ARAN

=Lab 4: Spring Constant=
 * Group Members:** Ani Papazian, Rachel Caspert, Sammy Wolfin, Ariel Katz
 * Class:** Period 2
 * Date Completed:** March 7, 2011
 * Date Due:** March 8, 2011

Materials & Procedure:
= = = = =Lab 3: Coefficient of Friction=
 * Group Members:** Ani Papazian, Rachel Caspert, Sammy Wolfin, Ariel Katz
 * Class:** Period 2
 * Date Completed:** January 14, 2011
 * Date Due:** January 19, 2011

The purpose of this lab is to determine the coefficient of static and kinetic friction between a wooden block and a penny and between a wooden circular plate and a penny. The other groups will use this value in their calculations. Also, we want to see how the maximum velocity to keep a car moving in a horizontal circle changes with the radius of un-banked turns, the radius of banked turns, and the banking angle.
 * Purpose:**


 * Hypotheses & Rationales:**
 * If the radius of the turn on an un-banked turn is increased, the maximum velocity will increase. This is because, according to calculations, there is a direct squared relationship between maximum velocity and the radius of the turn on an un-banked turn.
 * If the radius of a banked turn is increased, the maxim um velocity of a car on the turn will increase. When a car is turning a tight turn with a short radius, the velocity of the car must be closer to zero so that the car doesn't skid. On the other hand, wide turns with large radii allow the maximum velocity to be greater without the car skidding.
 * If the banking angle of the turn is increased, the radius will decrease to reach maximum velocity. According to the calculations, the radius and the banking angle are indirectly proportional.

aluminum track wooden angle blocks of different angles wooden circular plate balance penny protractor record player/turn table stop watch
 * Materials:**

General Set-Up Coefficient of friction (µ) between wooden block and penny Coefficient of friction (µ) between wooden circular plate and penny
 * Procedures:**
 * Coefficients of Friction:**
 * 1) Slide Protractor onto aluminum track
 * 2) Measure mass of penny
 * 1) Place angled wooded block on aluminum track
 * 2) Place penny on block
 * 3) Slowly raise track until penny barely starts to slide
 * 4) Measure angle track is raised
 * 5) Measure angle of incline of the wooden block, add to measured angle
 * 6) Plug data into excel, calculate µ
 * 1) Place wooden circular plate on aluminum track
 * 2) Place penny on wooden circular plate
 * 3) Slowly raise track until penny barely starts to slide
 * 4) Measure angle track is raised
 * 5) Plug data into excel, calculate µ

media type="file" key="My First Project.m4v" width="300" height="300"


 * Increasing radius on an un-banked turn:**
 * 1) Using a stopwatch, test the actual RPM of the record playing turn table for its slowest speed (16 RPM)
 * 2) With the record player set to its slowest speed, Test a set radius to see if the penny will stay in a uniform circular motion.
 * 3) Repeat step two and increase the radius until you are able to find the maximum radius of the circular path for this velocity.
 * 4) Repeat steps 1-4 using the next fastest speed for the record player and so on for the rest of the speeds.


 * Increasing radius on a banked turn:**
 * 1) Verify the speed of the record player.
 * 2) Choose one of the available speeds on the record player.
 * 3) Place the penny at the desired distance from the center and measure that radius.
 * 4) Turn on the record player and start the timer.
 * 5) Let it spin/turn on time.
 * 6) Turn the record player off and stop timer.
 * 7) Take note of the time and penny's position after the circle. Note whether or not the penny's position shifted/changed during the turn - did it slide down?
 * 8) Repeat steps 3-8 with the same radius but using the other three speeds.
 * 9) After all speeds have been used with one radius, change the radius and repeat steps 3-8 three times.
 * 10) In the end, you should have a trial for each radius with each speed (4 radii, 4 different speeds, so 16 trials).
 * Increasing banking angle:**
 * 1) Choose a record player and determine the rpm at different speeds.
 * 2) Next, find the angle for the four different wood blocks that you will be using as the banked turns.
 * 3) Attach the block to a cardboard disk on the record player suing velcro.
 * 4) Place a penny on the block and begin the rotation.
 * 5) Keep moving the penny up the block until you reach the minimum height that it falls off.
 * 6) Record your data and graph.


 * Data:**


 * Object || Max Angle w/out Sliding 1 Degrees || Max Angle w/out Sliding 2 Degrees || Max Angle w/out Sliding 3 Degrees || Max Angle w/out Slding 4 Degrees || Max Angle w/out Sliding 5 Degrees || Max Angle Avg || mass (kg) || weight (N) || ø radians || sinø || friction (N) || cosø || Normal (N) || µk ||
 * Wooden Block || 46 || 43 || 39 || 48 || 46 || 44.4 || 0.0025 || 0.0245 || 0.7745 || 0.6994 || 0.0171 || 0.7147 || 0.0175 || 0.9785 ||
 * Wooden Circular Block || 48 || 48 || 47 || 46 || 50 || 47.8 || 0.0025 || 0.0245 || 0.8338 || 0.7405 || 0.0181 || 0.6720 || 0.0165 || 1.1019 ||



Calculations used: wooden block:

wooden circular block:

This part of the lab was performed by Dylan, Spencer, Jillian, and Nicole. Through their graph we see that there is an inverse relationship between the banked angle and the radius.
 * Radius v Angle**

When there is a banked angle, there is an inverse square relationship between velocity and radius. This graph was done by Emily, Emily, Elena, and Amanda.
 * Velocity v Radius**

Roshni, Erica, and Amanda show that there is a square proportion between velocity and radius when it is an unbanked scenario.


 * Conclusion:**

Our lab group found the coefficient of static and kinetic friction between the wooden block and the penny for the other groups to use in their calculations. The coefficient of friction was used specifically when friction needed to be substituted by µN to isolate velocity by setting the normal equations of the x and y axis equal to each other.
 * Coefficient of Friction: **

**Relationship between radius and maximum velocity: unbanked turn:** As the radius increases, the maximum velocity will increase as well. There is a direct square relationship between maximum velocity and radius. By finding the maximum radius for multiple velocities, the relationship between radius and velocity was observed. The data found by the groups who performed this part of the lab show the direct square relationship. It is seen in the shape of the graph, which is a curve with a positive slope and the equation, which is quadratic. The R^2 values of the graphs were both very close to one (0.994, 0.996, and 0.9976), which means the results are very precise in respect to the line of best fit.

**Relationship between radius and maximum velocity: banked turn:** As the radius of the banked turn is increased, the maximum velocity will increase. By altering the speeds and radii multiple times, these results were observed. Velocity and radius have a direct square relationship. This is seen in one of the groups graphs. The line is curved upwards and the equation is quadratic. The results are extremely precise with the line of best fit with an R^2 value of 1.

**Relationship between banking angle and Radius:** As the banking angle increases, the radius will decrease. Banking angle and radius have an indirect relationship. This is seen in 2 of the 3 sets of data. The graphs show a linear negative slope, with a linear equation with a negative coefficient. This shows the indirect relationship. The results are pretty precise with the line of best fit with R^2 values of 0.96 and 0.9866.

Although our experiment and hypothesis proved to be fairly accurate, there are some sources of error to account for. Human error played a huge role in this lab, and could be why our calculations weren't as precise as they should have been. One thing that could have caused error would be the actual reading of the angles we used. We were reading the angle measuring tool, and it does not provide the most precise readings. Also, the penny seemed to move down the incline more immediately if we raised the track higher, quicker. If we raised the track slower, the penny stayed on for longer. This could have also caused some error. If the rate at which we raised the track to test the penny's friction was uniform, we could have gotten better results. Something we could have done to fix these problems would be to be more aware of small details like the ones just mentioned. If we added more trials, and more people reading the resulting angles, it could be possible our results would have been better. Also, if we raised the track higher and higher at a uniform speed without jerking the track at all, our angles could have been more accurate. Overall, our hypothesis proved to be accurate, and our results were pretty close to what we expected them to be.
 * Error Analysis:**

Data/Information for maximum velocity vs. radius of un-banked turn - Jae, Danielle, Jessica, Rebecca, Niki, Alyssa, Chloe, Steve, Andrew, and Justin Data/Information for maximum velocity vs. radius of banked turn - Eric, Sean Phil, and Chris Data/Information for banking angle vs. radius of turn - Anthony, Aaron, Jimmy, Navin, Nicole, Jillian, Spencer, and Dylan = = = = =**Lab 2: Minimum and Maximum Tension**=
 * Acknowledgments:**
 * Group Members: Ani Papazian, Ariel Katz, Rachel Caspert, Sammy Wolfin**
 * Date Completed: 1/10/11**
 * Date Due: 1/11/11**
 * Class: Period 2**

At what point in the circular path is the string at a minimum and maximum tension and what is the relationship between these tensions and different radius lengths.
 * Purpose:**

If a string is spinning in a circular motion, the maximum tension will be at the bottom of the circle and the minimum tension will be at the top of the circle exactly above or below the center. For minimum tension, if the radius increases, then velocity increases as well. According to the calculations, the velocity equals the square root of gravity times radius. This calculation serves as the rationale for our hypothesis. For maximum tension, if the radius increases, the velocity decreases. According to the maximum tensions calculations, derived from when the mass is at the bottom of the circle, the velocity equals the square root of tension minus mass times gravity times radius over mass. In these calculations, we see that the relationship between radius and velocity is an inverse relationship, so therefore as the radius increases, the velocity should theoretically decrease. (See Derivations of Calculations)
 * Hypothesis**:

Thread**,** masses (5g, 2g, 10g, 20g, 50g), meter stick, stopwatch
 * Materials:**

Minimum tension procedure:
 * Procedure:**
 * 1) Measure radius of thread
 * 2) Tie constant mass to the end of thread
 * 3) Spin thread at minimum possible speed without allowing circle/thread to go limp, time duration with stopwatch
 * 4) Record number of revolutions and total time
 * 5) Calculate velocity, compare to theoretical

Finding maximum tension:
 * 1) Tie a piece of thread (that will be used in experiment) to the end of a meter stick laying flat on the lab table
 * 2) Place meter stick so it hits just the end of the table, with the thread hanging down, not yet hitting the ground
 * 3) Attach a mass hanger to the end of the thread
 * 4) Add small mass values until the thread breaks
 * 5) Record maximum tension

Maximum tension procedure:
 * 1) Measure radius of thread
 * 2) Tie chosen, constant mass to the end of the thread
 * 3) Spin thread in a circle at maximum possible speed until thread breaks, time duration with stopwatch
 * 4) Record number of revolutions and total time
 * 5) Calculate velocity, compare to theoretical

finding maximum tension In this lab, we wanted to investigate both the minimum and maximum tension of a uniform circle. In order to do so, we looked at its relationship with the radius of the circle and how that affected the minimum and maximum tension needed. The minimum tension is found at the top of the circle, its the least amount of tension needed to just keep the circle in motion without it flopping. The maximum tension is found at the bottom of the circle, directly across from the minimum tension. The maximum tension is the circle's strings last ability to hold onto the object in motion without the object breaking the string and going off its circular path.
 * Introduction:**


 * Theoretical Calculations/Derivations of Calculations:**


 * Data:**

Maximum Tension and Experimental Velocity

Experimental Velocity vs. Radius for Maximum Tension

Maximum Tension



Minimum Tension and Experimental Velocity Experimental Velocity vs. Radius for Minimum Tension

In conclusion, our hypothesis supports our experimental values. We hypothesized that the minimum tension would occur directly above the center at the top of the vertical circle. We also thought that the maximum tension would occur directly at the bottom of the vertical circle, directly below the center. Both of these things proved to be true. In relation to the size of the radius, for minimum tension, as the radius increased, so did the velocity. This proved true in our hypothesis based on our data. When the size of the radius increased for maximum tension, the velocity decreased. This was also correct based on our hypothesis. In our experimental values, we did not get completely perfect results. This could be for many reasons. One reason could have been human error. We were timing the swings around the circle ourselves, and some time could have been gained or lost based on when the timer was started or stopped. Also, we were counting the swinging ourselves and some error could have occurred. Although some things could have gone wrong, and our R squared value isn't 1, our experiment was extremely successful and our hypothesis was proven correct.
 * Conclusion:**

=Lab 1: Changing Radius in Centripetal Motion=
 * Group Members: Ani Papazian, Ariel Katz, Rachel Caspert, Sammy Wolfin**
 * Date Completed: 1/4/11**
 * Class: Period 2**

As the radius increases, the force decreases. By looking at the circular motion laws, we are able to conclude that this statement would be valid. As the radius increases, the velocity decreases because of an inverse relation. And when the velocity decreases, the force decreases due to a direct square. Therefore leading to the statement, as the radius increases, the force needed decreases.
 * Hypothesis**:

1. Attach string to plug, and string to data studio data collector 2. Change the radius of the string while keeping all other variables constant 3. Spin the string in a circular motion 4. Measure data using data studio 5. Plug data into excel to calculate force while populating a graph 6. Determine the relationship between force and radius
 * Procedure**:

Force sensor, string, tape, rubber stopper (mass), data studio data collector, computer, meter stick
 * Materials:**

In this lab, we are investigating the 3 relationships of circular motion: centripetal force v velocity, centripetal force v radius length, and centripetal force v mass. Our group was specifically given the relationship of centripetal force v radius. We were looking to see if the tension force needed in the string would decrease or increase as the radius changed lengths. After testing different lengths and using data studio to record what the average tension needed was, we came to the conclusion that force was indirectly proportional to the length of the radius. Meaning, as the radius increased, the force decreased and as the radius decreased, the force increased.
 * Introduction:**


 * Data:**


 * Graph**:

(Tom, Tyler, Rory, Richie)

(Hallowell, Dember, Seigel, Pontillo)

The percent difference between our results and the results of the average of the other group's results were compared in order to determine the percent value of how much our results varied by.
 * Percent Difference Between Us and Class:**

In conclusion, we proved our hypothesis to be correct. We hypothesized that as the radius increases, the force will decrease. When we began our trials and recorded our data, exactly this happened. Force is indirectly proportional to radius, so it only makes sense that the force would decrease when the radius was increased or it would increase when the radius was decreased. We saw that when doing our lab, when the radius was shorter, we had to spin the string faster so that every trial would be uniform. When the radius was longer, we had to spin it much slower so that all the trials would be uniform, meaning they would have the same circumference every time. This also supports our conclusion. Because we had to spin it faster when the radius was smaller, more force was required. Spinning it slow with a longer radius could rotate with less force exertion. In conclusion, our experiment and resulting data fully supported our hypothesis that the radius and force are inversely related.
 * Conclusion:**

Our graph had an R^2 value of 0.9548. This value is relatively low, implying that our results are not extremely precise in respect to the line of best fit. However, the results were precise enough to determine an inverse relationship between force and radius.
 * Error Analysis:**

There are many potential sources of error in this experiment. The sources of error can cause the measured force to be wrong. We were able to determine that force and radius are inversely related, however our data may not have been perfectly accurate due to sources of error. If the string was held at an angle, the radius may not have been equal to the length of the measured radius. This would change the amount of force measured because the motion would not be perfectly circular. The speed may have changed slightly because of human error. An unknown change in speed would change the reading on data studio, and possibly lead to incorrect calculations. There was tape over the hook on the sensor, so the hook may not have been able to move as easily, causing the reading to not be perfectly accurate. Since there were not enough significant figures when using the mean, we had to use the y-intercept, which was difficult to get if the force was not constant. Also, the sensor may have not been perfectly precise.

In order to fix these errors, we can be careful and make sure the string was not held at an angle and measure the radius with a ruler as we are spinning to make sure that it is remaining the same. We can find another way to secure the string to the hook. For example, we can use less type. We could also monitor the speed more carefully with a precise stop watch in order to ensure that the speed remains constant. If we wanted to be more precise, we would need to acquire a more precise sensor. Additionally, we could have tried another way of using the sensor. For example, we can try using the sensor as the mass instead of the hanging masses.