Parallel axis theorem report PDF

Preview

Summary

Professor: Sachin Sachin...

Description

Parallel Axis Theorem Abstract: In the Parallel Axis Theorem experiment we analyzed how the moment of inertia is affected by the angular acceleration. This was demonstrated with a 3-step pulley system, which consisted of attaching a hanging mass to a string connected to a rotating disk. We then released the mass, pulling the string down and which in turn rotates the disk. There also were two adjustable cylinder blocks on a rod in which we would change the distance of. We used the equation I =

1 12

(mrlr 2) +

1 2

(mcRc2) +

1 4

(mclr2) +

1 4

(mc 2lr2) to represent the total moment of

inertia. When comparing our data to the model data to the model data it’s evident our results weren’t as accurate as they could be. We also were unable to calculate precision due to the lack of trials within the experiment. Introduction: The purpose of this lab is to exhibit how the distribution of mass affects the moment of inertia and to deepen our comprehension of the connection between total net force applied on a system and angular acceleration. To conduct our experiment we utilized a pulley system that contained a disk connected to a string with a hanging mass on it which was used to find the angular acceleration of each of the chosen masses. On the pulley there were also two cylinder blocks that were adjustable on a rod for us to find the angular acceleration at a chosen distance as well. By changing the hanging mass (m) as well as the distance from the pulley ( r ) the moment of inertia (I) ultimately changes. Since the rod is associated with the pulley it will pivot around an axis perpendicular to the axis of symmetry of the rod. The focal point of mass is amidst the middle of the rod and is also along the axis of rotation. In these trials we can utilize the Newton’s second law for force which yields the equation 𝐹𝑛𝑒𝑡 = 𝑚 * 𝑎 , where 𝐹𝑛𝑒𝑡 is the net force, m is the mass, and a is its acceleration. We additionally can use the equation for net torque in angular motion which is τ𝑛𝑒𝑡 = 𝐼 * α where 2

τ𝑛𝑒𝑡 is the net torque, 𝐼 is the moment of inertia (to obtain 𝐼 use the equation 𝐼 = 𝑚𝑟 ) and αis the angular acceleration. We can combine the equation given by Newton’s second law for force (𝐹𝑛𝑒𝑡 = 𝑚 * 𝑎 ) with Newton's law for mass (𝐹 = 𝑚ℎ𝑔 − 𝑇)to get 𝑚 ℎ𝑎 = 𝑚ℎ𝑔 − 𝑇. Since we want to solve for torque we can isolate torque and get the equation of τ = 𝑚 ℎ(𝑔 − 𝑎). Lastly we need to include the equation τ = 𝑅

𝑝

𝑇 since it gives the force acting on the pulley.

We can substitute it for tension into the equation giving us the final formula of of τ = 𝑅 𝑝𝑚 ℎ(𝑔 − 𝑎). In this experiment we also make the assumptions as follows; the brass cylinders are rotating with the rod instead of their own centers of mass, friction and air resistance are negligible, the cylinder is solid, the rope has negligible mass, and the string is inextensible. Theory: To determine the total theoretical moment of inertia expression we will consider the moment of inertia with different centers of mass. The first would be to consider a thin rod rotating around axis C and the moment of inertia of the rod is described as Icm =

1 (mrlr 2) 12

where

mr is the mass of the rod, lr is the length of the rod, and “Cm” is the center of mass. Next, the moment of inertia of a cylinder rotating around its own axis is represented by

1 (mala 2) 12

+

1 4

(maRa2) and ma is the mass of the cylinder, Ra is the radius of the cylinder, and la is the length of the cylinder. Since there are two cylinders you can multiply this equation by two to find the moment of inertia of both cylinders combined. Since the moment for inertia of the brass cylinders are rotating about an axis C, the axis of rotation for the rod system is parallel to the axis of rotation of each cylinder about its own center of mass. This means we can use the equation given by the parallel axis theorem. The parallel axis theorem states that given a mass M, which is rotating about an axis parallel to its center of mass, the moment of inertia of the system will be equal to the sum of the moment of inertia of the system about the center-of-mass axis plus the moment of inertia of a point particle of mass M located a distance d from the axis of rotation. This is represented mathematically by the equation: Isys = Icm + Md2 . We also will use 2 equations to help us calculate the total moment of inertia for the system which is represented by I =

1 (mrlr 2) 12

+

1 2

(mcRc2) +

1 (mclr 2) 4

+

1 4

(mc 2lr2) where mc2 is the

mass of the second cylinder. One of the equations we first would need is to find the moment of inertia of a system consisting of the rod and one cylinder placed at the center of mass of the rod when the system is spinning about axis C. This equation is given by I =

1 (mrlr 2) 12

+

1 (mcRc2) 2

where mr is the mass of the rod, lr is the length of the rod, mc is the mass of the cylinder, and Rc is the radius of the cylinder. Then the second equation needed is I =

1 (mrlr 2) 12

+

1 2 2 (mcRc )

+

1 4

(mclr2) which represents the moment of inertia of a system consisting of the rod and one cylinder placed a distance d from the center of mass of the rod when the system is spinning about axis C. Procedure: ● Construct an apparatus including a 3-step pulley and a rod consisting of two brass cylinders that rotate about an axis.

● Record the masses of both the cylinders and the rod. ● Measure the radii and the distance between both cylinders. ● Attach the PASCO rotary motion sensors to the pulley set-up and computer in order to accurately record your data. ● Using the formula I =

●

● ● ● ● ● ●

● ●

1 12

(mrlr 2) +

1 2

(mcRc2) +

1 4

(mclr2) +

1 4

(mc 2lr2) , which yields

moment of inertia values, create a model graph of distance squared vs. the moment of inertia. This will be used to compare how accurate/precise your data is. Before starting each trial ensure that the string is completely wrapped around the disks radius (so when you release it the driving mass pulls the string downward, completely unwinding the string) Attach 5 g of mass to the end of the string. This will be used as the hanging mass and it will remain constant throughout the trials conducted. For the experimental values, begin with the rods at a distance of .03m. This is the measurement from the center of rods to the cylinder. Press record on the PASCO software and release the hanging mass. The string should be pulled downward at this point and the rod should be spinning. Using the slope of the angular velocity, determine the angular acceleration at this distance for the cylinder. Use the values of the angular acceleration then to find the moment of inertia for each value. Repeat the same procedures for one trial at each distance of .07m, .05m, .025m, .17m, .12 m, .04 m, .15m, .14m, and .16m using the rotary motion sensor to find the angular acceleration Using the experimental values, construct a graph the distance squared vs. the moment of inertia Compare the model vs. experimental graphs.

Data: Distance (m)

Mass (g)

Angular acceleration (rad/sec^2)

0.025

5

6.25

0.03

5

5.59

0.04

5

4.92

0.05

5

4.03

0.07

5

2.71

0.12

5

1.32

0.14

5

1.06

0.15

5

0.953

0.16

5

0.836

0.17

5

0.761

Graph of distance vs. moment of inertia:

Analysis: Accuracy within data is defined as as the “correctness” of a measurement in comparison to its reference value. As you can tell based off our graph, our results aren’t accurate in comparison to the model. The slope of the model data is much steeper than the slope of our experimental. Although, when analyzing the calculated angular acceleration you can see how everytime the distance grows larger the angular acceleration decreases. This shows a progressive trend within our data which could mean we were on track with our results. Also, although our experimental data slope is much less steep, we can see that both slopes are increasing. This means that the greater the distance, the less the angular acceleration which in turn means there is more moment of inertia. Precision is different from accuracy in that it refers to the narrowness or broadness of how well we know a measurement. Since we only conducted 1 trial at different distanced with the hanging mass of 5 g, we can’t deviate how precise our measurements are. If we conducted more trials, such as 5, we could analyze the difference between our gathered results and then we would have the ability to look at the precision of the results. Conclusion:

The purpose of this lab is to exhibit how distribution of mass affects the moment of inertia and to deepen our comprehension of the relationship between total net force applied on a system and angular acceleration. We measured the angular acceleration after releasing a hanging mass attached to a string on a pulley. Then we used the angular acceleration to evaluate the moment of inertia. By looking at the graphs one can come to the conclusion that since as the distance increases the angular acceleration decreases which in turn makes the moment of inertia increase. Our results weren’t accurate since when comparing our experimental data to the model data our slope wasn’t steep enough (the moment of inertia’s varied). We also couldn’t gather information regarding how precise our results were since we only conducted 1 trial at each distance. Inaccuracies and precision could’ve been due to several factors including how we only conducted 1 trial. A systematic limitation we found was that we couldn’t move the cylinder blocks between the distance of 0.07-0.12 m because it would hit the wheel which in turn didn’t allow the wheel to rotate. Therefor, we couldn’t gather any data between the points of 0.07-0.12 m. Another systematic limitation regarded the PASCO system occasionally would freeze, messing up our data, so we would have to repeat trials until we ensured it was working properly. Lastly, we only chose 5 g as our hanging mass which could have affected our data. If we changed the mass or chose a greater mass (for example 50 g) our data could’ve varied. To improve the experiment, we could do 5 trials instead of 1. We then would have gathered more data and compared how much they vary to see if our results were precise. Also, we could have more advanced equipment this could improve our issue of not being able to gather data between 0.07-0.12 m as well as the freezing of the PASCO software. With that being said, one piece of apparatus that could be improved would be the height of the rod. It should be placed higher so we could record data within those ranges and not have it hit the rotating wheel. Lastly, we could have changed the mass if we did 5 trials at a specific distance. For example, at the distance of 0.025 m we could’ve attached 5 g the first trial and the increased the mass to 10, 15, etc....