Group4_2_ch6

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toc =Lab: Ballistic Pendulum= Caroline Braunstein, Julia Sellman, George Souflis, Maddy Weinfeld Period 2 3.21.12


 * Objective:** What is the initial speed of a ball fired into a ballistic pendulum?


 * Hypothesis:** Despite the three different ways initial speed of the ball is calculated, the initial speed should be the same because no variables are being changed. Due to error, they speeds will probably not all be identical. The photogate will probably produce the most accurate speed because it is recorded by the computer.

This lab was broken up into three parts. We first recorded the speeds using the ballistic pendulum method. For this part of the lab our materials were a projectile launcher, a ballistic pendulum, and a ball. We were able to record the angle the ball created when launching into the pendulum. Then, we used a photogate timer for the next part of the lab. We had the same materials, but instead of a ballistic pendulum, we used the photogate timer, which allowed us to get a time for the initial velocity. With this, we can just divide the diameter of the ball by the time to find the speed. Lastly, we used kinematics to find the speed of the ball. For this, we used a launcher, ball, carbon paper and a measuring tape. We recorded the height of the launcher and the distance the ball was launched so that we could use kinematics to solve for velocity. media type="file" key="Movie on 2012-03-21 at 08.19.mov" width="300" height="300" media type="file" key="Movie on 2012-03-21 at 08.32.mov" width="300" height="300" media type="file" key="Movie on 2012-03-21 at 08.24.mov" width="300" height="300"
 * Methods and Materials:**
 * Data:**
 * Sample Calculations:**

1. In general, what kind of collision conserves kinetic energy? What kind doesn't? What kind results in maximum loss of kinetic energy? 2. Consider the collision between the ball and pendulum. 3. Consider the swing and rise of the pendulum and embedded ball. 4. It would greatly simplify the calculations if kinetic energy were conserved in the collision between ball and pendulum. 5. Go to [] Select “Ballistic Pendulum” from the column on the left. What is the effect of increasing the mass of the ball? What is the effect of increasing the pendulum mass? Try it. (NOTE: You have to read “Student Notes” first before you can run the simulation.) 6. Is there a significant difference between the three calculated values of velocity? What factors would increase the difference between these results? How would you build a ballistic pendulum so that momentum method gave better results?
 * Analysis:**
 * Discussion Questions:**
 * Elastic collisions conserve kinetic energy. Inelastic collisions do not conserve kinetic energy. An inelastic collision causes the maximum loss of kinetic energy. For example, when a car crashes into another car at rest, and the center of mass does not change, the maximum kinetic energy is lost.
 * Is it elastic or inelastic?
 * inelastic
 * Is energy conserved?
 * it isn't conserved
 * Is momentum conserved?
 * it is conserved
 * Is energy conserved from the moment just before the ball strikes the pendulum to the moment the pendulum rises to its maximum height?
 * Energy isn't conserved because some is lost and transferred.
 * How about momentum?
 * Momentum is conserved throughout the experiment.
 * Calculate the loss in kinetic energy as the difference between the kinetic energy before and immediately after the collision between ball and pendulum.
 * [[image:Screen_shot_2012-03-22_at_11.15.15_AM.png width="173" height="118"]]
 * What is the percentage loss in kinetic energy? Find by dividing the loss by the original kinetic energy.
 * [[image:Screen_shot_2012-03-22_at_11.15.21_AM.png width="231" height="130"]]
 * According to your calculations, would it be valid to assume that energy was conserved in that collision?
 * There was very little energy lost throughout the experiment, although energy was definitely lost. Therefore, it wouldn't be valid to assume that.
 * Calculate the ration M/(m+M). Compare this ratio with the ratio calculated in part
 * Theoretically, these two ratios should be the same. State the level of agreement for these two quatnities for your data.
 * M/ (m+M)
 * .247/(.066+.247)
 * .789
 * When the mass of the ball increases, the height the ballistic pendulum moves increases as well. Though contrarily, when the pendulum's mass increases, the height decreases.
 * There is a small difference between the three calculated values of velocity. Factors which could have increased the difference between velocities could have been from the inconsistent launcher, air resistance wasn't accounted for, and measurements could have been off as well. We could build a ballistic pendulum which didn't have a significant amount of air resistance or that had a more specific measurement system for the angle so we could make sure out results were more accurate.


 * Discussion:**

Our hypothesis was correct for this lab. The three methods used to calculate the initial velocity of the steel ball yielded very similar results. Using the photogate, the calculated average velocity was 3.86 m/s. Using kinematics, it was on average 3.59 m/s. Using the Law of Conservation of Energy/Momentum, it was on average 3.80 m/s. That is below 10% difference between the three of them. The most precise of the three turned out to be the photogate method, yielding a 0% difference between 5 separate trials. The little error we did experience mostly came from the kinematics method. The exactness of measurement is always an issue; some guessing is always present because the changing height makes it difficult to measure from the barrel of the launcher to the carbon paper. This could be fixed if we had marked the exact positions with tape to help measure. Also, the launcher is fairly inconsistent. Whereas the photogate will register the exact time every single trial, the launcher shoots at varied positions. It took 10 or 11 shots from the launcher to make a decent cluster of 5 trials on the carbon paper. This definitely affected our results for obvious reasons. As far as real-life application of this concept, it can be seen in widely different situations as a game of billiards versus a car crash. In billiards, its important to understand how the momentum is conserved and transferred between balls in order to get a perfect shot. In a car crash, consider when a moving car hits one that is stationary. The momentum of the moving car can cause the stationary car to move quite significantly, similar to the ball hitting the pendulum.

=Lab: The Conservation of Momentum= Caroline Braunstein, Julia Sellman, George Souflis, Maddy Weinfeld Period 2 3.15.12


 * Objective:** What is the relationship between the initial momentum and the final momentum of a system? Which collisions are elastic collisions and which ones are inelastic collisions?


 * Hypothesis:** The initial momentum and the final momentum of a system should be equal. This is because of the law of conservation of momentum. Elastic collisions are when KE is conserved, and inelastic collisions are when KE is not conserved.


 * Methods and Materials:** Set up a //dynamics track// on a flat surface. On both ends, place //motion detectors// and connect them to two separate computers via //USB links//. Create experiments on both computers and set up a velocity vs. time graph. This will record the speed (and direction) of the cart. Place two //carts// on the dynamics track. Depending on the collision type, the procedure will differ from this point on. For "both in motion - sticking together", lightly push the first cart, and then push the other cart in the same direction so that it collides with the first cart and the two continue moving together (stuck by Velcro). Do several trials and add various //masses//. Record masses and velocities before and after the collision to calculate initial momentum and final momentum.

media type="file" key="Movie on 2012-03-14 at 08.20.mov" width="300" height="300"


 * Data (Spreadsheet+Class Data**):

//Spreadsheet:// //Class Data://

percent difference: percent difference/average:
 * Sample Calculations:**


 * Analysis:**

//Analysis Questions:// 1.Is momentum conserved in this experiment? Explain, using actual data from the lab. Momentum is conserved in this experiment. This can be seen by looking at the initial and final KE of each experiment. The values of initial and final kinetic energy are close enough that the difference between the two can be caused by human error. 2. When carts of unequal masses push away from each other, which cart has a higher velocity? Explain why this is. The cart of smaller mass has a higher velocity. The cars exert an equal amount of force on each other because of Newton's third law of motion. The force exerted on the smaller mass of the car makes the car go faster than the larger mass car.

3. When carts of unequal masses push away from each other, which cart has more momentum? The carts will have equal momentums. Momentum = mass*velocity. So the cart of smaller mass will have a higher velocity, but the momentums will be equal.

4. Is the momentum dependent on which cart has its plunger cocked? Explain why or why not. The plunger should not affect the momentum because momentum is based on two factors, mass and velocity. Nowhere would the plunger come into play in the equation.

 Throughout this experiment, we were testing the Law of Conservation of Momentum. We knew that the total initial momentum must equal the total final momentum, according to this principle. We wanted to find the mass and velocities right before the crash and right after, so that we could compare their total kinetic energy. We did these using cars and data sensors to find our results through excel. We included this theory in our hypothesis, and we were trying to see if the collisions were elastic or if they were not. We found that our hypothesis essentially was correct. This is because the collisions that we had which were elastic had the same initial and final total energy, while the inelastic collisions turned out to have different initial and final total energies. The collisions that were inelastic tended to have a great percent difference, while the elastic collisions usually or almost always had a percent difference less than 10 percent. Because we had been previously told that this percent error was approximately correct, we know that our results were pretty accurate.  There was definitely placement for error throughout the experiment. This included reading the data correctly off the graph, which we found become much easier to do when the initial velocity started off slightly greater than lower, because the data points were more prominent. We also needed to figure out the correct signs of negative or positive for the data points because each of our sensors had different positive and negative directions. Therefore, we had to have a general view and determine the signs ourselves when collecting the data in the table. Sources of error could have also included hands getting in the way when we were doing the trials. We didn’t account for friction and results could have varied from different places of contact due to different frictions. There is also a possibility that percent error stemmed from different initial velocities that we gave the carts manually. To decrease the percent error, we could have used a frictionless surface and we could have been more accurate when we pushed the carts initially, avoiding any contact with the sensors. All of our percent errors were below 10%, implying that we got pretty good results.  There are many places in everyday life to which this lab applies. This kind of trial when both cars are going in the same direction can happen in bumper cars and it can also happen in the game of pool, when one ball is moving and another is pushed into it to make it go into a goal.
 * Discussion**:

=Roller Coaster Project=

Top View:

Side View:

Video: media type="file" key="julia maddy caroline george coaster CLEAR VIDEO.mov" width="300" height="300"

Data: Link to Spreadsheet:

percent error:

A. A sample of each calculation included in the Excel spreadsheet.





Theoretical Calculations:

C. The minimum speed requirement a the top of the vertical loop. D.The minimum height requirement of the first hill

Discussion: The Energy Conservation Law states that the total amount of energy should remain constant throughout our roller coaster ride. In our roller coaster calculations, we didn't account for friction. Therefore, we didn't account for energy lost as heat transferred into another form. Air resistance and air pressure also weren't accounted for. Though, in real life, roller coaster designers must address many of these factors because those are factors, which riders will encounter when going on the roller coaster. There is a block system which prevents trains from colliding when more than one coaster is riding at the same time. There are operating computers, which control the ride, and these are inspected regularly. There are all kinds of sensors, which detect where the ride is at all times, and if problems occur, the proper precautions are taken. A backup break is always provided in the case that an emergency stop is necessary. These changes in the total energy on the roller coaster account for why there is a loss of energy in the system. We know through Newton’s second law that due to the unbalanced forces throughout the roller coaster’s course, there will be acceleration. The gravity is acting on the marble through the force of weight while there is also a friction force. These forces need to be proportioned by the roller coaster in a certain way so that the marble can make it through the vertical loops etc. Newton’s third law explains how every action has an opposite and equal reaction. The third law is exemplified in our ride because the marble must stay stay down on the track without falling out, which we were able to accomplish. The forces of gravity is pushing down on the marble while the normal force from the roller coaster is pushing up on it. Roller coaster builders also need to consider making a motor that will give the ride enough power to make it through its entire course. We had to imagine a motor to take the roller coaster to the top and we chose to make the ride to the top take 30 seconds because we feel like that is an average amount of time for most roller coasters. We then calculated the amount of power of our imaginary motor using this time and our calculations for work. We calculated that 00083 watts, which is small compared to real life roller coasters. For the bottom of the roller coaster, we created a hypothetical spring which would act as a backup stopping force. We used Hooke's law F=kx in order to find the spring constant. Because x represents the distance, we plugged in a reasonable distance and the force, and from there found the spring constant. K is the spring constant and we found this to be approximately 24 N/m. Hooke's Law implies that the amount that the spring extends will be in proportion to the weight causing this change in disposition.

Throughout the whole ride, the ball is accelerating, meaning its rate of velocity change over time is either increasing or decreasing. Whether it is going down hill, uphill, or through a loop, the ball is constantly accelerating at different rates as it goes through the roller coaster. This can be seen above where we calculate the acceleration and number of G's. At each of the key points, the acceleration is different and therefore so is the number of G's.

There were several sources of error in our project. First of all, our roller coaster was not perfectly built and should have had more supports. The lack of supports throughout made it difficult to do the analysis. The roller coaster shook a bit as the marble travelled down the coaster making it difficult to get exact measurements. To fix this problem, we should have redone the trials after adding more supports. It was also difficult to place a photogate timer in the perfect position to only get one reading. Therefore, some of our times may be slightly off. To fix this problem we should have kept the analysis more in mind while we were building. We could have left openings in the walls or built the walls lower down so that there were no pieces of paper in the way of the photogate, which could have changed our results.

In order for a roller coaster to be deemed "safe", its acceleration cannot exceed 4 G's or 39.2 m/s/s. Unsafe roller coaster accelerations can cause lightheadedness and blacking out. Some people can withstand more the 4G's, but the general limit for people is 4 G's. According to our calculated accelerations from the analysis, at no point was our roller coaster unsafe. The acceleration never exceeded 4 G's and therefore can be deemed safe acceleration wise. Though, there could always be improvements, which include using materials which are much more dense and sturdy. If we could use a sturdier material and set up the photogates in places which would allow us to take better measurements with no obstructions, we would probably get much better approximated results. This could have given us a much more realistic idea of how safe our roller coaster really is.

Using our knowledge about centripetal motion, we were able to determine the minimum speed at the top of the vertical loop by setting the normal force equal to zero. Then using the minimum speed determined, we were able to figure out what the minimum height requirement of the first hill was. We did this by making a total energy equation with GPE on one side and KE on the other. The velocity used in KE was the minimum velocity from the top of the loop. We then solved the equation for h or the height of the first hill. When the normal force is changed, that affects the apparent weight that an object has. For example, at the top of a hill the normal force is zero, making the person in the car feel weightless, this is their apparent weight. At the top of the loop in our roller coaster, the rider would feel weightless due to the low normal force.

Apparent weight is a measure of the downwards force. When there is a greater downwards force rather than just weight, or gravity and mass, this is apparent weight. At the bottom of the vertical loop, one would feel heavier than usual, while at the top, one would feel "weightless." This is because at the bottom of the loop, the normal force is larger than the weight. At the top of the loop, the weight is larger than the normal force.

<span style="font-family: 'Times New Roman',Times,serif;">Overall, our roller coaster was relatively safe, although there would have to be further analysis in order determine its strengths and weaknesses. As we discussed, there could have been error in our calculations, although as real roller coaster designers would have to do, we would have to create many backup devices which would come in handy if there was some kind of problem with the ride.

=The Law of Conservation of Energy= group members: George Souflis Caroline Braunstein Maddy Julia

What is the relationship between changes in kinetic energy and changes in gravitational potential energy?
 * Objective**

station 1: find the speed of the cart at the bottom of the incline A dynamics cart with a "picket fence" on the top was set on an inclined track with a photogate timer at the bottom. The cart was sent down the incline and the top bar of the picket fence went through the photogate timer. The "time in gate" was used to determine the final velocity of the cart by dividing the length of the bar by the time in gate.
 * Methods and Materials**

station 2: find the speed of the ball when it leaves the launcher and the final speed at the bottom of the path, just before it hits the ground A projectile launcher was set up to shoot the ball horizontally, a photogate timer was placed just outside the launcher's "mouth" so that it would record the time in gate just as the ball left the launcher. A photogate timer was also set up on the ground, to record the time in gate at the bottom of the ball's path. Using the diameter of the ball and the time in gate's, the initial and final velocity of the ball was determined by dividing the diameter by the time in gate.

station 3: find the speed of the mass at the lowest point of it's swing A mass was attached to a string, which was then hung from a ring stand. The mass was set so that it would swing right through a photogate timer at its lowest point of swinging. The mass was swung from 20 cm high through the photogate timer and the diameter of the mass was divided by the time in gate to determine the velocity at the lowest point of it's swing.

station 4: A ball was rolled down a ramp that curved up at both ends, but at different angles both having the same final height. The initial height of the ball was recorded along with the final height of the ball's path.

station 5: find the speed of the ball as it leaves the launcher A projectile launcher was set up to launch the ball vertically upwards. A photogate timer was set up at the mouth of the launcher in order to record the initial velocity of the ball (by dividing the diameter of the ball by the time in the photogate). The maximum height the ball reached was recorded using a meterstick.

station 6: find the speed of the "roller coaster" at the top of the loop A metal track with a loop at the end was set up with a photogate timer at the top of the loop to record the time in gate as a ball went past after being rolled down the track. The height of the top of the track and the top of the loop was recorded. Using this data


 * Data**




 * Total Energy Equations**



=Lab: The Law of Conservation of Energy for a Mass on the Spring= The spring which stretches the least with increased mass will have the largest k value, while the spring which stretches the most with increased mass will have the smallest k value. K value represents the spring constant.
 * Hypothesis:**
 * Objectives: **
 * To directly determine the spring constant k of several springs by measuring the elongation of the spring for specific applied forces.
 * To measure the elastic potential energy of the spring.
 * To use a graph to find the work done in stretching the spring.
 * To measure the gravitational potential and kinetic energy at 3 positions during the spring oscillation.
 * Methods and Materials:**

We attached two clamps with three hooks on each side to the metal stand displayed in the picture. We then hung four different springs from the stand. We hung different masses from the springs because we needed to overcome the inertia of the spring in order to find the spring constant by adding several masses and recording the different distances that the springs traveled. media type="file" key="IMG_0555.mov" width="330" height="330" For the second part, we put 500 g on the bottom of the red spring and there was cardboard at the bottom of the weight to create a surface to be detected by the motion detector. We then pushed down on it and released it. We recorded this on data studio, then finding the maximum displacement, the minimum displacement, and the equilibrium.

Data Tables and Graphs: Part A Data Table:

Graph:

Part B: Position v. Time Graph Velocity v. Time Graph Percent Error: Yellow Green Theoretical: 50 N/ m Experimental: 54.064 N/ m Percent error: 8.13% error Red Theoretical: 25 N/m Experimental: 26.335 N/m Percent Error: 5.34% error

Blue Theoretical: 30 N/m Experimental: 30.339 N/m Percent Error: 1.113% error

Percent Difference:



Yellow Individual Experimental: 35.5 Average Experimental: 36.63 Percent Difference: 3.08% difference Blue Individual Experimental: 30.339 Average Experimental: 30.87 Percent Difference: 1.72% difference

Green Individual Experimental: 54.064 Average Experimental: 50.67 Percent Difference: 6.7% difference Discussion Questions: Part B Analysis:
 * 1) Do the data for the displacement of the spring versus the applied force indicate that the data for the spring constant is indeed constant for this range of forces?
 * The data does suggest that the spring force constant (k) is constant for this range of forces because the graph is linear. The slope of these graphs indicates the spring force constant and the slope of a linear line is constant.
 * 1) How can you tell which spring is softer by merely looking at the graph?
 * You can tell that the red spring is softer because on the graph, you can see how they have the least steep slope, therefore having the lowest spring force constant. The softer springs have the greatest displacement because they are the easiest to manipulate.
 * 1) Describe the changes in energy of the hanging mass, beginning with it starting at rest, you pulling the mass down and then releasing it, and then the mass cycling through one complete period.
 * There are several energy transfers when a spring is pulled and released. When the total mass is at its minimum, there is elastic potential energy and gravitational potential energy. There is not kinetic energy because the spring is not in motion at the minimum and the velocity should be zero (in the lab, it was very close to zero). At equilibrium, there is kinetic energy, EPE, and GPE. At the maximum, there is only GPE because there is no motion and the spring is not being stretched. Despite the different types of energy present, the total amount of energy at each different position is equal according to the Law of Conservation of Energy.

Conclusion: We hypothesized at the beginning of the experiment that the spring with the smallest spring constant would extend the most and the spring with the largest spring constant would extend the least. For the green we found a spring constant of 54.064 N/m, the red we found had a spring constant of 26.335 N/m, the blue had a spring constant of 30.339 N/m, and the yellow had a spring constant of 35.5 N/m. This implies that the red would extend the most and the green would extend the least. We found this hypothesis to be correct because our graph shows that with the increased amount of weight, the red traveled further than the rest of the springs, implying that it had the highest spring constant. On the other hand, the green line shows that the spring traveled the least distance in relation to the forces that were applied to it. There was definitely some error throughout our experiment, although it was relatively minimal. In Part A, on the yellow spring, we had a percent error of 1.43%, on the green 8.13%, on the red 5.34%, and on the blue 1.113%. This was a difference fro out theoretical values listed on the box. We found these values through the slope of the line on our graph, which showed the approximate spring constant. Our results were relatively good, considering there was minimal percent error. In Part B of the experiment, we also had very good results, having less than 2% difference between all of our results. We got percent differences of 1.078% and 1.914%, which both show a minimal amount of error from our calculations. Error could have been from our measurements, which did come from a meter stick and therefore can only be measured to a certain decimal place. The springs could have also had the slightest velocity when we were taking measurements, which wasn't accounted for in our equations, and could have accounted for some error. Ways to improve this would be being in a room without any vents, which could help less air flow, and therefore not move the springs as much. We could have also all measured the displacements of the springs and averaged our measurements to account for anyone's error and get more accurate results. A real life application of this would be in bungee jumping. Whenever you go to do any kind of bungee jumping, weight is accounted for because a spring can account for only so much weight, and by knowing the spring constant, it can be figured out what spring is best for your weight to make a safe ride.