29-March-2017: Lab 8: Demonstration--Centripetal Acceleration vs. angular frequency

Eric Chong

Lab 8: Demonstration--Centripetal Acceleration vs. angular frequency

Lab Partners: Lynel Ornedo and Harvey Thai

Purpose: To determine the relationship between centripetal acceleration and angular speed

Introduction/Theory: Centripetal acceleration is acceleration that faces the center of uniform circular motion. Centripetal also means "center-seeking," so it is no wonder that the name fits this type of acceleration. Objects moving at constant speed can also be subject to acceleration, as seen in the case of circular motion. We can call the sum of forces that face towards the center of the circular motion the "centripetal force." In the centripetal force equation, the formula is related to the mass, velocity, and the radius of the circular motion. Here is the formula:

Fnet = (mv^2) / r

Since the velocity is part of the equation, it can also be said that angular frequency is related to the centripetal force. Thus, in this lab, we are finding the relationship between the centripetal acceleration, (v^2) / r, and the angular speed, ω. From the angular speed and velocity relationship, v = rω, we can derive an equation of centripetal force based on the angular speed. Here is the equation:

Fnet = mrω^2

As discussed, there seems to be a correlation with centripetal acceleration and angular speed, and after the completion of this lab, perhaps we will know more about this relationship.

Procedure: For setting up the apparatus, professor already did that for us. The set up consists of a giant disk with a force sensor placed close to the middle. A mass is tied to the force sensor, and the entire disk is run by a scooter motor, which makes the disk spin at a certain speed based on the voltage. There is also a photogate set up at the rim of the disk in order to measure the number of revolutions the mass would take. Here is what the setup looks like:

After setting up the force sensor to Logger Pro, we started with measuring the effects of changing mass. We first used a 300 g mass and tied it 47 cm from the center. The motor was set to "3" for its power setting. After starting the machine and waiting for it to go to constant velocity, we measured the force of the mass on the disk, and also measured the time it took to get to 10 rotations by setting an initial time and then looking at how much time has passed to get to 10 revolutions. Here is a sample picture of what the graphs looked like on Logger Pro:

(Note: This graph is the Force graph)

(Note: This graph displays the number of rotations in relation to time)

We made the radius and the angular velocity constant for this test. By just changing the mass, we can see the effects of mass on the centripetal force. After the measurements, we changed the mass to 200 g, 100g, and then 50 g and repeated the measurements.

For the next test, we made the mass constant at 200 g and the power setting constant at "3." In this case, we varied the radius, starting at 47 cm. from the center and varied it to 54.8 cm, 39 cm, and then 23 cm. We did the measurements the same way we did for the changing mass portion.

Finally for this part, we kept the mass constant at 200 g and the radius constant at 39 cm. We now vary the power setting of the motor, thereby changing the angular frequency. We started the power setting at "3" and then changed it to "3.4," "3.8," and "4.2." Again, we took the measurements the same way as the other measurements.

Here is a summary data table of the measurements we took:

After all of the measurements were taken, we graphed the force vs. mass graph (constant r and ω), the force vs. radius plot (constant m and ω), and the force vs. ω graph (constant m and r). Here are the graphs in order:

A quick note, there seems to be an error with one of the points of the force vs. radius graph. The point with the radius as 47 cm appears to be an outlier, and perhaps this has to do with the machine malfunctioning during the time we were recording the data, or perhaps we recorded the wrong measurements and these are not the actual values. Regardless, we do not know which value we mistook and we simply treated this point as an outlier. On the graph, this may have deviated our line downwards from its actual slope.

Conclusion: Based on the data taken and the plotted graphs, the first graph's slope was 4.115, and this is the same value as the product of the constants, the radius and the ω^2. We calculated the angular frequency from the period (we found the period by taking the difference of the final time and the initial time and dividing by 10), and used the relationship ω = 2π / period. From our calculations we found the angular frequency to be 4.406, which is somewhat close to what we got on the graph.

For the second graph, the slope is 0.01616, which is supposedly the same as the the product of mass and ω^2. However, calculating the product of those constants would be difficult, because the period seems to vary a lot in the data. All of the periods are different, and simply multiplying them with the constant mass would produce drastically different values each time. Hence, the second graph is flawed with errors, especially the mentioned outlier from before. The source of errors for this part could have possibly been from the motor, the changing inertia due to reducing the radius, and maybe some friction within the apparatus.

For the final graph, the slope is 0.09307, which is supposedly the value of the product of mass and radius. The calculated value is 0.078, which is somewhat close to the value based on the graph. Again, some sources of error for this graph could have been from the motor not running at the optimal speed, or friction that may have reduced the velocity of the spinning disk, thus reducing the angular frequency.

In conclusion, this experiment had much room for error, as seen in the second graph, and the error could have easily extended to the other two graphs as well. Nonetheless, this practical was still a success, in that we learned the relationship between centripetal acceleration and the angular frequency. A higher angular frequency would produce a higher centripetal acceleration, and also results in a higher centripetal force value. Furthermore, increasing the mass would also increase the centripetal force, as seen in the first graph and as supported by the equation mentioned in the introduction. Lastly, an increase in the radius would increase the centripetal force. This is also supported by the equation and the third graph, despite the error in the points and external sources. Through analyzing how centripetal force changes due to mass, radius, and angular frequency, we can support the claim that the force is equal to the quantity of mass, radius, and ω^2, furthering our understanding of how circular motion functions.