Simple Pendulum Lab
Title: Simple Pendulum Lab
Date: 4/9/14
Partner: Steph Kinsella
Date: 4/9/14
Partner: Steph Kinsella
Purpose
The purpose of this lab was to determine which of the following characteristics of a simple pendulum have an effect on the pendulum's period: mass, length, or amplitude.
Theory
A simple pendulum is described as "a hypothetical pendulum consisting of a weight suspended by a weightless spring." The period of this pendulum can be written as:
According to this equation, when the amplitude is limited to small angles, the period should only be affected by l, the length of the string.
Varying mass and length to confirm this fact do not require derivations; however the method used to vary the angle of the amplitude required the following equation, where d is the length of the horizontal that the pendulum will be raised to and l is the length of the pendulum to its center of mass (constant):
Varying mass and length to confirm this fact do not require derivations; however the method used to vary the angle of the amplitude required the following equation, where d is the length of the horizontal that the pendulum will be raised to and l is the length of the pendulum to its center of mass (constant):
Experimental Technique
Create a simple pendulum using a rod, string, and mass. The one used for this experiment is shown at left.
In order to test the effect of mass on the period, we must vary mass while keeping all other variables constant. Four different masses (below) were measured on a balance and then tested on the pendulum. |
In the mass experiment, the length, or the distance from the top of the pendulum to the bob's center of mass, was held constant by lining the center of mass up with a block (shown a right). The amplitude was also held constant using a protractor attached to the top of the pendulum. The period was measured by taking a video and using a frame-by-frame analysis.
In order to investigate length, the same bob (the silver one) was used consistently, and the amplitude was held constant using the protractor. The length was varied in constant intervals for ten trials and the period was measured using a stop watch. We measured ten oscillations and then used this to find the period of one full swing. |
Finally, to investigate amplitude, we held the mass and length constant and varied the angle that the pendulum was raised to before being released. In order to determine how far to raise the bob at smaller angles, the angles and the length were plugged into the equation from above and a distance was obtained. This distance was measured from the horizontal center of the bob and marked on masking tape on the lab table. A straight edge was held at each mark to ensure accurate angles. The period was measured with a stopwatch for ten oscillations, which was then used to determine the length of one full swing.
Data
In the mass experiment, the length was held at a constant 72.0 cm and the amplitude angle was 30 degrees. The following data was collected for each of the four masses in the mass-variation experiment:
For the length experiment, the mass was the silver bob at 23.7 g and the amplitude angle was a constant 30 degrees. For the varied lengths, the following periods were recorded:
For the amplitude experiment, the silver bob was used (mass 23.7 g) and the length of the pendulum was kept at a consistent 70.0 cm. Using the previously derived equation (sample calculation below-left), distances for the tape were determined and the following data was collected:
Analysis
For the mass experiment, the frames needed to be converted to seconds. Since the camera measured 30 frames per second, the following calculation was made to yield the data at right. Thus, mass does not affect the period of a simple pendulum.
In regards to the length experiment, the graph below shows the period as a function of the length with a correlation coefficient of .99, which is strong. The graph takes this shape because it has been linearized- that is, the T values have been squared. It is apparent that length does affect the period of a simple pendulum.
Because the T^2 value was plotted against the length rather than the simple T value, the following is true:
We can then use the graph to calculate g to be 8.93 m/s^2 with 8.88% error by using the relationship between the coefficients of l, as shown below.
The amplitude data was graphed, as shown below. A general upward trend is shown from left to right, but the low correlation coefficient does not allow for the determination of the definition of a "small angle" or any further accurate prediction.
The percent difference between the highest and lowest periods is shown below. Because it is so small, it can be concluded that amplitude does not affect the period of the pendulum.
Conclusion
In the mass experiment, each of the four trials were recorded to be exactly 51 frames, which comes out to 1.7 s. Because they are all identical, it can be confirmed that mass does not affect the period of a pendulum.
In the length experiment, it was determined that length does affect the period in a square root relationship. By linearizing the graph, the relationship between period, length, and gravity allowed us to determine the value of g with 8.88% error. Because measurements for this portion were taken using a meter stick and stopwatch, error can be attributed to reaction time and parallax error.
In the amplitude experiment, the angle did seem to affect the period, but upon further analysis, the low percent difference of 2.95% allowed us to conclude that amplitude does not affect the period. The error can be attributed to the reaction time discrepancies associated with the use of a stop watch. The error in the graph does not allow for us to determine what a "small angle" should be.
In the length experiment, it was determined that length does affect the period in a square root relationship. By linearizing the graph, the relationship between period, length, and gravity allowed us to determine the value of g with 8.88% error. Because measurements for this portion were taken using a meter stick and stopwatch, error can be attributed to reaction time and parallax error.
In the amplitude experiment, the angle did seem to affect the period, but upon further analysis, the low percent difference of 2.95% allowed us to conclude that amplitude does not affect the period. The error can be attributed to the reaction time discrepancies associated with the use of a stop watch. The error in the graph does not allow for us to determine what a "small angle" should be.
References
Boundless. “The Simple Pendulum.” Boundless Physics. Boundless, 03 Jul. 2014. Retrieved 11 Apr. 2015
Giancoli, D. (2009). Physics for Scientists and Engineers (4th ed.). Upper Saddle River, N.J.: Pearson Prentice Hall.
Lahs Physics (n.d.). Retrieved October 6, 2014, from www.lahsphysics.weebly.com
Giancoli, D. (2009). Physics for Scientists and Engineers (4th ed.). Upper Saddle River, N.J.: Pearson Prentice Hall.
Lahs Physics (n.d.). Retrieved October 6, 2014, from www.lahsphysics.weebly.com