Group1_2_ch6

Ryan Hall, Sarah Malley, Jenna Malley, and Kaila Solomontoc

=Law of Conservation of Energy Lab = **Objectives**: What is the relationship between changes in kinetic energy and changes in gravitational potential energy? **Hypothesis**: Due to the Law of Conservation of Energy, which states that energy cannot be created or destroyed, the energy in the beginning of the activity will be equal to the energy at the end of the experiment. **Procedure:**
 * __Station 1__: At the first station, we had to find the speed of the cart at the bottom of the incline. We measured the height of the ramp at the top and at the bottom. We measured the distance of the picket. We sent the cart down the incline with the picket fence on top of it, where it went through the photogate timer and we were able to record the time.
 * [[image:station_1_sjrk.jpg]]
 * __Station 2__: Our objective at this station was to find the speed of the ball when it left the launcher. To accomplish this, we measured the height of the launcher (and, by extension, the height of the first photogate) as well as the height of the second photogate above the ground. We also measured the diameter of the ball that served as the projectile. We recorded the times in the photogate and used those to calculate the velocity. We then found the mass of the ball.
 * [[image:station_2_sjrk.jpg]]
 * __Station 3__: The purpose of this activity was to find the speed of the mass at the lowest point of its swing. A wooden cylinder, affixed to a piece of string, served as our pendulum. We measured the diameter of the cylinder. We measured the initial height of the wooden cylinder above the photogate. Using this data, we calculated the velocity. We also found the mass of the ball.
 * [[image:station_3_sjrk.jpg]]
 * __Station 4__: Our goal here was to find the maximum height of the ball at the top of the incline. We first measured the minute height of the ball above the counter. We then sent the ball down the ramp and measured, on average, the max height the ball traveled (from the counter top).
 * [[image:station_4_sjrk.jpg]]
 * __Station 5__: The objective of this lab was to find the velocity of the ball as it left the launcher. We measured the diameter of the ball, found the mass of the ball, and found the time it took using the photogate timer. We used this data to calculate the velocity.
 * [[image:station_5_sjrk.jpg]]
 * __Station 6__: Our goal at this station was to find the speed at the top of the loop in the roller coaster. First we found the mass and the diameter of the ball. Then we measured the height from the counter to the top of the ramp. We let the ball go down the ramp and through the photogate timer, which gave us the time. We used this information to calculate the velocity.
 * <span style="font-family: Arial,Helvetica,sans-serif;">[[image:station_6_sjrk.jpg]]

<span style="font-family: Arial,Helvetica,sans-serif;">**Our Data**: <span style="font-family: Arial,Helvetica,sans-serif;"> <span style="font-family: Arial,Helvetica,sans-serif;">

<span style="font-family: Arial,Helvetica,sans-serif;">**Class Data**: <span style="font-family: Arial,Helvetica,sans-serif;">

<span style="font-family: Arial,Helvetica,sans-serif;">__Main Equation:__
 * <span style="font-family: Arial,Helvetica,sans-serif;">Analyzing the Data: **

<span style="font-family: Arial,Helvetica,sans-serif;">1.



2.

3.

4.

5.

6.

Our hypothesis that the energy in the beginning and end of the activity would be equal due to the Law of Conservation of Energy was not fully supported by the data we collected. Our percent differences came out to be 2.29%, 39.42%, 7.89%, 10.51%, 19.68%, 32.76% for stations 1-6 respectively. Had our hypothesis been supported, the percent difference for each activity would be 0%. While some of our values may seem as though they are off by a decent amount, there are different sources of error for each activity that can explain the percentages. For example, at Station 1, where the cart went down the incline, friction could be to blame for the small percent error. Additionally, for Station 2, where the ball was shot out horizontally, a slight angle on the shooter could have thrown off our results. For Station 3, where the wooden stick was sent through the photogate, we may have started swinging it at a position slightly higher or slightly lower than other groups due to inaccurate measurements. For Station 4, the percent difference may have just been an issue of where we started the ball as opposed to other groups, just like Station 3, height matters. Station 5's percent difference may be close to that of Station 2, where the shooter was off by a degree or two. For Station 6, the error may be contributed to possible friction on the ramp.
 * __Conclusion__**:

=The Law of Conservation of Energy for Mass on a Spring= part A: Based on Hooke's Law, the spring constant for each spring will be equal to the slope of their graph on a Position x Force graph. part B: Based on the Law of Conservation of Energy, the total energy at the max, min, and equilibrium points will be equal to each other. For the second part of the lab, tape a piece of cardboard to the bottom of a 500-g weight, which then attaches to the red spring. Use a DataStudio motion detector to record its position and its time as it springs up and down. Using the graph created from that data and the slope tools on DataStudio, find the max/min velocities and the velocity at equilibrium.
 * Objectives:**
 * Directly determine the spring constant k of several springs by measuring the elongation of the spring for specific applied forces.
 * Measure the elastic potential energy of the spring.
 * Use a graph to find the work done in stretching the spring.
 * Measure the gravitational potential and kinetic energy at 3 positions during the spring oscillation.
 * Hypothesis:**
 * Methods and Materials**: For the first part of the lab, attach springs to a ring stand, and attach the ring stand to a table with a clamp. Add weights to each of the springs, then measure the displacement in weight and in distance stretched. Plotted this data on a graph in order to find the slopes, which are equal to the spring constants.


 * Data for Part A**:
 * Graph for Part A**:

__Red Spring:__
 * Percent Error:**

__Green Spring:__ __ Yellow Spring: __ __Blue Spring:__ __Red Spring:__
 * Percent Difference for Spring Force Constants:**

__Green Spring:__ __ Yellow Spring: __ __Blue Spring:__


 * Graph for Part B**:


 * Data for Part B**:


 * Sample Energy Calculations:**



Yes. Each of the lines increases linearly and at a constant rate showing that the spring constant was the same for each range of forces. The one that has the smallest slope, or the least steep line, will be the "softest." This indicates that it had the most amount of position displacement each time a mass was added. The amount of energy never changes due to the Law of Conservation of Energy. When the hanging mass is at rest, it only has gravitational potential energy. When it's pulled on, more energy is added in the form of work and it keeps this energy as it goes up and down in the form of elastic potential energy and kinetic energy as it moves.
 * Discussion Questions:**
 * 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?**
 * 2. How can you tell which spring is softer by merely looking at the graph?**
 * 3. Describe the changes in energy of the hanging mass, beginning with it starting at rest, you pulling the mass down and then the mass cycling through one complete period.**

For part A of the experiment, our results came out fairly accurate. The highest percent error that we had for all four springs was 5.858%, and the largest percent difference that we had was 2.18% (compared to the class data). Our data definitely supported what the box told us the spring constant was, because it said there could be a plus/minus difference of 10%. The reason our experiment was probably so accurate was because this part of the lab left little room for error - it only consisted of adding weights and measuring the distance that the spring stretched. Our minor percent errors and differences can most likely be attributed to measuring errors. It can be difficult to make sure that the spring is completely still at the exact moment that it is measured, and the meter stick used to take the measurements is not always the most precise way to measure something. For part B, we were looking to see if the total amount of energy at the max, min, and equilibrium point was the same. After calculating them, these values came out to be somewhat close, but not exactly equal. After finding each individual value for total energy, we calculated the average and then did percent difference. At the minimum point, the percent difference was 30.28%, at the equilibrium point it was 1.54%, and at the maximum point it was 28.74%. There are certain sources of error that could be the cause of this occurring. One source of error is that the sensor only records at certain increments. Because of this, it may miss the actual maximum point. In order to help decrease this problem, we took data from multiple cycles and then averaged them.This is something that is difficult to avoid, and that is why it is important to collect data from more than one cycle.
 * Analysis/Conclusion:**

=<span style="font-family: 'Comic Sans MS',cursive;">Roller Coaster Project = Top View Side Views media type="file" key="Jenna Sarah Kaila Ryan Period 2 The Coaster of Doom.mov" width="300" height="300" Total energy in the roller coaster: Sample theoretical velocity (in this case, at the bottom of the first hill): Sample calculations for number of g's: Theoretical minimum starting height of roller coaster based on required energy:
 * Pictures**
 * Diagrams:**
 * Three Times Video!!**
 * Sample Calculations**

Theoretical spring system calculations: Power to get to the top of the coaster: (to solve for this, we made our time 30 seconds) Theoretical Min Speed at Top of Vertical Loop

Experimental Data The first thing we solved for was the total energy that the roller coaster would require to get going at the top of the hill. We knew that the total energy we solved for, would hypothetically remain constant throughout the marble's journey down the coaster. However, we also knew that this was not especially likely, as some energy in real life was likely to be lost along the way. This is what contributes to the difference in theoretical velocity and experimental velocity, as examined later on.
 * Data**
 * Theoretical Energy/Velocity at Various Points**
 * Number of G's at each Position:**
 * Point || Acceleration (m/s/s) || # of G's (g's) ||
 * Top of 1st Hill || 0.000 || 0.0000 ||
 * Way Down 1st Hill || 2.320 || 0.2367 ||
 * Bottom of 1st Hill || 2.253 || 0.2299 ||
 * Bottom of Vertical Loop || 0.864 || 0.0882 ||
 * Top of Vertical Loop || 0.238 || 0.0243 ||
 * Bottom of 2nd Hill || 1.430 || 0.1459 ||
 * Top of 2nd Hill || 1.010 || 0.1031 ||
 * Bottom of 3rd Hill || 0.712 || 0.0727 ||
 * Top of 3rd Hill || 0.617 || 0.0630 ||
 * Horizontal Loop || 0.608 || 0.0620 ||
 * The End || 0.429 || 0.0438 ||
 * Discussion**

We know that the **Law of Conservation of Energy** States that energy cannot be created or destroyed, but it can be dissipated - transformed into a different type of energy. Hypothetically, none of the energy in the project would be "lost" during our marble's journey down the roller coaster. Realistically, however, this is not particularly likely. It is common that because we did not have ideal circumstances, the energy changed forms. Due to any sort of shaking or human error, our experimental results will be different than our theoretical results. In order to try to eliminate this as best as possible, we added supports to the roller coaster at various points, and tried to keep the paper we were using for the track as smooth as possible, rather than crumpled and creased.

Although we didn't actually build an initial incline that would be there to get the marble to the first drop, we created a hypothetical one and calculated the **power** that would be necessary to get the marble to the top of the drop. Power is the rate at which energy is transformed, or in other words, the rate at which work is performed. The equation of power equals the work necessary to move the object over the total time this increment of the journey takes. This becomes GPE over the time it takes for the marble to get up the hill. We measured the mass of the marble, and the height it would need to travel. We then chose a logical time, of 30 seconds, of how long it would take for the marble to get to the top. The power value we then calculated is equal to 0.0099W.

Because our ball is moving, for the most part, in a downward slope, there is **acceleration**. However, as the slope changes depending on the location, the acceleration is different at different points throughout the roller coaster. Despite this, the total energy theoretically remains the same, no matter the steepness of the slope or any changes in velocity. In order to calculate acceleration, we used various techniques. We used what we know from kinematics, as well as what we know about centripetal forces. Due to the change in level and design of the roller coaster, there is acceleration at all the major points on it.


 * Newton's Laws of Motion** can be applied to the roller coaster we created. Newton's first law states that an object will continue in it's same state of motion, unless acted on by an outside force. This is true for the marble in the roller coaster. Without friction, normal forces, or gravity, the marble would continue to roll on at a constant rate. Due to the fact that there was friction on the roller coaster from the surface of the paper and tape, the marble was affected. Additionally, the upside-down loop provided a force of gravity acting upon the marble as well. Also, each time the marble came in contact with the sides of the paper where it was folded, a normal force was present, changing the marble's state of motion. Newton's second law states that net force is equal to (mass)(acceleration). This means that the mass of the marble along with the forces acting upon it determined its acceleration. For example, when the marble was traveling down an incline, the normal force, the force of friction, and the force of gravity acted on the marble to control its acceleration. Newton's third law of motion states that every action has an equal and opposite reaction. This was shown in the roller coaster in the case where the marble was at the top of the vertical loop. In a real-life situation where there are people in carts, the cart would exert a force on the people, who would then exert and equal and opposite force back onto the cart in order to remain in their seats. The normal force and the force of gravity oppose each other, allowing a person to not fall out.

At all points on the roller coaster, the **force of gravity** is -9.8m/s/s. Wherever the marble is, this is how much gravity is acting on it. The concept of **apparent weight** is also present at certain points on the roller coaster, and gravity and normal forces are the only two forces that affect apparent weight. When the normal force is equal to zero, this is when a feeling of weightlessness occurs. An example of this is at the top of a hill due to the fact that normal force and gravity are pointing in opposite directions. The object may also feel as though it is heavier than it is when normal force and gravity point in the same direction, such as at the bottom of a hill.

For the theoretical spring constant, we solved to find the spring constant necessary to stop our roller coaster in case of break failure.We knew that the elastic potential energy of this spring would need to be equal to the theoretical energy that we had previously solved for. We used the stopping distance the marble would need (as we have a sort of landing area included in our roller coaster design) to solve for this. In our case, the stopping distance was .00872 m. As a result, we discovered that the necessary spring constant was 33.9 N/m. We also used the idea of **Hooke's Law** to solve for the spring constant. Hooke's Law states that the extension of a spring is directly proportional to the force that is applied to it. The equation that represents this is F=-kx. In this equation, F is the force, k is the spring constant, and x is the amount of compression or stretching.

Due to the various loops and turns, **circular motion** can be applied to the roller coaster. In the vertical and horizontal loops, there is a centripetal force that points toward the center of the circles, creating an acceleration. This type of force allows the marble to remain on its path, which is important in a roller coaster so that people do not fly off.

=Lab: The Law of Conservation of Mass= media type="file" key="Movie on 2012-03-14 at 08.36.mov" width="300" height="300"
 * Objectives**: What is the relationship between the initial momentum and final momentum of a system? Which collisions are elastic and which ones are inelastic collisions?
 * Hypothesis**: We hypothesize that the initial momentum of a system will be equal to the final momentum of that system. We know that if a system conserves kinetic energy, it's an elastic collision, and vice versa. Because of this, we believe that the collisions in our system were inelastic.
 * Procedure**: We set up force sensors at either end of the track. We got two carts and measured their masses using a scale; we did the same with the weights that we got. We set up a velocity vs. time graph on the laptop hooked up to DataStudio. We pushed the carts into each other, each trial with various masses, and we recorded the results in a data table.
 * Data/Graphs**:

Percent Difference in KE Elastic or Inelastic? (where mass is measured in grams) KE percent difference sample calculation

Momentum on both sides: (where mass is measured in grams) Sample Calculations for Total Momentum:

Sample Calculations for Percent Difference: Momentum was conserved in this experiment. We collected enough evidence to support the Law of Conservation of momentum. While the final and initial momentums were not fully equal to each other, this can be due to human error, or other outside forces not taken into account. For example, in one trial where cart A was pushed into B at rest, the initial momentum was .18m/s and the final momentum was .17m/s. This is a percent difference of only 5.71%, allowing us to conclude that momentum was conserved.
 * Analysis**:
 * 1. Is momentum conserved in this experiment? Explain, using actual data from the lab. **

The cart with the smaller mass with have a larger velocity. This occurs because the force from the explosion created will push the lower mass harder and at a higher velocity than the other cart. The equation for momentum is mass times velocity, and if momentum is the same, the smaller mass cart will move faster.
 * 2. When carts of unequal masses push away from each other, which cart has a higher velocity? Explain why this is. **

In a situation like this, the momentum will be the same for each. Since the equation for momentum is mass times velocity, and the lighter cart will have a higher velocity while the heavier cart will have a lower velocity, the momentum will remain even when this is accounted for.
 * 3. When carts of unequal masses push away from each other, which cart has more momentum? **

No, momentum does not depend on this. Both carts still have the same force acting upon them, and so, the plunger will have the same effect on each. The equation for momentum is mass times velocity, so momentum is not dependent on the plunger.
 * 4. Is the momentum dependent on which cart has its plunger cocked? Explain why or why. **

We did five different trials that supported our hypotheses. The initial momentum of the carts during each of the trials was, when taking into account human error, nearly the same. We also did calculations do see if it was elastic or inelastic, and we discovered that all of the trials we did involved inelastic collisions.
 * Conclusion**:

Some of our percent differences are larger than others. Firstly, we can attribute this to the fact that the numbers we were working with were so small that even the littlest differences were very blatant. As well as this, there was, as always, some human error that played a role. Hands that lingered in front of the sensor could have skewed the results on our graph. Other than that, all measurements were computerized.

=Ballistic Pendulum Lab= Pendulum Photogate Kinematics
 * Objective**: What is the initial speed of a ball fired into a ballistic pendulum?
 * Hypothesis**: In order to do this lab, we must find the initial velocity in several different ways. We hypothesize that, no matter which way we calculate the initial velocity, they will all be similar to each other, if not the same. Using the photogate timer will likely give us the most accurate results.
 * Procedure**: First, we attached the pendulum to the launcher. We launched it into the ballistic pendulum and found the angle of every launch. We also measured the length of the pendulum and the mass of the After we were finished with this part of the data collection, we removed the pendulum from the launcher and put it off to the side. Then we attached the photogate timer to the launcher, and measured the time of the ball in-gate. We also took this time to measure the diameter of the ball. Finally, we removed the photogate timer from the launcher and attached a piece of carbon paper to the floor where it would land. We measured the height of the launcher above the ground, and the distance between the landing point and the launcher. We used all of this information to solve for velocity in three different ways.
 * Data:**

Deriving the equation to solve for velocity using the ballistic pendulum: Is energy conserved in the pendulum? Is momentum conserved? As shown in above calculations, the initial KE = .3295J and the final KE = .0548J. The difference in kinetic energy, and therefore the loss in kinetic energy, is shown by subtracting .0548 from .3295, which equals .2747J. The loss of Kinetic energy divided by the original KE, .2747/.3295 = .8337. The percentage lost in KE is 83.37%**.** Going by the calculations, it is definitely not valid for us to assume that energy was conserved in this collision. The difference between initial and final KE is .2747J which is an 83.37% loss in Kinetic Energy. These are large numbers that show that energy was not conserved. M=.246kg m=.066kg .246/(.066+.246) = .7885 This ratio is 78.85%**,** comparing to the 83.37% loss in KE calculated in part b) of this question. These values are close, they are off by about 4.52%.
 * Calculations:**
 * Discussion Questions**:
 * 1) **In general, what kind of collision conserves kinetic energy? What kind doesn't? What kind results in maximum loss of kinetic energy?**
 * 2) Elastic collisions conserve kinetic energy. These include head-on collisions and ballistic pendulum collisions. Inelastic collisions don't conserve kinetic energy. Explosions and Newton's Cradle are examples of inelastic collisions. The maximum loss of kinetic energy is in collisions where the objects in the system stay together after the collision.
 * 3) **Consider the collision between the ball and pendulum.**
 * 4) **Is it elastic or inelastic?**
 * 5) It is an inelastic collision.
 * 6) **Is energy conserved?**
 * 7) No - some is lost, according to our calculations.
 * 8) **Is momentum conserved?**
 * 9) Yes, according to these calculations:
 * 10) [[image:Screen_shot_2012-03-21_at_2.39.27_PM.png]]
 * 11) **Consider the swing and rise of the pendulum and embedded ball.**
 * 12) **Is energy conserved from the moment just before the ball strikes the pendulum to the moment the pendulum rises to its maximum height?**
 * 13) While the kinetic energy is not conserved, the total energy in the system, including both potential and kinetic energies, is conserved.
 * 14) **How about momentum**?
 * 15) Yes, the momentum is conserved.
 * 16) **It would greatly simplify the calculations if kinetic energy were conserved in in the collision between ball and pendulum.**
 * 17) **a)Calculate the loss in kinetic energy as the difference between the kinetic energy before and immediately after the collision between ball and pendulum.**
 * b) What is the percentage loss in kinetic energy? Find by dividing the loss by the original kinetic energy. **
 * c) According to your calculations, would it be valid to assume that energy was conserved in that collision? **
 * d)** ** Calculate the ratio M/(m+M). Compare this ratio with the ratio calculated in part 2. Theoretically, these two ratios should be the same. State the level of agreement for these two quantities for your data. **

Percent difference calculations for each method are shown below: Various factors that would increase the difference between these methods. The kinematics projectile one involves the most measurements done by hand, and the launcher is slightly inconsistent, making it possibly the least accurate. The photogate on the other hand is the most accurate, as the diameter of the ball is a standard value, and the value of the time is given through the computer. Also, in the pendulum, although we did what we could to decrease this problem, involves some friction. Although it is only a slight amount, it could affect the outcome of the velocity. As a way to get better results using the ballistic pendulum method, we could get a digital reading of the angle. This would eliminate the friction factor and give a more accurate value for theta.
 * 1) **Go to**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12px;">**[]** **. 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).**
 * 2) The higher the mass of the ball is, the higher the pendulum swings because there is a stronger force acting on the pendulum. The higher the mass of the pendulum, the less it travels because it's heavier and it involves more force to move it. The momentum and the kinetic energy are conserved in this elastic situation because ballistic pendulum scenarios nearly always conserve energy.
 * 3) **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?**

Our hypothesis was mostly correct for this lab. All three methods each gave us a very similar initial velocity of the ball, but the ballistic pendulum LCE method gave us the velocity closest to the average, not the Photogate method as we originally though. In fact, the Photogate gave us the result that was most off from our average calculated initial velocity. Our average initial velocity for the ball being shot at mid-range was 3.216 m/s, with the ballistic pendulum LCE method giving us a result of 3.160 m/s, the kinematics projectile method giving us 3.089 m/s, and the Photogate method giving us 3.399 m/s. These methods were off from the average by 1.74%, 3.95%, and 5.70%, respectfully. These errors could have arose due to numerous causes. Firstly, as the spring warms up after being used so often, it in fact begins to change the initial velocity of the ball. To fix this, time could have been spent waiting in between launches to allow the spring to cool. Also, in the kinematics method, the distances from the launcher to the spot where the ball hit the carbon paper on the floor each time could have been off, due the launcher being elevated on the table. This could've been solved by marking on the ground under the table exactly where the barrel of the launcher was positioned, and measuring more accurately from that spot. Also, fingers in the way of the Photogate's laser while holding the sensor in front of the launcher could have thrown of our times for the Photogate method. For next time, we would be sure that the laser has a clear path from one side of the sensor to the other, and is only obstructed due to the ball passing through it. This could be applied in a myriad of situations in real life. Car crashes, explosions, and even a simple game of pool involve momentum. It is important to understand these topics and how velocity, mass, and momentum all relate to each other in each instance, either to find the perfect shot in pool by knowing how the momentum is conserved and transferred when balls hit into each other, or in car crashes to find out how much energy will be transferred and how much the velocity will change, and if the crash is lethal or not.
 * Conclusion**