different lenths of the strin. Once you have your data for 10T at each L, you can then calculate T for each L by dividin by 10. For all of these values, you should use the same uncertainties and that you found earlier. σ L Makin a plot of your data Linear plot of L vs. From the pendulum equation, we expect that T = 2π to see this trend in our data. First, by re-arranin this relationship, we can find that L =. And, despite our non-ideal laboratory conditions, we may still expect, which indicates that we should obtain a decent linear fit to our data if we plot L aainst. Then, from the slope of this line, we can extract an estimate of the acceleration due to ravity. To do this, we first need to calculate the values of (and their uncertainties 2 ) from our current data. Findin is easy enouh. However, to obtain 2, you will need to propaate the uncertainty in a sinle period measurement. Usin the power rule from the uncertainty uide, we expect that the relative uncertainty in is 2 times the relative uncertainty in T, or: 2 = 2 T. Solvin this for the uncertainty in, you should find that 2 = 2T. Make sure that you understand how to et this equation from the equations in your uncertainty uide! L Plot your data points of (L, ), usin their uncertainties σ L and 2 as error bars on each data point. Althouh for most future experiments in this course, you will make plots on the computer usin the course Plottin Tool, you should practice drawin a line of best fit to your data and obtainin the slope value, as well as its uncertainty by drawin so-called max and min slope fit lines. Once your data is plotted on raph paper, with appropriate scales alon the horizontal and vertical axes startin at (0,0), you should use a straiht-ede to draw in a line of best fit that passes throuh the middle of your data points, as well as the oriin at (0,0). You should constrain your fit to pass throuh the oriin because, based on the pendulum equation, we expect that the period is T = 0 3/5

4 when the lenth is L = 0. From this line, you can extract its slope by takin the rise over the run of two points alon the line, usually written as m = Δy. Then, to find the uncertainty in your slope value, draw in lines with the maximum (most steep) and Δx minimum (least steep) slopes possible that still pass throuh the error bars of most of your data points, as well as the oriin at (0,0). Find the slopes of these lines, and call them max and min, respectively. You can estimate the uncertainty in your slope value σ m maxmin by usin =. 2 Lastly, to obtain an estimate (and its uncertainty) usin these values, note that the pendulum equation we are fittin has the form of a line y = mx + b, where y = L, x =, m =, and b = 0. Hence, if we refer to our slope value as m, we can set m =, re-arrane to et = m, and calculate an estimate for usin the slope value m. Also, to find the uncertainty in, we can apply the rules of the uncertainty uide to find that σ = σ m. Is your estimate for consistent with the accepted value of 9.81 m/s 2? What sources of error miht have affected your results? What aspects of the setup or methods can be improved to obtain more accurate or precise results? Lo-Lo Plot In the plot above, we used our knowlede of the pendulum equation to deduce that we should plot L vs. to find a linear relationship. However, if we had not known this, there is still a way to obtain the relationship between the quantities L and T of our data, assumin it is some kind of power-law relationship. A ood method of testin for a particular power-law dependency between two quantities is by makin a lo-lo plot (where lo indicates the natural loarithm.) For example, if we expect a quantity y to depend on another quantity x via a eneral power-law relationship of the form y = Ax n, and we take the natural loarithm of both sides of this equation, we would similarly expect that ln(y) = ln(a x n ) = ln(a) + ln( x n ) = ln(a) + n ln(x). Therefore, if we plot the natural loarithm of y (or ln(y) ) vs. the natural loarithm of x (or ln(x) ), we expect to see a linear trend with slope equal to n, which is the exponent on x in the oriinal relation y = Ax n! Also, from the intercept of this plot (or ln(a) ), we can extract the proportionality constant A in the oriinal relationship as well! To bein, you must take the natural loarithm of all of your values of L and T, to find their correspondin values of ln(l) and ln(t ). Next, to find their uncertainties, you must determine how to et the uncertainty in the natural loarithm of a quantity from the oriinal uncertainty in the quantity. You can do this by followin the eneral uncertainty propaation formula provided in the uncertainty uide. For a function f(x) of another quantity x, calculus tells us that the uncertainty of f is related to the uncertainty in x via: σ = df f σ df x. Here, is the derivative of the function f(x) with respect to the variable x. So, in the case where 4/5

5 f(x) = ln(x), the derivative is =, so that the uncertainty in ln(x) is =. Usin this eneral form, you can find the uncertainties and, which will be the error bars on your lo-lo plot. Either by hand or usin the Plottin Tool, plot ln(l) vs. ln(t ). Find the best fit slope value n, and its uncertainty σ n. These are your best estimate of the exponent n and its uncertainty σ n, in the relationship L T n. Does the accepted value of 2 fall within your experimentally determined estimate? If you convert the intercept ln(a) of the raph back into A, it should equal the proportionality constant in the re-arraned pendulum equation L =. Usin this information, you should find another estimate for the value of, and compare it to the value you obtained from the plot of L vs.. (Don't foret to convert back your uncertainty in the intercept ln(a) - which is provided by the Plottin Tool - to the uncertainty in A by reversin the uncertainty relationship you used before, i.e., σ x = x σ ln(x).) Which plot ives a more accurate estimate of? Which ives a more precise estimate of? Discuss this in your lab report! Number crunchin tool You can use this tool to convert your L and T values and their uncertainties into the quantities that you need to plot. Don't foret to appropriately round your values, just as you would if you were usin your calculator! (Note: if you leave rows blank, you will et an error messae, but don't be worried about this, since they won't affect the calculations for the other rows.) L1: +/- T1: +/- L2: +/- T2: +/- L3: +/- T3: +/- L4: +/- T4: +/- L5: +/- T5: +/- L6: +/- T6: +/- L7: +/- T7: +/- L8: +/- T8: +/- L9: +/- T9: +/- L10: +/- T10: +/- submit σ ln(l) 4π 2 df σ ln(t) 1 x σ ln(x) σ x x 4π 2 phy133/lab1pendulum.txt Last modified: 2016/06/21 14:20 (external edit) 5/5