The Pendulum

I. Simple Pendulum

a) Select eight lengths of string (L) ranging from a maximum with the bob (500 g mass) near the floor to a very short pendulum. Hang the pendulum from the force probe.

b) For the longest pendulum length measure the period of oscillation for initial displacements of 5, 15, 30 and 45 degrees. Use the force probe rotated 90 degrees to monitor the tension in the string, use a high collection rate (>2000 points/s) and measure T of one oscillation by measuring the time interval between successive zero crossings (going in the same direction) of the signal F(t). Notice that the force probe monitors the horizontal component of the tension in the string. You may want to make the force signal symmetric respect to zero by zeroing the probe when the pendulum is in equilibrium. Notice that F(t) = 0 every time the pendulum passes through the vertical (twice per cycle). You can use Analyze -> Examine to measure the zero crossing times directly on the graph.

Plot period T as a function of amplitude θ0 . According to theory the period approximately depends on the initial displacement , measured in radians, as:

(Eq. 1: "Large amplitude approximation")

Do your results agree with the large amplitude approximation theory?

Do you notice any change in the shape of the force probe signal at large amplitudes?

c) For the remaining lengths L determine the period of oscillation T for an initial displacement of 5 degrees (use the angle finder) by averaging the time for a number of oscillations. Use a much slower data acquisition rate.

For small amplitudes of oscillation the motion of the pendulum is expexted to be harmonic with period :

(Eq.2: "Small amplitude approximation")

Plot T2 vs L. Find g from the slope and compare with the standard value. Notice that the x-intercept value should be related to the position of the center of mass of the bob respect to the bob's hook (LTOT = L + dCM). Determine dCM from the graph and compare with its estimate obtained by hanging the bob in two different orientations and finding where the two vertical lines intersect.

Does the result for g agree with the standard value

Does the result for dCM agree with the measured value ?

II. Physical Pendulum

Use LoggerPro to measure the passage of a physical pendulum (meter stick + added masses) through an infrared beam. From these measurements determine the damping time (exponential decay time) and the period for the motion (note that the motion repeats after 2 successive passages through the vertical). Use the following procedure to do so.

The Photogate sensor is connected to Dig Source 1. The Data Collection Mode is Gate Timing , To obtain the correct speeds of passage you must enter the width of the stick in meters. To do so go to Data-> Column Options-> Velocity and change the default width (0.05 m) to the correct value.

a. Adjust the ruler support at the top so that the infrared beam is blocked by the center of the ruler when it is at rest.

b. Raise the physical pendulum until it lines up with the edge of the lab table.Click on Collect and release the pendulum to measure the speed v for successive passages of the meter stick through the beam until it stops.

Be sure to export (not copy and paste) your files as text before collecting the next set of data, and then open them in Excel to get times and speeds of passage. Then follow the Physical Pendulum Analysis described below.

Plot v2 as a function of time for the measured values and in the same plot an exponential fit. Use Solver to determine the damping time tau.

Repeat with a 200 g mass attached to the each side of the ruler (i) near its bottom, (ii) halfway up the ruler and (iii) near the top of the ruler. Measure their position in each case. Construct a Table to compare your values of v2-max, period T, and damping time tau for all four cases.

Do you identify any trends?

Physical Pendulum Analysis

You should clear the information in the top rows and keep the column headings. Select all cells (by clicking on the top left diamond icon) and use Data->Sort->(click the My List has Header row option) and select Sort by Gate State. This procedure will create a list without gaps of your times and speeds. Delete the extra data.

Use R1C1 type display. Delete the entire third column (Select the column and use Control-k). The measured v values are now in Column 3. In the second column create a set of times starting from 0 to replace the Gate State values, which you no longer need. Do this procedure by entering an equation in which you subtract from each time in Column 1 the first time value in Column 1. You will need to use both a relative cell reference and an absolute cell reference.

Set the values in Column 4 equal to v2 . The header for this column should be "v^2-meas".

Into the first two cells of Col 1 enter "Ampl" and "tau". Into the first two cells of Col 2 enter the initial value of v2 and a good guess for tau.

The header for Col 5 should be "v^2-expon". In Col 5 select cells from below the header down to the end of the data. Use absolute and relative addresses to insert into those cells the equation =A*exp(-t/tau). Enter into each of the cells in Column 6 the square of the difference between the corresponding cells in Col 4 and 5 {i.e., (v^2-meas - v^2-expon)2 }. Skip one row at the end of Column 6 and enter into the next cell down the sum of the cells in Column 6. This value is the sum of the squares of the differences between the measured and model values.

Create a graph of "v^2-meas" and "v^2-expon" vs time. Do this procedure as follows: Select the values of Columns 2, 4 and 5 from the headings to the end of the data. Because the columns are discontinuous, select 4 and 5, hold down the command key, and then select column 2. Go to Insert Chart and create an x-y scatter graph. Make sure that the model values are plotted as lines only (no markers), and the v^2-meas data are markers only (no connecting lines).

Go back to the spreadsheet and choose Solver. Solver is found under the Tools Menu in Excel 5 . For "Set Cell" enter the absolute address of the sum of squares in Column 6. Do this by typing the address or clicking on the cell. For "Equal To" select Min. For "By Changing Cell" enter R2C2,R1C2. After you hit Return, Excel will calculate the best values for tau and A to minimize the difference between the exponential model and experimental curves. Excel will provide the "best fit" to the data and plot the corresponding curve.

NOTE. You do NOT have to re-enter this procedure for each of the other three physical pendulum data sets. Having saved the original Excel analysis worksheet and graph, copy the data from Logger Pro into the spreadsheet starting at the Row 11 Col 1. The values in all Columns will automatically change their values according to the new data. Use either Clear or Fill Down to adjust to the number of rows on the new sheet. Enter a new cell for the sum of squares and then use Solver on the sheet. Save the new worksheet and new graph.