Moment of Inertia Lab
Lab Partners: Vince Mele and Kyle Higgins
Date Completed: 4-10-14
Purpose
In this lab we will be investigating the moment of inertia of a system including a pulley and a rod with masses hanging at various radii. We will compare the geometric moment of inertia with our rotational moment of inertia to see the validity of our calculations.
Theory
The sum of the of torques is equal to inertia multiplied by angular acceleration. We will use this equation as well as the sum of the forces equation shown below to find the Inertia. As shown below, the first two equation are solved for t and then combined to create the fifth equation. The final equation also needs a few other formulas to be added as shown below, such as the angular acceleration being equal to the tangential acceleration divided by the radius. Finally a substitution for weight is used and we have our final equation for the moment of inertia that we will calculate. The next equation will be our geometric inertia which will be compared to the calculated one we will find shortly. Shown below are two different moment of inertia equations one for a thin hoop and one for the long uniform rod. Keep in mind that in the geometric inertia equation the sub-scripted "w" stands for weights and all the masses of the wieghts were averaged together. The masses were each different as stated before and even though an average was used each mass was written separately to remind us that the masses were not actually all equal. Also, the sub-scripted "r" in the geometric equation stands for the rod the masses hang from. The final step to the experiment will be using the percent error equation and comparing our two calculations.
Experimental Technique
The first step of the experiment is to set up the apparatus as it was designed using a rotary motion sensor and the attachments shown to the right. As seen we will be attaching a pulley with a hanging mass to cause the acceleration of the pulley on top with the weighted rod. We decided to use multiple masses at varying radii for our apparatus. This is shown in the blue picture below. The rod is in neutral equilibrium and it will rest in any orientation we set it. This will ensure our results to be more accurate. The picture below in pink shows the apparatus just before launch where the string is guided by the attachment pulley to accelerate the top weighted pulley. Adding together the rod and various weights are what goes into the creation of our geometric inertia equation. The acceleration will then be found using tools in data studio. The graphs from these runs are shown below. The first graph shows the acceleration as the slope of the linear line. The second is a full graph showing how wind resistance effects the deceleration of the pulley which in turn means it had to have affected the acceleration(as you can see the faster the pulley spins the faster it slows down and the slower it is spinning the slower it will slow down because of the wind resistance being greater whenever the object is moving faster). The acceleration is taken and plugged into the equation shown later in the analysis section and it is the final piece needed to calculate the moment of inertia. Keep in mind that the acceleration is not actually negative as seen in the graph of the slope. It appears as negative because of the direction the rotary motion sensor spun but will be considered positive for our purposes. The rest of the values in the equation are measured using a vernier caliper or rulers as well as a scale to find the hanging mass.
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Basic Design for the Apparatus |
Data and Analysis
This section shows everything we have discussed so far with the final calculated values.
Conclusion
In this lab we have investigated moment of inertia and compared our calculated rotational inertia result to our geometric inertia value. We encountered various means of experimental error during this lab. As discussed above the wind resistance was a factor in impeding the acceleration of the pulley and that may have caused a slight amount of error to our calculation. Also if a smaller hanging mass is used causing a slower acceleration our answers would be more accurate. Another slight cause of our percent difference was that the masses were not all exactly equal and had we used the different masses paired with the radii they were at the geometric answer may have been closer to our calculated one. Overall we have successfully compared the different values with some accuracy. Reducing some of the errors in our experimental process may yield a closer result if the lab were to be recreated.
References
The Physics Classroom.com. Retrieved on April 22nd, 2014, from http://www.physicsclassroom.com