Group1_4_ch5

toc Chapter 5, Period 4, Lab Group 1:

Jonathan Itskovitch Max Llewellyn Kosuke Seki

=Swinger Stopper Lab= Task B, D: Jonathan Itskovitch Task C: Max Llewellyn Task A: Kosuke Seki



What is the relationship between System Mass and Net Force? What is the relationship between Velocity and Net Force? What is the relationship between Radius and Net Force?
 * Objectives:**

As System Mass increases, so does the Net Force, and in a linear fashion. As Velocity increases, so does the Net Force, and parabolically. As Radius increases, the Net Force Decreases, since they are inverses.
 * Hypotheses:**

Assemble a swinger stopper, where stoppers are attached at one end of a string to modify system mass. A tube will be placed in the middle of the string because, as the string whirls around, the only centripetal force is the tension. At the bottom of the string is a force meter, connected to the computer via a USB link.Three different variables are going to be tested upon whirling the string and stoppers: velocity, radius, and mass. To test the relationship between these and net force, they have to be separated into three data tables, and the other two variables must remain constant. Whirl the string around for 10 revolutions. Time this using a stopwatch.The net force is directly read with DataStudio, which calculates the mean force automatically. Make sure the graph is Force Pull-Positive. From here, the other variables can be found using algebra. To change velocity, just make the string go around faster or slower. For mass, add or remove stoppers at the end of the string. To change radius, make the distance between the stoppers and tube smaller or larger. Calculate this using a meterstick. When all is done, find the relationships by noticing the shapes of the graphs.
 * Methods and Materials:**

Data Table: Relationship of Velocity and Net Force Data Table: Relationship of Mass and Net Force Data Table: Relationship of Radius and Net Force We are using last year's class data for just the Radius/Net Force relationship. We were not able to finish our own due to time constraints.
 * Data:**


 * Graphs:**

The first graph is the relationship between net centripetal force and velocity. The graph appears to be parabolic. This can be derived from the following: The manipulation of the equation, assuming that the net force is y, and the velocity is x, shows several things. It shows that the coefficient before the x is mass over radius, making the graph shrink or stretch vertically. The velocity is squared, once again proving the parabolic state of the graph. The y-intercept is 0, because there is 0 velocity at 0 force. The R^2 value is 0.88, which is pretty decent. Ideally it should be 1, if all the points are perfectly aligned on the parabola. But an 88% accuracy is good. The second graph, showing the relationship between mass and net force, is linear. Again, the y intercept is 0 because there is no force needed with no mass added, to make the string go in a circle. The rest can be derived from the following: Net force is y, and mass is x. It shows a linear relationship. The slope is derived from velocity squared over radius. The R^2 value is 0.97, which is excellent. A 97% accuracy is very good; however, since it is linear, it should be closer to 1 than other graphs. As for the third graph, the relationship of radius and net force, the graph is an inverse. It is hyperbolic. There is no y-intercept of 0, and here is why: Net force is y, and radius is x. If x is 0, then the net force should be infinite or undefined. This equation shows an inverse relationship between net force and radius, with the constant above being mass times velocity squared. We got an amazing R^2 value for a hyperbolic graph, of 0.987.
 * Analysis of Graphs:**

Yes, our hypotheses have been satisfied very well.Our first hypothesis states that, as System Mass increases, so does the Net Force, and in a linear fashion. This is indeed the case. As seen by the graph above, there is a constant, linear relationship between system mass and Net Force. This is also proven by the direct relationship in the equation, F=ma. Our second hypothesis states that, as Velocity increases, so does the Net Force, and parabolically. Again, our graph above proves this. There is a parabolic shape to the velocity v net force graph, satisfying our statement. Beyond that, there is a parabolic relationship in the equation, F=(mv^2)/R. Our final hypothesis states that as Radius increases, the Net Force Decreases, since they are inverses. Again, this is correct. Our graph above shows a hyperbolic shape to the graph, proving the above. Also, from the equation derivation, F=(mv^2)/R, there is an inverse relationship between radius and net force.
 * Conclusion:**

Despite our good results in the lab, there have been several sources of error. The main source comes from the fact that the radius we measured is not the actual radius. The length of the string has a natural tendency to drop down, which means that the radius is just the horizontal component of the string length. Because we used the string length as radius, all of our calculations have error to them. To solve this error, estimate how far down the string went, and use the pythagorean theorem to find the horizontal component. Another source of error comes form the impossbility to maintain a constant velocity through trials. When testing for the relationships of radius and mass, velocity must remain constant but it is impossible to achieve exactly that. Therefore, there is some error in the graphs. To fix this, perhaps a machine can spin the string at a constant velocity. Another source comes from the timer. Even a one hundredth of a second can make a huge difference, so it is imperative to measure the time exactly from start to finish. However, pressing the button takes time, so it can add more time than the actual. To fix this error, use more than one timer and find the average.

=Minimum Speed Lab= Data Table: Minimum Speed Lab Class Tables:

Theoretical Velocity Calculation:

Percent Error Calculation (Theoretical and Experimental Velocity): Percent Difference Calculation

Conclusion: There is a good reason as to why there is such a huge experimental error. First off, it makes sense that the experimental tension we got is higher than the theoretical tension. That is because it is impossible to physically go to the bare minimum velocity required. Also, the time where velocity is truly at its minimum was at the top of the circle. We measured time for the whole circle, and there was acceleration. The rest of the time on the circle was faster than the top point. So the time was, off course, a little bit skewed there. Which also means it was impossible to obtain constant speed the entire time. Also, at the very minimum point, when tension is zero, the string will not go in a circle anymore. Also, it is impossible to keep the circle vertical the entire time, which means there was angles, components, and skewed data. Therefore, we have to go a little bit above. But the other reasons justify the huge error.

=Lab: Horizontal Circle= Task A: Max Task B: Max Task C: Jonathan Task D: Kosuke


 * __Objectives:__**
 * 1) What is the relationship between the radius and the maximum velocity with which a car make a turn?
 * 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?


 * Hypotheses:**
 * 1) Velocity increases with radius, although not at a constant rate. Since the circle is horizontal weight is not involved. The centripetal force is friction. Derived from the equation,[[image:Screen_shot_2012-01-04_at_12.16.10_PM.png]],we find that velocity equals [[image:Screen_shot_2012-01-04_at_12.17.06_PM.png]]which means there is not an inverse relationship between the two but it is not constant due to the square root.
 * 2) Banking decreases the value of the radius. Since there is a component of friction on the y axis as well, it allows more faster speeds without flying off. Because of this it allows for a lower radius to reach max velocity. This equation also proves this true.
 * 3) [[image:Screen_shot_2012-01-04_at_12.17.51_PM.png]]
 * 4) The steeper the angle the faster you can go. From the equation above, you can derive that a more banked turn will create more friction and allow higher velocities before flying off. This means that a lesser radius is required, so it goes down.


 * Methods and Materials:**

media type="file" key="Movie on 2012-01-06 at 11.51.mov" width="300" height="300"


 * Data Table: Individual Values of Velocity with Radius 0.1m**

The data shown here is the theoretical results we achieved while doing our labs. We did eight consistent trials and found that our average mu was .586 and our average velocity was .758 m/s. Our theoretical mu, however, was slightly different at .426. Between our average and theoretical, there is a 3.914% difference.




 * Data Table: Class Values of Velocity with Different Radii**

These are the class results. We calculated the average period, average velocity, average mu, and the experimental and theoretical mu.


 * Graph: Radius v Maximum Velocity**


 * Analysis:**


 * QUESTION 1:**

Percent Difference Between Class Average of mu and Our value of mu Equation for Maximum Velocity with a Banked Turn:
 * QUESTION 2:**
 * QUESTION 3:**
 * QUESTION 4:**

Due to a turn being banked, we are allowed to go faster while going around a smaller radius. Usually, it takes a larger turn if you go faster. In other words, banking decreases the value of the radius or increases the value of max velocity. This would make the value of A much bigger, and the y values of the graph (max velocity) higher for the same given radius. With this logic, it makes sense that the steeper the banking angle you can go even faster with the same radius. Also, as the equation shows, the numerator increases as denominator decreases, making a double case for a higher max velocity.


 * Conclusion:**

Yes, our hypothesis were correct. We said that velocity increases with radius, but not at a constant rate. This is true because of the square root relationship we found out. In addition, we know that velocity = circumference / time. Large radii means that there is a larger circumference to travel over. The nature of the turntable means that all points on a given radius will make one revolution in the same time. Hence, if the metal is traveling a larger distance over the same amount of time, making the velocity larger. We also hypothesized that banking will decrease the radius in which one reaches max velocity. This is also true because on a banked turn, there is more force going to the center of the circle due to the components of the normal force. Thus, we logically said that the higher the banking angle the more the radius decreases. This is true, as we found out. It is the converse of the previous reasoning.

Our percent error was 1.86%, and our percent difference was 3.91%. These numbers are very good but there are several sources of error that we could have fixed to make our experiment even better. The washer may not have been put at exactly 0.1 meters. We could have used a meterstick to make the radius more exact. Also, the coin fell off at a different time from when the period was recorded, which meant a slightly different period (probably lower). Because of our reaction time, it was entirely possible to use a period time that did not correspond to the specific revolution and velocity when the washer fell off. If we had a continuous tracker of period, or watch it more carefully and use a timer, then this error could have been reduced. We could have done this also by making more points where the photogate captured the speed for more accurate results. More points equals more data, which makes it easier to narrow down the specific one we want. Also, the voltage was turned up too quickly and inconsistently, skewing results. The knob was extremely sensitive to any movement at all, making it difficult to achieve a steady increase of voltage. Sometimes, a small adjustment would skyrocket the voltage by enough volts to make the washer just zoom off without recording a time for the period. A more accurate or less sensitive turntable would have reduced error by making adjustments much easier to make.

Highway engineers often have to do this kind of math and this kind of physics in order to determine how to make a turn that a car can safely make without skidding down the road or slipping. Using the same process, they figure out the maximum and minimum speeds a car can go on the turn. This helps them to figure out how to make a safe turn for a car.