18-Dec-12 PHYS Simple Pendulum. To investigate the fundamental physical properties of a simple pendulum.

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1 Objective Simple Pendulum To investigate the fundamental physical properties of a simple pendulum. Equipment Needed Simple Pendulum Apparatus with Meter Scale and Protractor Bobs 4 (Aluminum, Brass, Lead, and Plexiglass) Table Clamp Introduction An ideal simple pendulum consists of a particle of mass m suspended by an un-stretchable, weightless string of length L, as shown in Figure 1. The particle is known as the bob of the pendulum, and is free to swing back and forth to the left and right of the vertical line through the pendulum s pivot point. The time taken for one complete oscillation is called period T. One oscillation is the motion taken for the bob to go from position A (initial position) through C (center position) to B (the other extreme position) and back to A. Pivot point θ θ String, length L Bob, mass m Figure 1 For small oscillations (θ less than 10 ), the period T of a simple pendulum is given by T = 2π ( L / g ) 1/ 2 where, g is the free-fall acceleration and is equal to 9.80 m/s 2. Notice that the period of a simple pendulum is (a) independent of the mass m of the particle and (b) proportional to the square root of the length of the pendulum. 1

2 In this lab you will be making measurements on a real-world pendulum, in which the string is light, but not weightless, and the mass is not a point, but has some dimensions. Nevertheless, if you measure the length L from the pivot point to the center of mass of the bob, it will turn out that the measured periods for small oscillations will agree very well with the equation given above. The experimental set up is shown in Figure 2. 2

3 Pivot position, y 1 L = y 1 - y 2 Lead Aluminum Brass Plexiglas Center of the Bob position, y 2 Figure 2 3

4 Exercise 1 Feel for small angles In this exercise, you will get an idea of small oscillations for which the approximation sin θ θ is valid. You must convert θ to radians for this comparison. You will compare the period T for small oscillations against that for large oscillations. Since T will be a small value, you will measure the time t for 10 oscillations, and then calculate T = t /10. This increases the accuracy of your measurements. 1. Open Excel 2007 and enter the information as in Figure 3. Change the letter q in cell A1 to θ by double clicking on cell A1, selecting q, and changing the font to Symbol in the pop-up menu. Do the same in cells B1 and C1 as well. Figure 3 2. To convert θ = 0 to radians, type in cell B2 =radians(a2) and press Enter key. 3. To find sin 0, type in cell C2 =sin(b2) and press Enter key. Recall that in Excel, the angles used as arguments of sin and cos functions should be expressed in radians. 4. In cell D2, calculate the percent difference between θ in radians and sin θ, using the formula θ sin θ Percent difference = 100 θ The cell D2 will now display #DIV/0 as the denominator in the formula calculates to zero; leave it as it is and continue to the next step. 4

5 5. To repeat the calculation for θ = 2, select cells B2, C2, D2 and use the left mouse button to press on the small square at the lower right corner of cell D2 and drag it down to the next row (row 3) only. 6. You will now repeat the calculation for θ = 4, 6, 8, 10,..90 in steps of 2 as follows: Figure 4 Select cells A2 to D3 (rows 2 and 3 together) and use the left mouse to press on the small square at the lower right corner of D3 (see Figure 4) and drag it down until you reached 90 in the cell A. 7. Use Decrease decimal button in Number group to keep only one decimal place in Column D. 8. Select Columns A, B, and C and Insert Scatter with Smooth Lines plot (see Figure 5). 9. As usual choose Layout 1 (from Design Tab Chart Layouts Group) and change to appropriate titles, add major gridline in the x-axis as well, BUT do NOT delete the legend. 5

6 Figure Copy your worksheet and paste it in your report. 11. Copy your graph from Excel and paste it in your report. 12. Write your observation on the graph and on the % difference from the worksheet. 13. Hang any of the small bob to pendulum and adjust the length of the pendulum, L to be 70.0 cm. Measure the length L from the pivot point to the center of mass of the bob. You may assume that the center of mass is at the center of the bob (see Figure 2). 14. Calculate the period T using the formula T = 2π ( L / g ) 1/ Displace the bob of the pendulum from the vertical line by θ = 10 and let it go. 16. From the Start menu, open the program Free Stopwatch. This displays a Stopwatch in your desktop. 17. Using this stop watch, measure the time t for 10 oscillations and then calculate the period T = t / Repeat Steps 13 and 14 with θ = 30, 50, and 70. 6

7 19. Find the percent difference between the measured and calculated values of T, using Measured value Calculated value Percent difference = 100 Calculated value 20. Record your results in your report. 21. Can you conclude, from your results, that the formula only for small oscillations? T 1/ 2 = 2π ( L / g ) is accurate Exercise 2 Fixed Length, Variable Mass In this exercise, you will check whether the period T is independent of the mass m for a fixed value of length L. You will be able to vary the mass m by attaching various objects made of aluminum, brass, lead, and Plexiglas to the string. 1. Measure the masses of the objects using a triple beam balance. 2. Attach one of the objects to the string, and fix the length L from the pivot point to the center of mass of the object to 70.0 cm. 3. Displace the object from the vertical line by θ = 10 and let it go. 4. Using a stop watch, measure the time t 1 for 10 oscillations. 5. Repeat Steps 3 and 4 twice more, and measure the times t 2 and t 3 for 10 oscillations. This step is required to reduce the error caused by your reaction time in starting and stopping the timer. 6. Record your data in a new Excel sheet as shown in Figure 6. To write t 1 in cell C1, first type t1 then double click on the cell, select 1, right click, choose Format cells, and then select subscript in the Format cells window. Do the same in cells D1, E1 and F1 as well. 7

8 Figure 6 7. Repeat Steps 2 to 5 for the other three objects as well, and record the data in the Excel sheet. 8. To calculate the average value t av for aluminum, type in cell F2 =average(c2:e2) and press Enter key. 9. To calculate T for aluminum, type in cell G2 = F2 /10 and press Enter key. 10. To calculate t av and T for the other three objects, select cells F2, G2 and use the left mouse button to press on the small square at the lower right corner of cell G2 and drag it down. 11. Copy the table from Excel sheet and click on the space provided in the report (word document) where you need to paste it. In the Home tab and Clipboard group, choose Paste Special. In the Paste Special Dialog box select Microsoft Office Excel Worksheet Object and click OK. 12. What can you conclude from your results? Exercise 3 Fixed Mass, Variable Length In this exercise, you will verify that the period of a simple pendulum is proportional to the square root of the length of the pendulum. You will also determine experimentally the freefall acceleration g. 1. Choose any one of the four objects used in Exercise 2 as the bob of the pendulum. 2. Change the length L from the pivot point to the center of mass of the object to 40.0 cm. 3. Displace the object from the vertical line by θ = 10 and let it go. 4. Using a stop watch, measure the time t 1 for 10 oscillations. 5. Repeat Steps 3 and 4 twice more, and measure the times t 2 and t 3 for 10 oscillations. 6. Record your data in a new Excel sheet as shown in Figure 7. 8

9 Figure 7 7. Repeat Steps 3 to 5 for L = 50.0 cm, 60.0 cm, 70.0 cm, 80.0 cm and 90.0 cm as well, and record the data in the Excel sheet. 8. Calculate t av and T as you did in Exercise To calculate L 1/2, type in cell G2 =sqrt(a2) and press Enter key. Then select cell G2 and use the left mouse button to press on the small square at the lower right corner of cell G2 and drag it down. 10. Copy the table and use Paste Special as you did in step 11 of Exercise 2 to paste it in your report as Microsoft Office Excel Worksheet Object. 11. Plot T versus L 1/2 and add trendline as you did in the previous labs. Make sure you have plotted T on the y-axis and L 1/2 on the x-axis. 12. Copy the graph and paste it in the space provided. 1/ Recalling the formula T = 2π ( L / g ), determine g exp from the slope and calculate the percent difference between g exp and the accepted value, g acc = 9.80 m/s Comment on the sources of error in this experiment. 9