Part 1: Relationships
To perform this lab, we set up rotating system that would comprise of two disks, a pulley, and a hanging mass. The idea was to attach the mass to the system by a string, have it attached some radius from the center of the disks, and allow it to fall. This would in turn cause the system to accelerate. We would then measure this in LoggerPro and record several trials with different hanging masses, disk masses, and rotating radii. (see Fig. 1).
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| Fig. 1 The apparatus consisted of a hanging mass, rotating disks, an air hose connected to an air valve and a rotational motion sensor. |
To begin, we first had to measure all of the equipment we were planning on using. The materials included:
- Two steel disks (one for the top and another for the bottom)
- An aluminum disk (used to change the mass of the rotating disk)
- A small and large torque pulley (used to find the effect of changing the radius at which the force acts on the object)
- A hanging mass (used to cause the system to begin rotating)
Using calipers, we measured the diameter of the two steel disks, aluminum disk, and torque pulleys. (see Fig. 2). We then found the mass of the two steel disks, aluminum disk, torque pulley and hanging mass. (see Fig. 3) We recorded all this data on our white board.
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| Fig. 2 Using calipers, we were able to gain precise measurements of each disk. |
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| Fig. 3 Data Table |
As seen in Fig. 1, the apparatus is hooked up to our laptop, allowing us access to data that LoggerPro could extract for us. The idea is that the black and white dashes around the disk could be counted while moving by some external sensor on the rotational apparatus. This would allow us to find angular velocity and acceleration. To properly set up the sensor, we had to input how many marks there were on the top disk (200). Once we had this, we were ready to begin collecting data.
We tied the first mass (24.7 g) by string to the small torque pulley (r = 1.26 cm) atop the steel disk (1348 g). We began collecting data with LoggerrPro as we released the mass and allowed it to accelerate down. We collected data as the mass dropped and as it rose again, giving us an angular acceleration down (cw) and up (ccw). (see Fig. 4) However, we had to note that there was some friction within the system. The apparatus allowed for a virtually "frictionless" surface for the disk(s) to rotate, however in reality this was not true. Thus, in order to counter the effects of friction, we would simply take the average angular acceleration. This average acceleration is what we would use when analyzing and evaluating our data.
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| Fig. 4 The spike at the top of this graph indicates the turn around point of the hanging mass. To find the angular acceleration, we had to take the slope of this graph. |
We ran the same experiment six times, with some slight changes each time. To find the effect of different tension pulling on the system, we ran the first three trials with the same small torque pulley, same top steel disk, but instead added additional mass to the hanging mass each time. To find the effect on using a different rotating disk on the system, we ran the next two trials with a large torque pulley, but with the top steel disk in one trial and the top aluminum disk in another. Finally, for the final trial, we used the large torque pulley and allowed the top and bottom steel disks to rotate. We received data through LoggerPro (see Fig. 5) and recorded it (see Fig. 6).
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| Fig. 5 Here are the graphs for all six of our trials. We took the slope of one only as an example. |
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| Fig. 6 |
NOTE: During the data collection, we made an error when calibrating the machine to 200 marks per rotation. Our graphs represent angular accelerations that are slightly off. However, to correct the issue, we multiplied our values for angular acceleration by 360/200.
With all this data, we drew a couple of conclusions. First, we found that when we almost doubled the hanging mass, we almost doubled the angular acceleration. Same went for tripling the mass. Second, we found that when we used the large torque pulley rather than the small torque pulley (doubling the radius at which the tension acted on the disk) we also doubled the angular acceleration. Third, we found that when we decreased the mass of the rotating disk three-fold (using the aluminum disk rather than the steel disk) we INCREASED the angular acceleration roughly three times. Fourth, we found that when we doubled the rotating mass (allowed two steel disks to rotate), we cut the angular acceleration in half.
The final step in part one of this lab was to find the relationship between the linear velocity of the hanging mass and the rotational velocity of the system. To do this, we simply ran one run the exact same way as we did the other six trials. We used a large torque pulley for the tension to act on. The mass of the disks spinning and mass falling become irrelevant, as we are simply analyzing velocity in the angular and linear sense. Using a motion sensor and note card, we began collecting data with logger pro, in both rad/s and m/s. When we graphed linear velocity in the y-axis and rotational velocity in the x-axis, we obtain the graph in Fig. 7. When we took the slope of this graph (positive or negative), we obtained a value of 0.02513 m which translates into 2.513 cm, which is practically exactly the radius of the large torque pulley.
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| Fig. 7 We found that the slope of the linear vs angular velocity was exactly the radius of the large torque pulley. |
Conclusions for Part One:
For part one of this lab, we were able to verify the magnitude of the hanging mass and the radius at which a force was acting on a disk were directly related to the angular acceleration of the system. This meant that if we increased the hanging mass, we increased the angular acceleration. (same for radius). We were also able to verify that if we increased the mass that was rotating, we decreased the angular acceleration. All of these relationships were numerically proportional (ie. if you double the hanging mass, you double the angular acceleration and if you tripled the mass that was rotating, you cut the angular acceleration into 1/3 of what it used to be). As for linear and angular velocity, we found that they were related by the radius at which the force acting on them was being applied. Through this, we were able to verify the relationship v=rω.
Part 2: Moment of Inertia of Each Disk:
For the second part of the lab, we had to use our tools to find the moment of inertia of the single top steel rotating disk, the two steel rotating disks combined, and the top aluminum disk. To do this we had to derive an expression for the moment of inertia of this system using the force and torque equations. We are finally left with the equation I=(mgr/α)-mr^2 To verify if this is true, we used the given equation for the moment of inertia of rotating disk as our "actual" value. I=(1/2)MR^2 Our calculations can be seen in Fig. 9
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| Fig. 9 |
In all, we found that we had a percent error of less than 5% for the single steel disk, both steel disks, and aluminum disk. This indicated that our data and formula were good.
Conclusion:In all, the lab was a success and allowed us to find the multiple relationships between angular acceleration and the factors that change it. We were also able to verify our equation for the moment of inertia of a rotating solid disk with our data. However, we had to account for a few areas of uncertainty. This included all our areas of measurements as well as the frictional torque we did not account for. We assumed that averaging the angular acceleration would completely rid the friction, which is not entirely true. However, in all we were able to prove our moment of inertia and relationships.
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