Pendulum Periods

Lab 9: Pendulum Periods

Aryan Toosi

Trent Blanchard

Lily Kuang

Jaime Tan

TA: Ben Folsom

PHY 113: Tuesdays at 1:30

Objectives:

An introductory impression of the motion of a pendulum may leave one to think that the period of oscillation is independent of amplitude, and depends simply on the length of the pendulum. In order for this determination to be appropriate, two standard conditions must be met: the amplitude is small (less than 1 radian) and the mass of the system is concentrated at the end of the string.

For this particular experiment, the objective is to examine the behavior of a pendulum when the aforementioned conditions are not present. In addition, one will simplify the analysis of the pendulum and determine when and why these approximations begin to break down.

Materials:

The following materials were utilized during the experimentation process: computer, vertical support rod and clamp, right-angle clamp, Vernier Rotary Motion Sensor, protractor, Vernier Rotary Motion Accessory Kit, and a metric ruler.

Procedures:

Prior to the initiation of the experiment, the apparatus for conducting the experiment was constructed by the lab assistants. For the first part of the experiment, the distance between the point of attachment of the rod to the 3-step pulley and the center of mass associated with the weight at the end of the rod was measured. The sensor was then connected to the interface, and the data-collection program was opened to develop one graph depicting the change in angle over time. After starting the data collection program, a protractor was used to measure the angle of the rod being pulled through 5 degrees, and then the rod was released. The previous step was repeated for amplitudes of 10, 15, and 20 degrees. The second portion of the experiment required that the previous procedures for releasing the rod be reproduced 14 additional times. The first amplitude measurement was 25 degrees and additional trials requiring an increase in increments of 5 degrees were performed up to 60 degrees of displacement. After reaching an amplitude value of 60 degrees, the addition of 10 degree increments were used up to 120 degrees of rod displacement.

Data Analysis:

1. Calculation of the mean:

(sum of the radian value) / (number of samples) = average radian value

2. Calculation of standard deviation:

The calculations associated with the standard deviation were obtained through the use of Microsoft Excel.

3. Standard deviation of the mean:

standard deviation of the mean = (standard deviation of the radian values) / (square root of N), which provides the uncertainty of the measurement.

4. Calculation of Theoretical ω:

ω = square root of (g/l)

example: square root of (9.81m/s2 / 0.31 m) = 5.625 rad/sec

5. Percent discrepancy between calculated slope and graphically derived slope:

(derived value – accepted value) / (accepted value) x 100% = Percent Discrepancy

6. Expression for Period of a Pendulum:

T = (2 π)(1/ ω)

Figure 1: Angle vs. time graph

Figure 2: Relationship between Period and Amplitude

Figure 3: Relationship between Period and Amplitude

Figure 4: Force diagram

Results:

Table 1: Expression for theoretical angular frequency, which is ω.

ω = square root of (g/l)

g= gravitational acceleration, 9.81 and l= 0.31meters

Square root of (9.81/0.31)=5.625

Table 2: Values relating the angle vs. time data obtained for portion I.

Run Angle measured from vertical B Parameter

1 5 5.752 ± .001197

2 10 5.749 ± .001066

3 15 5.573± .008301

4 20 5.711 ± .008755

Average 5.696 ± .005023

Table 3: Percent Discrepancy for the experimental and theoretical angular frequency.

(Experimental value – theoretical value) x 100% (5.696 – 5.625) X 100% = 1 %

(Theoretical value) 5.625

Table 4:Additional analysis for Experimental and theoretical angular frequency when the length is altered, l = 0.28m

Experimental ω 5.85 rad/sec Percent discrepancy:

Theoretical ω 5.92 rad / sec 1.20%

Table 5: Formula for the Period of a Pendulum

T = (2 π)(1/ ω)

Table 6: Values relating the angle vs. time data obtained for portion II.

Angle in Degrees Angle in Radians ω Period

25 0.4363 5.756

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