Group5_2_ch5

toc =Swinging Stopper Lab=

Task A - Jessica Smith Task B - Sammy Caspert Task C - Maddy Weinfeld Task D - Jake Aronson


 * Objectives**

In this lab, we are trying to determine the relationship between:
 * centripetal force and mass
 * centripetal force and velocity
 * centripetal force radius


 * Hypotheses and Rationales**

We believe that the relationships outlined in "Objectives" will be as follows:
 * As mass increases, so does centripetal force, and vice versa. Centripetal force is equal to mass times "g," so a greater mass will yield a greater centripetal force.
 * As velocity increases, so does centripetal force, and vice versa. As the velocity increases, so will the tension in the string, which is the centripetal force.
 * As radius increases, centripetal force decreases, and vice versa. A greater radius would cause more slack in the string, so the tension (and thus centripetal force) would be less.



Our materials are a string, a stopper, a hollow tube, a paper clip, a timer, and masses.
 * Methods and Materials**

1. Put together the contraption. On one of the string but the weights and a paperclip and the other a stopper. In the middle of the string put a hollow tube around it and form a 90 degree angle while spinning (see in video). 2. Test for mass vs. centripetal force. Do a trial for a certain amount on the hanging mass and then add more hanging mass and do the trial again. Repate multiple times. Use a stopwatch to measure the time taken for 10 revolutions. Record data. 3. Although we didn't have enough time, you would graph velocity vs. centripetal force and radius vs. centripetal force. For velocity, keep radius and mass constant while only changing velocity. For radius, keep velocity and mass constant while only changing radius. Record data.

media type="file" key="New_Project_-_Medium_1_2.m4v" width="300" height="300"
 * Procedure (swinging of the stopper):**

Our data for changing velocity**:**
 * Calculations, Data, & Analysis**

Data taken from last year:

We did not have enough time to complete the lab so we had to use some data from last year's class. All of the data both from our own tests and from last year, matched what they should have. The graph of mass v. centripetal force was linear because as mass increases so does the force. This makes sense and is easy to visualize because something with a bigger mass clearly requires a greater net force, which is ultimately what the centripetal force is. The graph for velocity vs centripetal force is a polynomial function. Centripetal force is proportional to the velocity squared. Lastly, the graph for radius vs centripetal force is an inverse function because the centripetal force is equal to 1/radius. This means that as the radius increases, the centripetal force decreases because it is easier to make a wider turn than a tighter one.
 * Analysis**

Overall, our data and the data from last year support our hypotheses. The graph of mass versus centripetal force was a linear graph, just like we predicted. We learned in the lab that system mass and centripetal force are directly related, which was our hypothesis: as mass increases, so does centripetal force, and vice versa. Centripetal force is equal to the net force, which is equal to mass times "g," so a greater centripetal force would require a greater system mass. Next, the graph of velocity versus centripetal force was an exponential function. Originally, we predicted that the graph would also be linear, but we were slightly mistaken. We learned that centripetal force is equal to velocity squared, which is an exponential (not linear) function. As the centripetal force increased, the velocity was squared, causing the J-curve (or at least close to it) of our graph. Last, the graph of radius versus centripetal force is an inverse exponential function. We learned that centripetal force and radius are inversely related, meaning that centripetal force is equal to 1 over the radius, which is a downward-sloping exponential function. As the centripetal force increased, the radius decreased.
 * Conclusion**

There were a few possible sources of error. First, there was the angle at which we spun the string. We did not spin the string horizontally, lest the string fly out the tube, so there we should have calculated components of the centripetal force vector. Second, there was the velocity at which we spun the string. We did not spin the string at a completely constant velocity, because the motion in our hands inevitably changed a bit during each trial. Third, there was the radius of the spinning string. The string slid up and down the tube as it was spinning, because it was impossible with our basic resources to keep measurements completely constant. All of these sources of error skewed the data, so the measured forces are imprecise. However, because the same sources of error affected each trial, the data were fairly accurate in their consistence for each variable. ==

== =Minimum Velocity Activity=

Objective: Using a 1 meter long string and 1 washer, calculate the minimum velocity required to make a complete circle. Theoretical Minimum Velocity:

Experimental: We calculated the time it took for ten revolutions and divided it by ten to get the amount of time it took for one revolution. We then found velocity by doing C/t.

Calculation to find theoretical velocity: Experimental Velocities (group work):
 * **Time(s)** || **Velocity(m/s)** ||
 * 1.2 || 5.24 ||
 * 1.14 || 5.487 ||
 * 1.15 || 5.435 ||
 * 1.31 || 4.804 ||
 * 1.35 || 4.833 ||

Sample Calculation (to find velocity):

Percent Error and Percent Difference:

The reason we had a percent error was because we were not able to move the string at the exact minimum velocity. We were not consistent and it was difficult to get the least possible amount of tension.

Class Average Velocity

=Conical Pendulum Lab=

Purpose: Find the relationship between the radius of a conical pendulum and its period.

Hypothesis: As the radius of the conical pendulum increases, the period gets smaller because as the radius gets bigger, the pendulum bob will move faster and thus take less time to complete one period.

Method and Materials: Method and Materials: First, we labeled and sketched a diagram of the conical pendulum with the length of the string, mass of the pendulum bob, radius of the circle, and the angle between the x axis and the pendulum string. Using a meter stick and a scale, we found the length of the string and the mass of the pendulum bob. We then created our hypothesis. As our first radius, we chose 0.20 meters because it was small. We pushed the pendulum to create a circle with a fairly consistent radius all around. Using a stopwatch, the class recorded the amount of time it took for one period and then we found the average time for the radius. We then repeated these steps for different radii: 0.50 m, 0.75 m, and 1.00 m. The available materials were string to hang the pendulum, a heavy mass, a meter stick to measure, and stop watch to time.

Experimental: This is a data table for the class data. For each radius, we did three trials and everybody in the class recorded the time for one period. The average for each radius is written on the bottom.

Sample Calculation for Theoertical:

Sample Calculation



Data Table: ^This data table shows our theoretical and experimental values as well as the percent error for each radius.

Analysis:
 * 1) Calculate the theoretical period.
 * 2) See above
 * 3) Calculate the average experimental period for each radius.
 * 4) See above
 * 5) Discuss the accuracy and precision of your data.
 * 6) Our results are both accurate and precise. We achieved accurate results, which we know because we had a percent error within a range of 3% for all four radii. We also achieved precise results, which we know because we had a percent error of less than 3% for each of the four radii calculations.
 * 7) Why didn’t we use the tangential axis at all in this lab?
 * 8) The tangential axis is in the direction of motion at one point, but we wanted to solve for a constant velocity per revolution. We could only solve for that constant velocity using the centripetal axis.
 * 9) What effect would changing the mass have on the results?
 * 10) Changing the mass would have no effect on the results. Mass is factored into the equation solving for tension, and then again in the equation solving for theoretical velocity, so the change is negligible and cancelled out.
 * 11) How did period change as the radius increased? Is it a linear relationship? Why or why not?
 * 12) The period got smaller as the radius increased. The relationship is not linear. R is a factor in so many parts of the equation. It comes into play when finding the centripetal force, velocity, and the angle of the tension. The relationship is inverse but not proportional.
 * 13) What are some sources of experimental error?
 * 14) First, the string changes heights in the middle of the lab because it fell. This could create a source of error because not all of the trials actually had the same length of sting. Another possible source of error was if people started to time the period later in the trial. The radius would no longer be the intended one, so the period would be larger than if the radius were still the intended distance. The last possible source of error was that the pendulum was not exactly conical. Because it was being thrown by hand, and the string was so long, it was hard to tell the exact shape that the string was moving in.

=Moving in a Horizontal Circle Lab=

Task A: Maddy Weinfeld Task B: Jake Aronson Task C: Sammy Caspert Task D: Jessica Smith


 * Objectives and Hypotheses**
 * 1) What is the relationship between the radius and the maximum velocity with which a car makes a turn?
 * 2) As the radius of a turn increases (assuming no bank), the maximum velocity with which a car can make that turn increases as well (and vice versa). If the radius is too small and the car's velocity is too great, then the so-called "centrifugal force" will be stronger than the centripetal force. In other words, the centripetal force will not be strong enough to balance the force of the car's weight and momentum in a straight line. A greater radius allows for a greater velocity because the turn is wider.
 * 3) How does the presence of banking change the value of the radius at which maximum velocity is reached?
 * 4) The presence of banking allows for a smaller radius at which maximum velocity is reached. With a bank, the friction force of the road towards the center balances the force of the car's weight and momentum into the turn, allowing for a greater velocity. The banking-created friction force is the centripetal force that keeps the car in its turn.
 * 5) How does changing the banking angle change the value of the radius at which maximum velocity is reached?
 * 6) As the banking angle increases, the radius at which maximum velocity is reached decreases (and vice versa). With greater velocity, the car will need a greater normal force and friction force to balance its weight and momentum into the turn, which are pushing the car to continue moving straight (tangential to the circle). If the normal and friction forces increase, then so does the centripetal force and so the car's ability to make the turn.

In this lab, we used a rotational turntable, power supply, photogate and mass. We attached one end of a DataStudio USB cable to the rotational turntable's photogate, and we plugged the other end of the USB cable into the computer. On the computer, we clicked "Create an Experiment" and then "Recordable Timer." We placed the mass on the 40 cm mark of the rotational turntable, then increased the voltage until the turntable increased velocity. We clicked "Start" on the DataStudio activity, watched as the turntable spun, and clicked "Stop" as soon as the mass flew off of the table. Repeat about 8 times for accurate and consistent results.
 * Materials and Methods**

media type="file" key="Movie on 2012-01-04 at 08.45.mov" width="300" height="300"
 * Procedure (video)**


 * Data**


 * Calculations and Analysis**

Because velocity is equal to the "square root of velocity*radius*mu", the square root, or 0.5 power, leads us to believe that the shape of this graph best fits a power function. With our "x" as the radius and our "y" and the maximum velocity, it is clear that as the radius increases so do the maximum velocity. However, as the radius gets bigger, the amount the velocity increases gets smaller and smaller. The coefficient 2.0672 is equal to the "square root of radius*mu". From this we were able to find a different way to calculate the coefficient of friction from our data, which we got to be 0.44.
 * 1. Discuss the shape of the graph and its agreement with the theoretical relationship between R and v.**


 * 2. Derive the coefficient of friction between the mass and the surface.**

**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.)** **4. A “car” goes around a banked turn.** >> ** and leads to  ** >> andleads to
 * Radius || Mu ||
 * .10 || 0.68 ||
 * .20 || 0.59 ||
 * .25 || 0.64 ||
 * .30 || 0.55 ||
 * **.40** || **0.56** ||
 * Average || 0.60 ||
 * 1) **Find an** __**expression**__ **for its maximum velocity, in terms of variables only.**
 * 1) **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?**

By including trigonometric equations, using an angled surface changes a lot. In this equation compared to 4-1, you can have a higher velocity with a smaller radius. Because radius is on the "x" and velocity is on the "y", the graph would have a steeper slope.

__% Difference between Our Mu and Graph Mu__
 * Percent Difference/Error:**



__% Error between Our Exponent and Theoretical Exponent__

Our and the class's results support our first hypothesis: as the radius increased, so did the maximum velocity at which the mass could make the turn. With a greater magnitude (radius), the centripetal force attracting the mass increased, allowing for a greater velocity. Our graph displays the equation y=2.0672x^0.3847, which is a power function defined by y=Ax^b. The shape of the graph is close to that of a square root power function, as it should be. We learned that v=sqrt(g*Mu*R), which mirrors the equation of our graph: the y-value is the maximum velocity, the A-value is the sqrt(g*Mu), the x-value is the radius, and the B-value is the power of the radius. Because we are trying to achieve the square root of the radius, B should be close to 0.5. Our B is equal to 0.3847, which shows that some sources of error affected our experiment. We had a percent error of 23%, which was very large. This 23% error occurred between our calculated B-value and the theoretical B-value. As stated before, the B-value should have been 0.5, but many sources of error affected the data, which affected our graphed equation. Some of the error occurred with the inconsistencies of the voltages each time and therefore the periods and velocities are slightly off. For example, we could have turned the voltage up too quickly, which would have negated the centripetal friction and made the mass fly off of the turning table too quickly. To try to eliminate this problem and other problems, we could have performed more trials and gained experience with the new tools that we had never used before. Another major source of error was the amount of time between one group member saying stop and the other clicking stop on data studio. Our reaction time affected our data, which was apparent in the different values that we achieved for each trial.
 * Conclusion**

We would have changed the lab in two main ways. First, we would have instructed students to acclimate themselves with the new equipment. If we had been familiar with the rotating turntable going into the lab, then we may have achieved more precise results with more consistency (precision and accuracy). Second, we would have attached a DataStudio sensor directly to the mass, so that it would indicate the precise instant when the mass flew off of the turntable. With more precise results, our B-value would have been much closer to the theoretical 0.5-value. These sources of error are ones that we experience in our everyday lives, especially as seniors, because we now drive cars. In a car, if you accelerate too fast around a turn, then your car will skid or even flip over from the lack of balance between the centripetal and tangential forces. That is why we are required to take driver's education and supervised driving practice before we earn our licenses: practice behind the wheel makes perfect, just as practice with the turntable would have made nearer-to-perfect results.