Circular Motion - The TA\'s name was Michael Schott. This is a full lab report. PDF

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The TA's name was Michael Schott. This is a full lab report....

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Woods

Circular Motion Lab Report

Dana Woods

Lab Partner: Katherine Andersh Course: PHYS181-015 TA: Michael Schott Due Date: 5:00 PM on 10/27/16

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Abstract The overall goal of this experiment was to determine how different variables affected the tension of a string attached to a mass as this mass rotated in a circle. Specifically, the three variables tested were angular velocity, mass, and radius. It was observed that as each of these variables increased, the tension in the string also increased. This is seen by the positive, linear slopes of our graphs. When the angular velocity, mass, and radius were altered, the slopes (which represented the product of the other two variables that remained constant) were 0.0435, 4.1071, and 6.1045, respectively. The trends seen are consistent with what is known about how these variables should affect one another. Also, because there is a known value for each of these trials, a percent error calculation can be done to determine how far off our actual results were from what they theoretically should be.

Introduction An object is undergoing circular motion when it is moving in a perfect circle and is able to remain in that circle. The “force” that keeps the object moving in a circle is termed centripetal force. To be more accurate, this “force” is determined by the mass of the object and its acceleration, specifically, its centripetal acceleration. No matter where in the circle the object is, it will always be accelerating towards the center. This is also the case even when the object has a constant velocity (angular velocity for circular motion). This is because there is acceleration when velocity changes its magnitude or direction. Therefore, even if the magnitude of the angular velocity remains constant, the direction is constantly changing, and gives the object an acceleration. Newton’s second law can be used to relate the force, mass, and acceleration of an object traveling in a circle. For circular motion, the mass of the rotating object, the radius of the circle, and the angular velocity squared are all directly proportional to the net force. By altering only one variable at a time, we were able to verify that this equation pertained to the experiment, and the relationship between the variables is coherent with what Newton’s second law states.

Theory Circular motion still follows the same principle as linear motion with regards to Newton’s second law. Newton’s second law states that the net force is equal to the mass of the system times the acceleration of the system. In the case of an object traveling in a circle, the acceleration is pointing towards the center of the circle and is known as centripetal acceleration. The centripetal acceleration is equal to the tangential velocity squared divided by the radius, and can be shown by the equation (1) where Fnet is the sum of the net forces applied to the system, m is the mass, ac is the centripetal acceleration, v is the tangential velocity, and r is the radius. In this case, the only force acting on the mass (other than normal force and weight, which cancel out) is tension, which is why in the equation above, tension is shown to equal the product of the mass and the centripetal acceleration. For this lab, however, we are looking at angular velocity, not tangential velocity, because of the equipment we have available. The angular velocity equation is

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(2) where w is the angular velocity. Because we are not measuring tangential velocity in this lab, this can be solved for in Equation 2. Then, this can be combined with Equation 1 to get (3) to give us the relationship between the tension on the system, the mass of the system, the radius of the circle, and the angular velocity of the system. These variables, as previously explained, can be changed to determine how the tension changes in response. After the experimental values are determined, the theoretical values can also be calculated to determine the percent error in our calculations. This can be done using the equation (4)

in order to determine how close the values we obtained from the experiment were to the actual calculated value.

Procedure

Figure 1. Picture of apparatus, showing the force and rotary motion sensors, the mass and its counterweight, the string connected to the mass, and the rotating arm.

Figure 1 shows the apparatus for the experiment, which remained the same during all three parts. The force sensor was used to determine the force, in this case tension, acting on the

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Woods system and the rotary motion sensor was used to find the angular velocity of the system as it rotated. In part one of the lab, the relationship between tension and a changing angular velocity was determined. This means that the mass and the radius were kept constant. The mass was determined to be 0.25105 kg and the radius was set to 0.16 m. The counterweight was placed at exactly the same radius on the opposing side of the rotating arm to balance the apparatus as it rotated. For each part of this lab, the PASCO Capstone computer system was used to record the data. The rotating arm was spun until its angular velocity was at least 10 rad/s. Once we hit record on the computer, it graphed an angular velocity vs. tension graph. The data was recorded until the rotating arm came to a stop. Because the tension is directly proportional to the angular velocity squared, as seen in Equation 3, a tension as a function of angular velocity squared graph was made to analyze the data. Part two required us to change the mass of the system, each time adding approximately 0.05 kg to the slider. This gave us a total of five different masses. The radius remained constant at 0.16 m, and the angular velocity we decided to look at was 5 rad/s (the closest value to this was 4.987 rad/s, so this is the angular velocity we ended up basing our calculations on, and will be referred to this number from now on). The rotating arm was spun until it was at least 4.987 rad/s, and then data was recorded. After the data points passed 4.987 rad/s on the angular velocity axis, the recording was stopped, and the tension at the 4.987 rad/s mark was determined. A graph of tension as a function of mass was created for data analysis. For part three, the radius varied, while the mass and angular velocity remained the same. It was determined that the largest radius we could go while keeping the entire mass on the rotating arm was 0.22 m. Because we were required to do 5 different radii, we divided 0.22 m by 5 and got the radii to be 0.044 m, 0.088 m, 0.132 m, 0.176 m, and 0.22 m. The mass was kept at 0.25105 kg and the angular velocity we looked at was 4.987 rad/s. The rotating arm was, again, spun until it was going at least 4.987 rad/s and the data started to be recorded. After the data points passed 4.987 rad/s, the recording was stopped, and the tension was found at that angular velocity. Tension as a function of the radius was graphed to analyze the data.

Sample Calculation and Results For each part of the experiment, Equation 3 was used to determine the relationship between tension and the variable that was altered. However, this equation was not directly used when analyzing the data because the x-axis and y-axis of each graph as well as the slope gave us the numbers for the values seen in the equation. It was used, however, to calculate what our values should have been and then utilized for a percent error calculation. In part one, the angular velocity was changed while the radius of the circle and the mass of the object remained constant. This data is seen in Graph 1. For this graph, the equation can be manipulated to determine what the slope represents. T =mr=0.0435 kg ∙ m 2 ω

(5)

This equation says that the slope for Graph 1, which represents the mass multiplied by the radius, is 0.0435 kg m. The trend seen by the graph is that as the angular velocity increases, the tension in the string increases.

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Woods For the second part of the experiment, the mass varied and the radius and angular velocity of the circle remained the same for each trial. The data for the five different masses can be seen in Table 1 and Graph 2. The slope is represented by the equation

( )

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T rad =ω2 r =4.1071 ∙m s m

(6)

which shows that the slope of 4.1071 (rad/s)2 m is equivalent to the angular velocity squared times the radius of the circle. The trend line in Graph 2 indicates that as mass is increased, the tension also increases. For the final part of lab, the radius was altered while the mass as well as the angular velocity were constant for each trial. These results can be seen in Table 2 and Graph 3 for the five radius measurements. Again, Equation 3 can be rearranged to become

( )

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T rad ∙ kg =ω 2 m=6.1045 r s

(7)

and therefore shows what the slope signifies. According to this equation, the mass multiplied by the square of the angular velocity is equal to 6.1045 (rad/s)2 kg. The trend for this graph is that as the radius increases, the tension of the string also increases. Equations 5-7 above use the slopes given in the graphs. However, these are not the theoretical values, which is due to the errors explained later. Because we are able to calculate a theoretical value for each of these slopes, a percent error calculation can be done for each graph, and can be seen in the caption of each graph. For an example, the percent error for tension as a function of radius will be calculated. First, the theoretical slope had to be calculated by multiplying the square of the angular velocity by the mass. ω2 m=6.24

( )

2

rad ∙ kg s

(8)

This theoretical value, along with our experimental value (the slope of Graph 3) can be used to determine the percent error. Value−Experimental Value |TheoreticalTheoretical |×100= 2.23% value

% Error=

(9)

This calculation indicates that there is a 2.23% error with our slope compared to the calculated slope.

Discussion and Conclusions The goal of this experiment was to use the apparatus shown in Figure 1 to determine the relationship between the tension in the string, the angular velocity of the mass, the radius of the circle, and the mass itself, which was undergoing circular motion. By looking at Equation 3, it can be seen that tension is directly proportional to the mass, the radius, and angular velocity

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Woods squared. The results we obtained followed this equation, as shown in Graphs 1-3 with their positive slopes. As stated previously, an object traveling in a circle has an acceleration, specifically centripetal acceleration, that points towards the center of the circle. This can be seen specifically for our experiment in Figure 2. This free body diagram shows all the forces acting on the mass. Mg comes from the weight of the mass, n is the normal force of the rotating arm on the mass, and T is the tension from the string attached to the mass. The normal force and the weight are the same size and cancel out. This can be explained by the fact that the mass never moves in the vertical direction. That just leaves the tension force. By using the logic of Newton’s second law, any time the net force in one direction is not zero, it must be accelerating in that direction. Because tension is the only force in the horizontal direction, this must mean there is an acceleration. This force is also pointing towards the center of the circle, so therefore it must contribute Figure 2. Free body diagram of the to the centripetal acceleration of the mass. mass on the rotating arm. The centripetal acceleration of our mass was put into terms that we could easy calculate using the apparatus, as seen in the theory section. The first part of the lab required us to look at tension as the angular velocity changed. The slope obtained for this graph was 0.0435 kg m. During part two, it was observed how a change in mass affected the tension of the string. The slope of this graphed data set was 4.1071 (rad/s)2 m. Finally, we had to vary the radius of the circle. This slope turned out to be 6.1045 (rad/s)2 kg. Even though the trends for our data sets were accurate according to the equation, there was still some error when we calculated what the slopes should be. The percent error for the change in angular velocity, the change in mass, and the change in radius were 8.30%, 3.21%, and 2.23%, respectively. These percent error values are not extremely high, which is good because there was little error in our data. However, the error that is seen is still relevant, and the possible reasons for this are discussed shortly. One of the errors in this lab could have been due to not measuring the radius of the circle correctly. This is because the radius was to be measured at the furthest point the string would “stretch” without the force sensor bending downwards. When measuring to the correct radius, they may not have been exact. This could also have led to the error of the counterweight not being placed in the correct position on the rotating arm. During the part of the lab where the masses varied, the counterweight was not put on the scale, and therefore it might not have been acting as an effective counterweight, especially to the lower masses. Another error could be that the apparatus was not completely level. Even though we used a level, it was still hard to get it perfectly centered on the table we were using. Also, at one point near the end of the experiment, the apparatus was bumped slightly, and a level was not used to rebalance the apparatus. This could have caused it to wobble a bit and slightly skew the points in the software, specifically in the third part of the experiment when it was bumped.

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Mass vs. Tension Angular Velocity: 4.987 rad/s Radius: 0.16 m Mass (kg) Tension (N) 0.0515 0.087 0.10145 0.347 0.1513 0.578 0.20125 0.722 0.25105 0.924 Table 1. Tension as a function of mass (constant radius and angular velocity). As the mass increases, the tension increases.

Radius vs. Tension Angular Velocity: 4.987 rad/s Mass: 0.25105 kg Radius (m) Tension (N) 0.044 0.52 0.088 0.607 0.132 0.838 0.176 1.155 0.22 1.589 Table 2. Tension as a function of radius (constant mass and angular velocity). As radius increases, tension increases.

Tension as a Function of Angular Velocity Squared 18 16 14 Tension (N)

12 10 8 6 f(x) = − 0 x + 5.3 R² = 0.5

4 2 0

0

2000

4000

6000

8000

10000

12000

-2 Angular Velocity Squared (rad/s)2 Graph 1. Tension as a function of angular velocity squared. The black line indicates the best fit line for the data. As the square of the angular velocity increases, the tension increases. The percent error for the slope was calculated to be 8.30%.

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Tension as a Function of Mass 1

f(x) = 4.11 x − 0.09 R² = 0.99

0.9 0.8

Tension (N)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.05

0.1

0.15

0.2

0.25

0.3

Mass (kg) Graph 2. Tension as a function of mass. The black line indicates the best fit line for the data. As the mass increases, the tension increases. The percent error for the slope was calculated to be 3.21%.

Tension as a Function of Radius 1.8 1.6

f(x) = 6.1 x + 0.14 R² = 0.94

1.4

Tension (N)

1.2 1 0.8 0.6 0.4 0.2 0

0

0.05

0.1

0.15

0.2

0.25

Radius (m) Graph 3. Tension as a function of radius. The black line indicates the best fit line for the data. As the radius increases, the tension increases. The percent error for the slope was calculated to be 2.23%.

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