Lab 8
Centripetal Acceleration vs. Angular Acceleration
3-27-17
Ricardo Gonzalez
Josue Luna
The goal of this lab is to find a direct relationship between the angular speed and the centripetal force. Angular speed represented as (⍵) can be related to the Force Centripetal.
1. Introduction/Theory
For our experiment, we will find a direct relationship between Angular speed and Force centripetal. To do so, we have to understand what is happening. As a surface spins with a mass placed on it, the mass experiences a few forces. As the mass is spun at a constant angular speed, the force wanting to pull a mass our of its surface is due to a Force centripetal. When we tie a string from the center of the surface to the mass and increase the angular speed, the force wanting to pull the mass out of the surface is going to be equal to the tension in the string. This is best shown by a free body diagram below.
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| Free body diagram for a mass as the surface rotates with a constant angular speed. |
We know that as long as the surface is rotating only in the horizontal direction, the only forces acting on the mass is its weight due to gravity and the normal from the surface to the mass. Using Newtons Second law, the F(centripetal)=ma(centripetal). We also know that the acceleration centripetally is also equal to the radius times angular speed squared (ac=rw^2). By substituting rw^2 for the a(centripetal) in Newtons second law, we get an equation of F(centripetal)=mrw^2. Assuming the radius and mass are held constant, we are left with an equation that, in hopes, relates angular speed and centripetal force.
2. Procedure
By using a power supply generator and a previously constructed apparatus in which a circular surface is spun using a series of wheels and belts that is powered by a power generator, we are able to simulate a rotating disk at constant angular speed. By using the logger pro software, we zero-d the forces both vertically and horizontally. On top of the surface, we tightened down the force sensor and taped a scale onto the surface. We also placed a piece of tape at the edge of the surface in such a way that by using a photogate, we would be able to measure the period of one revolution. The photo gate was connected and monitored as well on the the logger pro software. One end of a string was tied to the force sensor and the other end was tied to the mass with a radius that would vary and is measurable using the taped scale on the surface.
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| A 2-D representation of our experiment |
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We measured our radius in inches and then converted those measurements to meters in order for the measurements to be in SI units.
The period was measured by taking the average of 10 revolutions.
To calculate W^2, we set the equation W^=(2*pi/T)^2 where T is our period for one revolution.
4. Results
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| A graph of the Force vs Mass |
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| A graph of the Force vs W^2 (angular speed) |
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| A graph of Force vs Radius |
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| The equations solved for F/M= RW^2 and F/R=MW^2 |
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| The equation solved for F/W^2=MR |
5. Analysis
To prove the mass is held as a constant, we want to graph the Force vs mass. To do so, we then assume the angular speed and radius are held constant. By doing so, we can graph the trials that contain the same angular speed (the same power supply) and radius and varying masses. The graph as shown in the photo above, shoes a linear slope of 8.937 N/kg, and a correlation of 0.994. When we mathematically calculate the right side of the equation Force/Mass = Radius*Angular spped squared, we get 9.424 m*rad^2/s^2. It is obvious that 8.937 does not equal to 9.424. However the correlation of the results shown in the graph are very close to 1 and thus proves that as the mass increases, so does the force and so we can then say the mass can be held as a constant.
To prove the radius is constant, we want to graph the Force vs radius where the radius is changing. Similarly, the trials we will use will will be when the mass is held constant and the power supply is held constant, and thus assuming the angular speed is constant. By analyzing the graph, we can see the slope of the points is 2.649 N/m and a correlation of 0.981. When we solved the right hand side of the equation Force/Radius=Mass*Angular Speed squared, as shown in the photo above, we get an answer of 6.11 kg*rad^2/s^2. The slope of the line of the graph should be equal to the right hand side of the equation, but is once again not the case. With a correlation close to 1, we can then assume the radius can be held as a constant.
Lastly, to prove the angular speed is directly related to the Force applied when assuming the mass and radius are held constant, we graph of Force vs W^2. The trials used were those that contained the same mass and same radius from the center and the trials where the power supply changes. As we analyze the graph, we can see that the slope of the points gives us 0.128 N*s^2/rad^2 and a correlation of 0.987. When we solve the right hand side of the equation Force/Angular speed squared=Mass*Radius, we get an answer of 0.141 kg*m. For the first time we have found that the slope of the graph is equal to the right hind side of the equation, with a very minimal amount of error. This alone tells us that the Force is directly related to the angular speed squared.
6. Conclusion
To prove the mass is held as a constant, we want to graph the Force vs mass. To do so, we then assume the angular speed and radius are held constant. By doing so, we can graph the trials that contain the same angular speed (the same power supply) and radius and varying masses. The graph as shown in the photo above, shoes a linear slope of 8.937 N/kg, and a correlation of 0.994. When we mathematically calculate the right side of the equation Force/Mass = Radius*Angular spped squared, we get 9.424 m*rad^2/s^2. It is obvious that 8.937 does not equal to 9.424. However the correlation of the results shown in the graph are very close to 1 and thus proves that as the mass increases, so does the force and so we can then say the mass can be held as a constant.
To prove the radius is constant, we want to graph the Force vs radius where the radius is changing. Similarly, the trials we will use will will be when the mass is held constant and the power supply is held constant, and thus assuming the angular speed is constant. By analyzing the graph, we can see the slope of the points is 2.649 N/m and a correlation of 0.981. When we solved the right hand side of the equation Force/Radius=Mass*Angular Speed squared, as shown in the photo above, we get an answer of 6.11 kg*rad^2/s^2. The slope of the line of the graph should be equal to the right hand side of the equation, but is once again not the case. With a correlation close to 1, we can then assume the radius can be held as a constant.
Lastly, to prove the angular speed is directly related to the Force applied when assuming the mass and radius are held constant, we graph of Force vs W^2. The trials used were those that contained the same mass and same radius from the center and the trials where the power supply changes. As we analyze the graph, we can see that the slope of the points gives us 0.128 N*s^2/rad^2 and a correlation of 0.987. When we solve the right hand side of the equation Force/Angular speed squared=Mass*Radius, we get an answer of 0.141 kg*m. For the first time we have found that the slope of the graph is equal to the right hind side of the equation, with a very minimal amount of error. This alone tells us that the Force is directly related to the angular speed squared.
6. Conclusion
In conclusion, we have found that when holding the mass and radius constant, increasing the angular speed by changing the power supply, the both sides of the equation F/W^2=MR is held true. This is because when we set the R&W^2 and M&W^2, the angular speed was not the same any given instance in either scenarios. Only when the Mass and radius were held constant did both sides of the equations very closely equal one another. The apparatus used was not by any means perfect and thus has created deviations within the periods of all trials.
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