Group1_2_ch5

Ryan Hall, John Chiavelli, Sarah Malley, and Jenna Malleytoc

=Lab: Centripetal Force = Task A: John Task B: Jenna Task C: Ryan Task D: Sarah



**Objective**: What is the relationship between net force centripetal and mass, net force centripetal and velocity, and net force centripetal and the radius of the circular motion?

**Hypothesis**: We believe that the relationship between the net force centripetal and mass will be linear. This is because as the mass increases, the net force centripetal increases. We predict that the relationship between the net force centripetal and velocity will be a positive exponential curve. This is due to the fact that the net force centripetal is proportional to the velocity squared. WE predict that the relationship between the net force centripetal and the radius of the circular motion will be inversely exponential or a downwards polynomial curve. This prediction is based on the notion that the net force centripetal is relative to the inverse of the radius.

**Materials and Methods** The materials we used included a long piece of string, a hollow plastic tube, a rubber stopper, stopwatch, meter stick, duck tape, and assorted washers. After attaching the string to the rubber stopper, through the plastic tube, and attaching the washer masses to the bottom as hanging masses, we were able to start conducting trials. About 2 cm below the plastic tube we attached a piece of duck tape as a reference point for maintaining constant speed. We altered the variable of hanging mass after each trial because it is equal to the centripetal force. As we added or removed washers, the radius and length changed, as well as the time. Additionally, the velocity is dynamic as the mass changes.

**Video**  media type="file" key="Movie on 2011-12-14 at 08.27.mov" width="300" height="300"

<span style="font-family: Georgia,serif;">**Procedure** <span style="font-family: Georgia,serif;">First, attach the rubber stopper to one end of the string .<span style="font-family: Georgia,serif;"> Then feed the string through the hollow tube and center it roughly in the center of the string. Next, attach the washer masses to the other end of the string. Hold the string at the plastic tube so that the rubber stopper end is above the fist. Then add the masses of washers in intervals, only changing one variable at a time. After that, swing the rubber stopper around, attempting to limit and eventually stop the vertical and horizontal motion of the hanging washers. Use a piece of duck tape as a reference point below the plastic tube in order to maintain constant speed. Record the time it takes for 10 revolutions to occur, and then find the average time. After 3 trials, change the mass of the washers at the bottom of the string. Then plug in the data to an Excel spreadsheet and create graph for analysis.

<span style="font-family: Georgia,serif;">**Data** <span style="font-family: Georgia,serif;"> <span style="font-family: Georgia,serif;"> <span style="font-family: Georgia,serif;">

<span style="font-family: Georgia,serif;">**Graph** The above graph shows the relationship between net force and mass when dealing with a centripetal force. A linear equation is used for the line of best fit. \ The above graph shows the relationship between net force and radius when dealing with a centripetal force. A power fit is used for the trend line. The above graph shows the relationship between net force and velocity when dealing with a centripetal force. A polynomial trend line is used for the line of best fit.

<span style="font-family: Georgia,serif;">**Analysis** <span style="font-family: Georgia,serif;">__Net Force vs. Mass graph:__ <span style="font-family: Georgia,serif;">For our Net Force (a.k.a. centripetal force) vs. mass graph, we got a linear line of best fit, with the slope of 9.8. This makes sense, because due to the equation F=ma, force and mass are directly proportional, so as one increases, the other does as well, and vice versa. The slope of 9.8 also works out, because that is equal to acceleration due to gravity. This is because the equation Fg=mass*g, and since the net force in this case is equal to the weight of the hanging mass, Fc=Fg. Therefore, Fc=mass*g, and since y=mx (y-intercept is zero), then g = Fc/mass, which on the graph is y/x, which is also the slope of the line. This shows that g equals the slope of the, which is true because g = 9.8

<span style="font-family: Georgia,serif;">__Net Force vs. Radius:__ <span style="font-family: Georgia,serif;">For our Net Force vs. radius graph, we got the line of best fit to be a decreasing exponential function. This is because centripetal force (Fc) is equal to 1/radius. This means that as the radius increases, the centripetal force decreases, exponentially

<span style="font-family: Georgia,serif;">__Net Force vs. Velocity:__ <span style="font-family: Georgia,serif;">For our Net Force vs. velocity graph, we got the line of best fit to be an increasing polynomial function. This is due to the fact that Fc=v^2, meaning that as velocity increases, Fc increases polynomially.

<span style="font-family: Georgia,serif;">Out of the 5 groups in our class, 2 had the correct radius vs. Fc graph, 3 had the correct mass vs. Fc graph, and 4 had the correct velocity vs. ∑Fc graph. Since we had all three correct, our results were better than most of the rest of the class, possibly even the most accurate.

<span style="font-family: Georgia,serif;">**Conclusion** <span style="font-family: Georgia,serif;">In this lab, we were able to reserve the relationships between mass, velocity, and radius, and centripetal force. Our hypothesis was proved correct. We originally predicted that the mass-centrifugal force graph would be linear, which was supported by our data and later by in-class analysis. We discovered that as the mass increases, so does the net centripetal force. The same goes for our prediction about our radius-centrifugal force graph; we hypothesized that it would be a positive exponential graph, which also turned out to be true. Net centripetal force is directly proportional to velocity squared. Finally, we predicted that the relationship between the net centripetal force and the radius would be inversely exponential (or a downward polynomial). This is because, as we learned in class, net centripetal force is directly proportional to 1 over the radius. <span style="font-family: Georgia,serif;">Our data seems to match with that of the class's; if anything, it seems we fared slightly better than most groups in the class. Out of 5 groups, 2 obtained the correct radius vs. ∑Fc graph, 3 obtained the correct mass vs. ∑Fc graph, and 4 obtained the correct velocity vs. ∑Fc graph. We obtained all three with relative accuracy. <span style="font-family: Georgia,serif;">As with any lab, there are several sources of error that could have slightly altered our results. By far the most blatant was the inaccuracy that came with measuring the radius of the circle. Because we could not measure the radius unless the system mass was actually moving in a circle, it was hard to measure the exact distance with 100% precision. We do think that we did the best we could, and that our results were not too thrown off by this. If we ever did this lab again, I would suggest we use the force sensor, as it would likely allow us to obtain more accurate results - it can measure weight (and, in this case, tension) far better than we, using only a balance and some simple calculations, could.

=<span style="font-family: Georgia,serif;">Minimum Speed Activity =

<span style="font-family: Georgia,serif;">**Objective**: Determine the minimum speed an object attached to a 1-meter string could go. <span style="font-family: Georgia,serif;">**Hypothesis**: Based on the equation we solved before the activity, we hypothesize that the velocity will be 3.13 m/s. <span style="font-family: Georgia,serif;">**Methods and Materials**: We used a meter-stick to measure out a 1-meter long piece of string. We had a weight that we tied to the end of the string. We swung the weight around in a horizontal circle as slowly as we could without letting the string go slack. Using a stopwatch, we timed how long it took to complete five revolutions. We then repeated this process two more times. <span style="font-family: Georgia,serif;">**Data** <span style="font-family: Georgia,serif;">Our Data <span style="font-family: Georgia,serif;"> <span style="font-family: Georgia,serif;">Class Data <span style="font-family: Georgia,serif;">

<span style="font-family: Georgia,serif;">**Calculations** <span style="font-family: Georgia,serif;"> <span style="font-family: Georgia,serif;"> <span style="font-family: Georgia,serif;"> <span style="font-family: Georgia,serif; line-height: 0px; overflow: hidden;">

<span style="font-family: Georgia,serif;">**Analysis/Conclusion** <span style="font-family: Georgia,serif;">Our percent error was 47.8%; our theoretical velocity was > 3.13 m/s, while the number we actually obtained experimentally was 4.63 m/s. While this seems off, it's important to remember that, as people doing this experiment, it's virtually impossible to be completely precise. The math we did showed us that the minimum velocity //should// be > 3.13 m/s, but obtained such accuracy in a real-life experiment is a different story. As humans, it would be almost impossible for us to even get to a speed that was exactly greater than 3.13 m/s, let alone to maintain it as we continued to swing the weight in a circle.It was also very hard to know just how slow we could go - by the time we got to that point, it was likely we were going //too// slow. The radius could have changed, too, while we were doing the activity - it likely became smaller, which would have increased the acceleration, thus contributing to our error. <span style="font-family: Georgia,serif;">When compared with the class data, our value fit in with the cluster of numbers, all of which were also greater than 3.13 (between 4.01 and 5.97). The class average was 5.08, which made our percent difference 8.86%. Relatively speaking, this is not all that high.

=<span style="font-family: Georgia,serif;">Lab: Conical Pendulum =

<span style="font-family: Georgia,serif;">__**Purpose:**__ To find out what the relationship between the radius of a conical pendulum and its period is.

<span style="font-family: Georgia,serif;">__**Hypothesis:**__ The relationship between the radius of a conical pendulum and its period will be inversely linear. As the radius increases, so will the angle of tension, and therefore its period will subsequently decrease as its velocity will increase as well.

<span style="font-family: Georgia,serif;">__**Rationale****:**__ We believe that this will happen because if the period decreases, its velocity must increase.

<span style="font-family: Georgia,serif;">__**Materials and Methods:**__ The materials we used to set up the laboratory were a long piece of string and a heavy mass. We hung the mass from the string to eventually create a conical pendulum. Next, we used tape to secure three meter sticks on the floor to create a source of measurement for the radius of the conical pendulum. The stopwatches were used to time the the amount of time it took for the mass to complete one full cycle, or period as per a certain starting point.

<span style="font-family: Georgia,serif;">__**Procedure:**__ <span style="font-family: Georgia,serif;">The variable being changed after each set of time trials is the radius. We changed the radius four times and recorded three time trials of a period. The radius lengths we used were 0.20m, 0.50m, 0.75m, and 1.00m. To establish an accurate conical pendulum, a gentle tangential push was applied to the mass to set it in orbit. The time recorded was the time it took for the mass to complete one full revolution around a specific point. Class data was averaged so that each radius was accompanied by an experimental mean.

<span style="font-family: Georgia,serif;">__**Data:**__

<span style="font-family: Georgia,serif;">__**Calculations**__ <span style="font-family: Georgia,serif;">Example Theoretical Period Calculation <span style="font-family: Georgia,serif;">Example Percent Error

<span style="font-family: Georgia,serif;">__**Analysis Questions:**__
 * 1) **<span style="font-family: Georgia,serif;">Calculate the theoretical period. **
 * 2) <span style="font-family: Georgia,serif;">For the circle with the radius of .2 m, it was 3.16 seconds. For the circle with the radius of .5 m, it was 3.14 s. For the circle with the radius of .75 m, it was 3.08 s. For the circle with the radius of 1 m, it was 3.02 s.
 * 3) **<span style="font-family: Georgia,serif;">Calculate the average experimental period for each radius. **
 * 4) <span style="font-family: Georgia,serif;">For the circle with the radius of .2 m, it was 3.25 seconds. For the circle with the radius of .5 m, it was 3.14 s. For the circle with the radius of .75 m, it was 3.13 s. For the circle with the radius of 1 m, it was 3.06 s.
 * 5) **<span style="font-family: Georgia,serif;">Discuss the accuracy and precision of your data. **
 * 6) <span style="font-family: Georgia,serif;">For the most part, our results appeared to be very accurate - for the circle with the radius of .5 m, we even obtained a 0% error. The results we obtained in class were relatively precise, especially when it came to the later trials. For the first few trials, especially when dealing with the circle with the radius of .2 m, our results were less precise. Perhaps this is because we were still asleep!
 * 7) <span style="font-family: Georgia,serif;">**Why didn't we use the tangential axis at all in this lab?**
 * 8) <span style="font-family: Georgia,serif;">We didn't use the tangential axis at all in this lab because there were no forces acting on that axis. The only significance of the tangential axis is that it represents the direction of motion, but since forces are not present on the axis, we cannot use it to find out the value of any other forces.
 * 9) <span style="font-family: Georgia,serif;">**What effect would changing the mass have on the results?**
 * 10) <span style="font-family: Georgia,serif;">A change in mass would have a few changes upon the lab. If it was increased, the period of one revolution would also be increased because of the slower velocity. On the other hand, if the mass was decreased, the period would be shorter because of the higher velocity. Tension on the chord would also change depending on the change in mass. The period for each radius would be effected as well, depending on the mass change, it would be similar to that of velocity.
 * 11) <span style="font-family: Georgia,serif;">**How did period change as the radius increased? Is it a linear relationship? Why or why not?**
 * 12) <span style="font-family: Georgia,serif;">Our hypothesis was incorrect in predicting this. We said that there would be an inverse linear relationship between period and radius, however, there was actually an inverse exponential relationship. The relationship between these two isn't linear because the radius is changing.
 * 13) <span style="font-family: Georgia,serif;">**What are some sources of experimental error?**
 * 14) <span style="font-family: Georgia,serif;">The possible sources of error in this laboratory were significantly controlled as far as our percent error was concerned. The major source of error would be the variations in timing. Since the class timing was not always precise and typically contained some outliars, the accuracy of our results in general would be slightly off. However, our percent error values were extremely good. They included: 2.75%, 0%, 1.60%, and 1.31%. These values are indicative of sound results, with an average percent error of 1.42%. Another origin of error could have been when measuring the radius of the conical pendulum. Although a meter stick was used, the disparity between the human eye and reality can always be a factor of inaccuracy. Also, when the string broke during our fourth trial, the string seemed to have become longer after hanging it up a second time. This difference could have affected our calculations including the period and percent error.

=<span style="font-family: Georgia,serif;">Lab: Moving in a Horizontal Circle = <span style="font-family: Georgia,serif;">Task A: Sarah <span style="font-family: Georgia,serif;">Task B: John <span style="font-family: Georgia,serif;">Task C: Jenna <span style="font-family: Georgia,serif;">Task D: Ryan

<span style="font-family: Georgia,serif; font-size: 110%;">__**Objectives**__ <span style="font-family: Georgia,serif; font-size: 110%;">1. What is the relationship between the radius and the maximum velocity with which a car makes a turn? <span style="font-family: Georgia,serif; font-size: 110%;">2. How does the presence of banking change the value of the radius at which maximum velocity is reached? 3. How does changing the banking angle change the value of the radius at which maximum velocity is reached?

<span style="font-family: Georgia,serif; font-size: 110%;">__**Hypotheses**__ <span style="font-family: Georgia,serif; font-size: 110%;">1. We hypothesize that the larger the radius of the turn, the higher a velocity is possible. <span style="font-family: Georgia,serif; font-size: 110%;">2. By adding the presence of banking, a higher maximum velocity can be reached at a lower radius. <span style="font-family: Georgia,serif; font-size: 110%;">3. By having a higher banking angle, a higher maximum velocity can be reached at a lower radius and vice versa.

<span style="font-family: Georgia,serif; font-size: 110%;">__**Materials and Methods**__ <span style="font-family: Georgia,serif; font-size: 110%;">The materials we used included a rotational turntable acting as a horizontal circle, which was powered by a regulated power supply. A small rod protruded from the top of the turntable, which enabled the photogate timer hovering above the turntable to collect the period after each turn. The mass we used was a small washer, which was placed 10 cm away from the center of the circle. As the velocity of the turntable was manually increased, the washer would fly off and therefore the maximum velocity could be determined from the period.

<span style="font-family: Georgia,serif; font-size: 110%;">__**Procedure**__ <span style="font-family: Georgia,serif; font-size: 110%;">First, we turned on the regulated power supply and connected DataStudio to a laptop. Then, we measured a radius of 10 cm on the rotational turntable and placed the washer mass in that specific position. As the voltage was gradually increased, the period was being recorded on DataStudio through the photogate timer. When the washer flew off the turntable, we recorded the last recorded time between the photogate transmission. We repeated these steps for roughly eight or nine trials until we had established concurrent results for the period. Lastly, we performed the calculations in order to determine the maximum velocity.

<span style="font-family: Georgia,serif; font-size: 110%;">__**Video**__ media type="file" key="Movie on 2012-01-04 at 08.25

<span style="font-family: Georgia,serif;">__**Graph**__

<span style="font-family: Georgia,serif;">__**Data That Needs to Be Collected:**__ Our Data: Class Data:



__**Calculations**__ __Finding graph µ__

Finding our µ __Percent error between our exponent and the theoretical exponent (.5, since it is squared to the one half power):__ __Percent difference__ between graph µ and our µ
 * Between class µ and our µ


 * __To find maximum velocity__**

__Analysis**__ <span style="font-family: Georgia,serif;">1. Discuss the shape of the graph and its agreement with the theoretical relationship between R and v.

<span style="font-family: Georgia,serif;">The graph should be a power line with the equation y = Ax^b. In this situation, Y would be equal to velocity, A would be equal to acceleration due to gravity times the coefficient of friction, and f R would be equal to X. Since velocity is actually equal to the square root the right side of the equation, the right side of the equation must be raised to the one-half power. Therefore, B is equal to one-half.

<span style="font-family: Georgia,serif;"> 2. Derive the coefficient of friction between the mass and the surface. <span style="font-family: Georgia,serif;">

<span style="font-family: Georgia,serif;"> 3. Compare your coefficient of friction with that of all groups doing this lab. (Be sure to post a data table with the class values.)
 * 0.680 ||
 * 0.590 ||
 * 0.640 ||
 * 0.550 ||
 * 0.560 ||
 * Average = 0.604 ||

As caclulated above, the percent difference between our coefficient of friction to that of the average of the class was 12.3%. Compared to the other groups' values, our coefficient of friction was the highest, and was .04 higher than than the second highest value, and .13 higher than the lowest value.

4. A “car” goes around a banked turn.
 * 1) <span style="font-family: Georgia,serif;">Find an __expression__ for its maximum velocity, in terms of variables only.
 * 2) [[image:Screen_shot_2012-01-09_at_10.13.25_AM.png]]
 * 3) <span style="font-family: Georgia,serif;">How do you think the graph would change if you performed the same procedure but with an angled surface, instead of the level surface we used?

If banking is present, a higher velocity is able to be attained at a lower radius. Therefore, since the graph is radius plotted on the X axis against velocity plotted on the Y axis, with banking, the line would be much steeper, since a certain radius would be able to have a higher velocity reached.

<span style="font-family: Georgia,serif;">__**Conclusion**__ We hypothesized that the larger the radius of the turn, the higher a velocity is possible. Using our data and the class data, we discovered that this is true. This is most likely because the farther away the mass is from the center, the more distance it can cover with the some number of revolutions. Additionally, we hypothesized about banking angles. Although we didn't do any experiments about these, we can possibly conclude that what we said was true. We stated that due to the presence of banking, as well as the larger the banking angle, the higher the velocity can be at a lower radius. This is likely true because the banking angle would hypothetically make the object slide toward the center at a velocity that's too slow, so it would take a larger velocity to make the mass fly off.

For our percent error of our exponent (B) was 24%, and our percent differences for our coefficient of friction was 59.6% against the graph coefficient of friction, and 12.3% against the average class coefficient of friction. This possible difference and error could have been caused by human error in reaction time for pressing the stop button on DataStudio once the washer fell off, the device being bumped into while spinning, or the voltage being turned up too quickly. This can be applied to real life because