Introduction. A pendulum is an interesting and versatile device which possesses many unique properties and uses. This particular exercise involves the use of a pendulum for the purpose of measuring the local gravitational acceleration or "g." As employed, appreciated and described many years ago initially by Galileo and Newton hence, a pendulum's motion is similar to that of a freely falling body. Due to the fact that it is connected to and suspended from a fixed point by a constant length string, the falling motion of a pendulum is restricted thereby causing it to swing laterally in a periodic (or cyclic) fashion. According to Galileo and Newton, pendulums which possess identical distances from the pivot to their centers of mass behave in the same mechanical fashion regardless of differences in mass or angles of oscillation (provided that the angles of oscillation are relatively small). Galileo and Newton discovered that altering string lengths had a direct affect on the times required for the pendulum to complete a single full swing cycle to the original point of release. In other words, the period of a pendulum is directly related to its length and not upon mass or any other feature, provided that air resistance is neglected. According to Newton, the period of a pendulum can be determined from the following equation:
Pendulum Motion and the Measurement of
Local Gravitational Acceleration
Laboratory Exercise
How can a pendulum permit us to determine local gravitational acceleration?
How can we better understand the factors which affect a pendulum's period of swing?
How does a pendulum's length affect its behavior?
Cycle II Regents Physics
John Dewey High School
Mr. Klimetz
T = 2 x pi x sqrt(L/g)
where T is period (s), pi is a constant (3.14), L is pendulum length (m), and g is local gravitational acceleration (m/sexp2).
By reorganizing the variables in the above equation to isolate g we obtain
g = 4 x piexp2 x L/Texp2
thereby permitting us to determine local gravitational acceleration (g) directly merely by precisely measuring a pendulum's period of swing (T) in seconds and its length (L) in meters from the pivot to the center of mass of the suspended pendulum bob.
Objectives. This exercise will require you to measure and average period of three sequential episodes of twenty swings of a pendulum set at five different lengths with a stopwatch. Dividing the average of the three episodes of twenty swings of a particular pendulum length by twenty will yield the time of a single swing period of the pendulum. The measured length and period data are then substituted into Equation (2) and the value of g is then derived. Your calculated values of g are then to be compared with the accepted value of g (9.80 m/sexp2) via a percent error analysis.
(1)
(2)
Equipment
Brass spherical pendulum bob
Ring stand
Nylon fishing line
Meter stick
Stopwatch
Three-position horizontal cord clamp
C-clamp
Procedure.
1. Place the ring stand so that the upright is close to the edge of the lab table.
2. Securely affix the base of the ring stand to the lab table with the c-clamp.
3. Attach the three-position horizontal cord clamp to the upright of the ring stand,
approximately 3 cm from the top.
4. Thread a 1.3 m length of fishing line through a pendulum and attach by knotting the
protruding end of the line. Remove any excess line with scissors.
5. Firmly attach the pendulum line to the outermost fitting on the three-position horizontal
cord clamp and suspend the bob.
6. Set the pendulum to the lengths instructed.
7. Move the pendulum bob approximately 20 cm from the equilibrium (rest) position and let
swing. Be sure not to push the bob in any way. It must start it's trip from rest. Using the
stopwatch, determine the time for the pendulum to complete 20 full swing cycles. Record your
data into the data table. Repeat this step twice more.
8. Calculate the average of the three 20-swing measurement cycles and enter into the data
table. Remember to show all work.
9. Calculate the time of a single period of swing by dividing the answer obtained from Step 8 by
20. Enter this into the data table. Remember to show all work.
10. Repeat Steps 6 through 9 for the remaining pendulum lengths.
11. Substitute your data into Equation (2) and solve for g.
12. Perform a percent error analysis of your best results. Show all work
Data Table
Line Length (m)
0.25
0.25
0.25
0.50
0.50
0.50
0.75
0.75
0.75
1.00
1.00
1.00
Time of Twenty Swings (s)
______
______
______
______
______
______
______
______
______
______
______
______
Average of Twenty Swings (s)
______
______
______
______
Time of One Swing (s)
______
______
______
______
Calculations
Line Length (m)
0.25
0.50
0.75
1.00
g (m/sexp2)
______
______
______
______
Equation
Questions
1. Discuss any possible sources and causes of error in your experiment. How might air resistance have affected your apparatus? Would air resistance have changed the period of swing? If so, how would it have affected your value of g?
2. Suppose our pendulum apparatus was dropped down an airless elevator shaft while it was in the middle of swinging. Do you think it would continue swinging while freely falling? Why or why not? (Think!)
3. Explain why the mass of the pendulum has no influence on period of swing. Would a pendulum of a given length on Earth have the same period as one on the Moon? (The Moon possesses 1/6 the mass of Earth.) Explain.
4. At which position(s) in its path is the pendulum moving at the fastest speed? Why does it keep moving when it reaches the bottom of its swing path? At which position(s) in its path is the pendulum moving at the slowest speed? Where in the pendulum does the greatest acceleration occur?
By reorganizing the variables in the above equation to isolate g we obtain
