Abstract:

In this investigation, a mass was attached to a string and was swung horizontally for certain number of rotations (ten). The sole goal for this investigation was to find and verify the relationships between Centripetal Force, Frequency and Radius of circular path. In order to get relationships between the variables mentioned above, this experiment was divided into two parts. In Experiment A, the radius of the path (length of the string), along with the mass was kept constant, and the relation between centripetal force and square of frequency was determined. In Experiment B, the Centripetal force and the mass were kept constant, and the relationship between square of frequency and the Radius was determined. For Experiment A, the results showed that the relationship between the centripetal force and square frequency was linear, such that FC∝0.96f2. On the other hand, for experiment B, the results showed that the relationship between the radius and square of frequency is Inverse, such that 1r∝0.71f2. Purpose:

The purpose of this part in the investigation was to verify the relationships between frequency, centripetal force and radius of rotation for an object moving in a horizontal circular motion. Hypothesis:

Part A:

Assuming that the mass and the radius will be constant in this part of the experiment and considering the base theoretical expression FC=m4π2rf2, it can be stated that FC=kf2, where k is the combined value for all the constants in the expression. After analyzing the new equation, it can be stated that the centripetal force and square of frequency will have a linear relationship and will be directly proportional to each other. Moreover, to further classify the equation, value of constant should be found. An equation to represent constant would be k= m4π2r, after the mass and the radius is available, the value of k can be calculated using the equation mentioned. Part B:

Assuming that the centripetal force and the mass will be constant in this part of the experiment and rearranging the base theoretical expression FC=m4π2rf2, it can be stated that 1r=(m4π2FC)f2, where m4π2FC is the value for the constant. Therefore this equation can be stated as 1r=kf2. After analyzing the new equation, it can be stated that the rotational radius and the square of frequency will have a inverse relationship. Moreover, to further classify the equation, value of constant should be found. An equation to represent constant would be k= m4π2FC, after the mass and the centripetal force is available, the value of k can be calculated using the equation mentioned. Materials:

Apparatus| Uncertainty|

Newton Spring Scale| ±0.1N|

Meter Stick| ±0.1cm|

Mass Balance| ±0.01kg|

Stop Watch| ±0.1s|

Safety Goggles| -|

Fishing Line| -|

String| -|

Rubber Stopper with hole| -|

Diagram:

Procedure:

PART A: Frequency vs. Force

1. The experimental adjustment above was set up using the apparatus listed 2. The length of the string was set at 60.cm, which made up the radius of rotation 3. Then, the string was swung horizontally to practice it’s movement 4. Then the stopper was spun at a constant centripetal force of 1.0N for 10 rotations. The time period it took for 10 rotations was noted in the observation chart 5. Step 4 was repeated for centripetal force value starting at 0.5N, reaching 2.5N by increments of 0.5 6. The time interval it took for each trial was noted in the observations chart PART B: Frequency vs. Radius

1. The experimental adjustment above was set up using the apparatus listed 2. The radius of the string was set at 80cm

3. The stopper was rotated at a constant rate, so that it maintained a force of 1.0N 4. The stopper was rotated ten times, at a constant force of 1.0N 5. Step 3-4 were repeated for various radii values starting from 80cm and reducing to 30cm by increments of 10 6. The time interval it took for each trial was noted in the observations chart Observations:

PART A:

TABLE 1- Rotational frequency versus Centripetal force, with a constant mass of 23.2g Radius (cm) (±0.2cm)| Centripetal Force (N) (±0.2N)| Time for 10 rotations (s) (±0.1s)| Period (s) (±0.01s)| Frequency of (Hz) Rotation(±0.01Hz)| Square of Rotational Frequency (±0.01Hz2)| | | Trail 1| Trial 2| Average| | | |

60.0| 0.5| 7.4| 7.6| 7.5| 0.75| 1.33| 1.78|

60.0| 1.0| 6.5| 6.9| 6.7| 0.67| 1.49| 2.23|

60.0| 1.5| 6.3| 6.1| 6.2| 0.62| 1.61| 2.60|

60.0| 2.0| 5.5| 5.7| 5.6| 0.56| 1.79| 3.19|

60.0| 2.5| 5.1| 5.3| 5.2| 0.52| 1.92| 3.70|

* Qualitative Observations:

* Keeping the radius constant, it can be observed that as the Centripetal force increases, the square of frequency increases. Referring back to the hypothesis, the observations made verifies with the hypothesis. However, the square of frequency should be linear to the Centripetal force, and this will be confirmed graphically in the analysis section. PART B:

TABLE 2 – Rotational frequency versus Radius, with a constant mass of 23.2g Radius (cm) (±0.2cm)| Centripetal Force (N) (±0.2N)| Time for 10 rotations (s) (±0.1s)| Period (s) (±0.01s)| Frequency of (Hz) Rotation(±0.01Hz)| Square of Rotational Frequency (±0.01Hz2)| | | Trail 1| Trial 2| Average| | | |

80.0| 1.0| 6.8| 7.0| 6.9| 0.69| 1.45| 2.10|

70.0| 1.0| 6.6| 7.0| 6.8| 0.68| 1.47| 2.16|

60.0| 1.0| 6.5| 6.9| 6.7| 0.67| 1.49| 2.23|

50.0| 1.0| 6.1| 6.3| 6.2| 0.62| 1.61| 2.60|

40.0| 1.0| 5.8| 5.6| 5.7| 0.57| 1.75| 3.06|

30.0| 1.0| 5.4| 5.2| 5.3| 0.53| 1.89| 3.57|

* Qualitative Observations:

* Keeping the centripetal force constant, it can be clearly observed that as the radius of the path (length of string) decreases, the square of rotational frequency increases. Referring back to the hypothesis, it was stated that the square of frequency would be inversely proportional to the radius. However, this statement can be further proved by graphical analysis.

Analysis:

* Sample Calculations for Table 1, centripetal force of 0.5N: *

* average time=trial 1+trial 22

average time=7.4+7.62

average time=7.5 seconds

* period=average time for 10 rotations10

period= 7.510

period=0.75 seconds

* frequency=1period

frequency=10.75

frequency=1.33 Hz

* square of frequency=frequency2

square of frequency=1.78 Hz2

Graphical Analysis

PART A:

Graph 1: Square of Frequency vs. Centripetal Force, for a constant radius of 60.0cm

As clearly seen in Graph 1, Centripetal Force and Square of frequency are directly proportional and follows a linear relationship. As discussed earlier in the Observations, as the Centripetal Force increases, the square of frequency increases. However, this relationship is further proved using graphical techniques Referring back to the hypothesis, after rearranging the theoretical formula, it was supposed that FC=kf2, where k is the constant of proportionality. To reflect on the efficiency of the experiment, experimental and theoretical constant values must be determined. To find the Experimental Constant Value, slope of Graph 1 will be needed. To find the Theoretical Constant Value, the equation of constant found in the hypothesis will be needed. After both these values are found, they can be used to find the percentage difference. This percentage can be used to conclude the efficiency of our experiment and the depth of the errors made during the experiment. Table 3: Constant of Proportionality Values for Graph 1

Experimental Constant Value| Theoretical Constant Value|

slope=k= y2-y1x2-x1k=3.70-1.782.5-0.5k=0.96| k=m4π2rk=0.02324(3.14)2(0.6)k=0.55| Now that the experimental constant value and the theoretical constant value are obtained, the percent difference between the two constant values can be calculated. To calculate the percentage difference, the following formula can be used: %difference= |measurement 1 – measurement 2|(measurement 1+measurement 2)/2×100% %difference=|0.55-0.96|(0.55+0.96)/2×100

%difference=54%

The percent error of the constant term in this part of the experiment is 54%, the percent difference is this high because of several systematic errors performed during the experiment. These errors will be discussed in detail under the sources of error section of the lab report.

PART B:

Graph 2: Square of Frequency vs. Radius, for a constant centripetal force of 1.0N

As clearly seen in Graph 12, Radius of the path and Square of frequency are inversely proportional to each other. As discussed in the Observations, as the Radius of the path decreases, the square of frequency increases. However, this relationship is further proved using graphical analysis. From the shape of the graph of Square of Frequency (Y-axis) verses the Radius of the path (X-axis), it appears that f2 is inversely proportional to rn(f2∝1rn) . Firstly, we need to alter the x-values accordingly. In this case a graph of f2∝1r`1 should be plotted.

Table 4: Graphical Analysis for relationship between Square of Frequency and Radius r (±0.2cm)| r (±0.002m)| 1r (±0.01m)| f2 (±0.01Hz2)| 80.0| 0.800| 1.25| 2.10|

70.0| 0.700| 1.43| 2.16|

60.0| 0.600| 1.67| 2.23|

50.0| 0.500| 2.00| 2.60|

40.0| 0.400| 2.50| 3.06|

30.0| 0.300| 3.33| 3.57|

Graph 3: Square of Frequency versus 1/radius (Graphical Analysis of Graph 2)

After analyzing the structure of Graph 3, it can safely be stated that square of frequency and inverse of radius are directly proportional and follow a linear relationship(1r∝kf2). In order to evaluate this relationship, the constant of proportionality of Graph 3 must be determined. In order to find the constant of proportionality, the slope of the line of best fit must be used. To find the other constant value, which is the Theoretical Value, the equation of found in the hypothesis must be used. After both these values are determined, they can be used to find the percentage difference. This percentage can be used to conclude the efficiency for this part of the experiment. Table 5: Constant of Proportionality Values for Graph 3

Experimental Constant Value| Theoretical Constant Value|

slope=k= y2-y1x2-x1k=3.57-2.103.33-1.25k=0.71| k=m4π2FC k=40.0232(3.14)21k=0.91| Now that the experimental constant value and the theoretical constant value are obtained, the percent difference between the two constant values can be calculated. To calculate the percentage difference, the following formula can be used: %difference= |measurement 1 – measurement 2|(measurement 1+measurement 2)/2×100% %difference=|0.71-0.91|(0.71+0.91)/2×100

%difference=24%

For the second part of this experiment, the percentage difference between the experimental constant of proportionality and the theoretical constant of proportionality is 24%. There will be a detailed explanation about the possible systematic errors made during the experiment that caused such a high percent of difference. These errors will be discussed under the sources of error section of the lab report. Discussion:

Comparing experimental relations to theoretical predictions PART A:

In the hypothesis it was stated that the Centripetal Force and Square of Rotational Frequency will have a linear relationship and will be directly proportional to each other. This outcome was reached by analyzing all the constants in the theoretical formula and combining them as k. For this part of the experiment, the theoretical formula was rearranged as FC∝kf2, where k was a combined value for m4π2r. Furthermore, during the graphical analysis, it was found that the theoretical value for k is 0.55 kg∙m.

Therefore an equation to represent the theoretical relation between Centripetal Force and Square of Rotational Frequency would be FC=0.55f2. Moving on to the experimental relationships, according to the observation analysis and graphical analysis, it was confirmed that Centripetal Force and Square of Rotational Frequency had a linear relation and were directly proportional. However the two constants were not exactly the same, after examining the slope of the graph of this part, it was found that the experimental value for k was 0.96 kg∙m. Even though the experimental constant is not similar to the theoretical constant and has a difference of 56% (refer to graphical analysis for Part A), the graphical relationships were found as predicted initially. Therefore the experimental relationships agree with the theoretical predictions made during the hypothesis.

PART B:

In the hypothesis it was stated that the Radius of rotation and Square of Rotational Frequency will have an inverse relationship. This outcome was reached by analyzing all the constants in the theoretical formula and combining them as k. For this part of the experiment, the theoretical formula was rearranged as 1r∝kf2, where k was a combined value for m4π2FC. Furthermore, during the graphical analysis, it was found that the theoretical value for k is 0.91 kg∙m. Therefore an equation to represent the theoretical relation between the radius and the Square of Rotational Frequency would be 1r=0.91f2. Moving on to the experimental relationships, according to the observation analysis and graphical analysis, it was confirmed that the radius and Square of Rotational Frequency had a inverse relation. However the two constants were not exactly the same, after examining the slope of the graph of this part, it was found that the experimental value for k was 0.71 kg∙m. Even though the experimental constant is not similar to the theoretical constant and has a difference of 24% (refer to graphical analysis for Part B), the graphical relationships were found as predicted initially. Therefore the experimental relationships agree with the theoretical predictions made during the hypothesis. Conclusion:

PART A:

The Hypothesis stated that “the Centripetal Force and Square of Rotational Frequency will have a linear relationship and will be directly proportional to each other and could be represented by the equation FC∝0.55f2”. The observations showed that the same graphical relations but was represented by the equation FC∝0.96f2. Since the experimental relationship shows the same relationship as theoretical predictions, and the constant values are not varied by a big margin. It is safe to conclude that the hypothesis was accepted. PART B:

The Hypothesis stated that “the radius of rotation and Square of Rotational Frequency will have an inverse relationship and could be represented by the equation 1r∝0.91f2”. The observations showed that the same graphical relations but the constant was a little varied and the graph was represented by the equation 1r∝0.71f2. Since the experimental relationship shows the same relationship as theoretical predictions, and the constant values are not varied by a big margin. It is safe to conclude that the hypothesis was accepted.

Sources of Error:

During any lab, there are certain sources of errors. Similarly, even in this lab, there were several systematic errors that caused inaccuracies during the results. These results include: * Irregular Circular Movement: during the experimental procedure, a mass had to be attached to a string and it was to be rotated horizontally above the performers head. During this part, the movement of the string was not perfectly horizontal and had some meander patterns. This error could’ve increased the time interval it took to complete 10 rotations and must’ve varied our observations. This error can be controlled by being more attentive while performing rotating the string and practicing the movement before performing the experiment. * Centripetal Force not Constant: during the experimental procedure, a Newton spring scale was attached to the string. A certain value on the Newton Spring scale was required on each trial, but however due to irregularities in the path (discussed in point 1); the centripetal force values were not perfectly constant throughout the experiment. This could’ve altered the time interval as well which would have eventually affected the frequency value.

This error cannot be completely perfected but can be improved. Attempting a few trials before the actual experiment can be helpful in order to cause a reduction in this error. * Mass of String: during the experiment, a rubber stopper was attached to a string and that string was attached to a Newton. The force value showed on the spring scale reflected the mass of the rubber stopper and the mass of the string, however in our observations and analysis, only mass of the stopper was used and mass of the string was assumed zero. Even though the value of the string was not that big, but it could’ve caused some alterations in the lab results and certainly is a systematic error. This error can be improved by measuring the mass of the string used and plugging it in the theoretical formula. By improving all the systematic errors stated above, there is a much higher chance of getting the experimental constant to be closer to the theoretical constant value. This way, the percent difference would be reduced (got closer to 0%) and the experimental relationships would agree with the theoretical values a lot more.

References:

* http://www.calculatorsoup.com/calculators/algebra/percentagedifference.php * http://www.physicsforums.com/showthread.php?t=322249

* http://www.slideshare.net/wsautter/circular-motion-3163645 * http://aplusphysics.com/courses/regents/circmotion/regents-circular-motion.html