Pendulum Lab

Pendulum Lab (Period vs Length) Purpose: Using a simple pendulum the purpose of this lab experiment is to find if the period of oscillation is dependent to its length, and how the gravitation acceleration effects the period at different lengths. The results are then compared to the true value of gravitation acceleration to show the accuracy of the experiment. Apparatus: For this investigation the equipment used is as listed: Tall ring stand String/spring holder String Bob Meter stick Protractor Computer with Logger Pro Timer Independent variable: is the Length Dependent variable: is the period of oscillation. Procedure: Period vs Length 1. Set up a free swinging pendulum using the string, bob, pendulum holder and ring stand. The string needs to be adjustable to lengths of 10 cm to 100 cm. 2. Start with a length of 10 cm and pull to an angle of your choosing using a protractor (always use the same angle for every trial), and release the pendulum at the same time you will be using the timer to measure the time of five oscillations. Enter you data in the table, and repeat for two more times. 3. Now set the length to 100 cm and repeat step two. 4. Continue to measure at four different lengths of your choice and repeat the process of step two for each new length. 5. Finally use computer with Logger Pro to graph the data. Data: Table 1 Length L(m) Time for Five Oscillation (s) Average of the Period of Five Oscillations Oscillation T (s) Trial three (s) Trial One Trial Two .1 3.79 3.83 3.80 3.81 .762 .2 4.73 4.79 4.75 4.76 .952 .4 6.82 6.90 6.88 6.87 1.37 .5 7.39 7.45 7.43 7.42 1.48 .7 8.55 8.60 8.62 8.59 1.72 1 10.18 10.30 10.29 10.30 2.06 Sample calculation: Average of 5 oscillations: Trial one + two + three/3 = 3.79+3.83+3.80/3=3.81s Period of oscillation: Avg of 5 oscillations/5= 3.81s/5=.762s Data Analysis: Data: Table 2 Length L(m) 0.1 0.2 0.4 0.5 0.7 1.0 Period Square T2(s2) .58 .91 1.88 2.19 2.96 4.24 Sample calculation: Period Squared: t2= (.762s)2=.58s2 For data table one calculations we graphed the relationship between the period vs the length of the pendulum as show on graph 1. The graph shows to be a curve fit with formula t=2.04L0.450 which would appear to be a square-root relationship which shows that a pendulums period will vary depending on the length of said pendulum which can be calculated using the period formula: t=2π√L/g. The .450 is close enough to .50 that shows it supports the data that this is a square-root relationship and this can be seen in graph two for the period squared versus the length. Graph two is a linear relationship for the period of oscillation squared and the length of the pendulum. The formula obtained for graph two is t2=3.92L+.229 which is represented of linear equation y=mx+b where the slope of the line is 3.92 and the y-intercept is .229 which would account for the error in the experiment because the y-intercept should be at zero. A pendulum with zero length cannot oscillate there for if L=0, t=0. Using the slope we calculate the gravitation acceleration. gcal=4π2/slope 4π2/3.92=10.07 %error=|9.81m/s2- gcal|/9.81m/s2*100% %error=|9.81-10.07|/9.81*100%=3% Results show 3% error which would indicate that the period is dependent on the length of the pendulum and shows simple harmonic motion. Conclusion: Simple harmonic motion (SMH) is a body in motion when acted on by a force, F, that is always proportional to the body’s displacement, x, from a fixed point and always directed toward that point (cited from Lab Introduction). An example of this is a simple pendulum, which contains a period of oscillation that is the time it takes for the pendulum to make one swing forward and back to the same point. Gravitational acceleration is the acceleration on an object caused by force of gravitation (Wikepedia.org/wiki/Gravitational_acceleration) which is used to calculate the period of oscillation. The relationship between the time of oscillation and its length shows that as the length is increased the period also increases. The slope indicates for every one meter the time increases by 3.92 seconds. The equation for this is t2=3.92L+ 0.229(taken from Logger Pro). My analysis for this experiment shows that the relationship between the length of the string and time of oscillation, is as the length increased the period of oscillation also increased. This would represent simple harmonic motion. Problems that may have affected the results of this experiment would be human error during the measuring of the data. Sometimes after measuring the string to the correct length for measuring it would tend to drop or pull up, and the string would need to be re-measured again. This would make the time of oscillation inaccurate for that length because it wasn’t the true length being measured. It was also uncertain whether the angle the string was pulled to and released from of 30 degrees was always the same for each length as it was difficult to be sure it was at that angle, because the protractor could not be placed directly on the string so it was sort of a floating measurement. This would cause the measurements to be inaccurate or have a greater percent uncertainty because there was no way to be sure if the angle was bigger or smaller then we believed it to be. The ring stand wobbling during the measurement would be a contributing factor to the error as well. Finally, the measuring of the time accurately was difficult because there was one person holding the string and one person timing the oscillation so there was no way of knowing if the timer was actually pressed at the exact time as the bob was being released. This would cause a difference in how much time was actually going by during that instant. So the percent error of 3% may indicate why the y-intercept was not equal to zero, and may be due to the inaccuracy of the measurements taken. The slope indicates that for every 1 meter the time of oscillation increases 3.92 seconds.