Simple Pendulum -...
9
18-Dec-12 PHYS101 - 13 © KFUPM – PHYSICS Department of Physics revised 22/12/2012 Dhahran 31261 1 Simple Pendulum Objective 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. Figure 1 For small oscillations (θ less than 10°), the period T of a simple pendulum is given by 2 / 1 ) / ( 2 g L T π = 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. Pivot point String, length L Bob, mass m θ
Transcript of Simple Pendulum -...
18-Dec-12 PHYS101 - 13
© KFUPM – PHYSICS Department of Physics revised 22/12/2012 Dhahran 31261
1
Simple Pendulum Objective 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.
Figure 1 For small oscillations (θ less than 10°), the period T of a simple pendulum is given by
2/1)/(2 gLT π= where, g is the free-fall acceleration and is equal to 9.80 m/s2. 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.
θ
Pivot point
String, length L
Bob, mass m
θ
18-Dec-12 PHYS101 - 13
© KFUPM – PHYSICS Department of Physics revised 22/12/2012 Dhahran 31261
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.
18-Dec-12 PHYS101 - 13
© KFUPM – PHYSICS Department of Physics revised 22/12/2012 Dhahran 31261
3
Figure 2
Center of the Bob position, y2
L = | y1 - y2 |
Pivot position, y1
Aluminum Plexiglas Brass
Lead
18-Dec-12 PHYS101 - 13
© KFUPM – PHYSICS Department of Physics revised 22/12/2012 Dhahran 31261
4
Exercise 1 – Feel for small angles In this exercise, you will get an idea of small oscillations for which the approximation
θθ ≅sin 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
𝐏𝐞𝐫𝐜𝐞𝐧𝐭 𝐝𝐢𝐟𝐟𝐞𝐫𝐞𝐧𝐜𝐞 = �𝛉 − 𝐬𝐢𝐧 𝛉
𝛉� × 𝟏𝟎𝟎
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.
18-Dec-12 PHYS101 - 13
© KFUPM – PHYSICS Department of Physics revised 22/12/2012 Dhahran 31261
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.
18-Dec-12 PHYS101 - 13
© KFUPM – PHYSICS Department of Physics revised 22/12/2012 Dhahran 31261
6
Figure 5
10. 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 2/1)/(2 gLT π= .
15. 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 /10.
18. Repeat Steps 13 and 14 with θ = 30°, 50°, and 70°.
18-Dec-12 PHYS101 - 13
© KFUPM – PHYSICS Department of Physics revised 22/12/2012 Dhahran 31261
7
19. Find the percent difference between the measured and calculated values of T, using
𝐏𝐞𝐫𝐜𝐞𝐧𝐭 𝐝𝐢𝐟𝐟𝐞𝐫𝐞𝐧𝐜𝐞 = �𝐌𝐞𝐚𝐬𝐮𝐫𝐞𝐝 𝐯𝐚𝐥𝐮𝐞 − 𝐂𝐚𝐥𝐜𝐮𝐥𝐚𝐭𝐞𝐝 𝐯𝐚𝐥𝐮𝐞
𝐂𝐚𝐥𝐜𝐮𝐥𝐚𝐭𝐞𝐝 𝐯𝐚𝐥𝐮𝐞� × 𝟏𝟎𝟎
20.
Record your results in your report.
21. Can you conclude, from your results, that the formula 2/1)/(2 gLT π= is accurate only for small oscillations?
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 t1 for 10 oscillations.
5. Repeat Steps 3 and 4 twice more, and measure the times t2 and t3 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 t1 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.
18-Dec-12 PHYS101 - 13
© KFUPM – PHYSICS Department of Physics revised 22/12/2012 Dhahran 31261
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 tav 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 tav 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 free-fall 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 t1 for 10 oscillations.
5. Repeat Steps 3 and 4 twice more, and measure the times t2 and t3 for 10 oscillations.
6. Record your data in a new Excel sheet as shown in Figure 7.
18-Dec-12 PHYS101 - 13
© KFUPM – PHYSICS Department of Physics revised 22/12/2012 Dhahran 31261
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 tav and T as you did in Exercise 2.
9. To calculate L1/2, 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 L1/2 and add trendline as you did in the previous labs. Make sure you
have plotted T on the y-axis and L1/2 on the x-axis.
12. Copy the graph and paste it in the space provided.
13. Recalling the formula 2/1)/(2 gLT π= , determine gexp from the slope and
calculate the percent difference between gexp and the accepted value, gacc = 9.80 m/s2.
14. Comment on the sources of error in this experiment.
