Group4_6_ch4

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= Gravity and the Laws of Motion. = Group members: Rachel Knapel (task B), Jake Greenstein (task A), Lerna Girgin (Task C), all split up part D Period 6 11/15/11

1. Find the value of acceleration due to gravity 2. Determine the relationship between acceleration and incline angle 3. Use a graph to extrapolate extreme cases that cannot be measured directly in the lab 4. What is the relationship between the mass of the rolling ball and its acceleration.
 * Objective:**

1. You can expect acceleration due to gravity to be 9.8 m/s because we have measured it before when doing free fall and projectiles. 2. Gravity and normal forces, on a flat surface, are at equilibrium. This is because gravity points straight down, while normal points straight up, making them opposite each other. A ball on a flat surface, therefore, doesn't move. However, if we make the surface the ball is on at an angle, the normal force will not be directly opposite gravity. This is because normal force is always perpendicular to the surface. Therefore, the resulting force will be along the declining surface, making the ball roll down. Therefore, the greater the angle the greater the acceleration. 3. We can use the slope we find for the graph we make to extrapolate extreme cases. 4. The mass of the ball should not affect the acceleration is we didn't have air resistance. Since we do have air resistance the heavier the mass the greater its acceleration would be.
 * Hypothesis:**

Start by gathering all the materials. You will need a ramp, a clamp, a ring stand, a ball, a stop watch, and someplace to record your data. You begin by setting up your ramp. Do this by taking the ramp and putting the clamp of the ramp to be able to connect it to the ring stand. Once this is done you measure 15cm for you height from the bottom of the ramp to the table, then you tighten the clamp to the ring stand to ensure that the ramp will stay up at this height. After this, you take your ball and start at the point where the height is 15 cm and then take the ball letting it go down the ramp until it gets to the bottom, as you are doing this you must time how long it takes to get from the top to the bottom. Once this set is done record the data you found, the height, the change in distance it traveled, and the time it took. After, keep the clamp tight and measure another height along the ramp and repeat the above steps to find the time it took to get from this point to the end. Repeat this one more time and record all of your data. Once this is done loosen the clamp and change the height to 20 cm. Repeat the above steps by doing three trials at three different height. Once that is done and the data is recorded change the height one for time to 30 cm so that you will end up measuring data for 3 different angles. Repeat the above steps by doing three trials at three different heights, finding the time it takes to travel in that distance you chose. After this is done make sure all of you data is collected and you are ready to start your calculations.
 * Methods and materials:**

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media type="file" key="Movie on 2011-11-15 at 13.23

__Trial 1: Calculating Final Velocity__
 * Calculations:**

Δd = 120 cm Δt = 1.83 s

Δd = 1/2 (V f + V i )t 120 = 1/2 (V f + 0)1.83 120 = .915V f V f = 131.15 cm/s

__Trial 1: Calculating Acceleration__

V f = 131.15 cm/s V i = 0 cm/s Δt = 1.83 s a = ?

Δd = v i t + 1/2at 2 120 = 1/2a(1.83) 2 120 = 1/2a(3.3489) 120 = 1.67445a 71.67 cm/s/s = a

__Trials 1,2,3 : Calculating Average Acceleration__

71.67 + 70.31 + 65.84 / 3

Average acceleration = 69.27 cm/s/s

__Trials 1,2,3: Calculating Average Sinθ:__

.125 + .123 + .120 / 3

Average Sinθ = .123°

__Percent Error of Acceleration in Trials 1,2,3:__

-- x 100 theoretical
 * theoretical - experimental|

|9.8 - 6.1228| -- x 100 9.8

= 37.52%

__ Calculating Percent Difference: __

|Average class experimental - individual experimental| -- x 100 average class experimental

average class experimental = 6.76006

|6.76006 - 6.1228| x 100 6.76006

= 9.43 %


 * Data Table:**

Measured Time vs. Incline Distance (with calculated variables)

Class Data of Mass vs. Slope (acceleration) at 15 cm
 * Mass (g) || Slope: || Acceleration @ 15cm (m/s/s) ||
 * 225 || 10.4 || 1.19 ||
 * 535 || 5.9427 || 1.025 ||
 * none posted || 6.5937 || .51 ||
 * none posted || 4.7411 || .94 ||
 * 16 || 6.1228 || .72 ||


 * Chart:**




 * Analysis:**

1. The acceleration due to gravity is 6.1228.

2. The percent error between 6.1228, our experimental value for free fall acceleration, and 9.8, the theoretical value for free fall acceleration is 37.52%.

3. Group 1 had a mass of 225g and an acceleration of 10.4. Group 2 had a mass of 535g and an acceleration 5.9427. Group 3 had an acceleration 6.5937. Group 5 had an 4.7411. Judging from these results, Newton's laws prove to be correct because it shows that the bigger the mass, the smaller the acceleration.

4. Gravity is the force that causes the ball to roll down the ramp. However, it is only a part of it; the vector of the normal force and weight cause the ball to roll with horizontal as well as vertical motion.
 * F = ma
 * g= 9.8 m/s 2
 * m = 16g
 * SinƟ = .179
 * W x = ma x
 * m *g (sinƟ) = m a
 * 9.8*.179 = a
 * a= 1.757 m/s 2


 * Discussion Questions:**

1. Is the velocity for each ramp angle constant? How do you know? The velocity for the three ramp angles were not constant and varied with each angle. This is because as the incline became higher and higher, the ball had a bigger acceleration, making the velocity higher as the ball reached the end of the ramp.

2. Is the acceleration for each ramp constant? How do you know? The acceleration for the three ramps were not constant. The relationship between an angle and and the acceleration of an object appears to be that the steeper the incline is, the bigger the acceleration. This is a result of the vector of normal force and gravity. On a flat surface they are at equilibrium, resulting in no change of motion. However, since normal force is perpendicular to the surface, the resulting vector would increase in magnitude at a steeper angle, until the only force acting upon the ball is gravity, putting the ball in free fall.

3. What is another way that we could have found the acceleration of the ball down the ramp? Another way to find the acceleration of the ball down the ramp would have been to set up two ramps that faced each other, like Galileo's experiment. We could find the length of the second ramp, the one the ball would go up, time the ball until it reaches the point where it was let go, and find the initial and final velocity.

4. How was it possible for Galileo to determine g, the rate of acceleration due to gravity for a freely falling object by rolling balls down an inclined plane. He placed two planes facing each other and performed an experiment similar to our answer for question 3. We could find the length of the second ramp, the one the ball would go up, time the ball until it reaches its highest point on the opposite ramp, and find the initial and final velocity.

5. Does the mass of an object affect its rate of acceleration down the ramp? Should it affect the motion of an object in freefall in the same manner? Yes, the mass of an object does affect its rate of acceleration down the ramp because the bigger the mass, the harder it is for the object to change its state of motion. If the ball's mass had been very large, it would have been less likely that it would have accelerated as quickly. It cannot, however, affect the motion of an object in freefall because the acceleration due to gravity for a freely falling object is always 9.8 m/s/s.

Upon completing the lab, two of the hypotheses made at the start of the lab were proven to be false while one of them proved to be true. The hypothesis concerning the acceleration due to gravity was that it would be 9.8 m/s/s because of background knowledge pertaining to free fall and projectiles. However, this was not the case when the ball was going down a ramp set at an angle. The acceleration achieved from performing the lab was 71.85 cm/s/s for the first trial. Our second hypothesis seemed to be true though. We discovered that the acceleration did change based on the angle of the ramp. The greater the angle the faster the acceleration. The first angle we had the ramp at was .123 degrees and the acceleration was 69.27cm/s/s. The next angle was .166 degrees and as we expected the acceleration increased to be 95.42 cm/s/s. For our third trial the average angle was .249 degrees and the acceleration was 159.62 cm/s/s. This proves that the greater the angle the large the acceleration is. our next hypothesis was that we could use the graph to find extreme points. We could do this by plugging numbers in for x seeing what value comes out for the y or plugging a value in for y and seeing what comes out for x. X is the degree of the angle while Y is the acceleration. Our last hypothesis was that Mass should not affect acceleration, but due to air resistance the heavier the object the greater its acceleration would be. After comparing our data with the rest of the classes we can conclude that our hypothesis was correct. This is because a group with the weight of .225 kg had an acceleration of 1.19 and a group with the weight of .282 kg had an acceleration of .94, while a group with the weight of .0047 had an acceleration of .803 which is less than the others. Also, when comparing the date it was found that the heaviest ball had the lowest acceleration, .535kg and acceleration of .746, but the lowest weighing ball did not have the height acceleration. Overall most of our hypothesis were good, but some were a little off. Experimental error occurred in a few areas, which affected the accuracy and precision of our results. The physical act of releasing a ball, starting a timer, catching the ball and immediately stopping the timer leaves too much room for human error. Each trial at a given angle should yield the same acceleration. However, as can be seen in our results, some of our trials were inaccurate. Instead of relying on a stop-watch, an experiment involving one-of-those-devices-measuring-distance-that-plugs-into-computer would have effectively removed human error from the equation entirely! It would measure the distance at given time intervals, providing us with the same information in a simpler and better way. Also, our experiment did not account for friction. The fact that the ball rolled, not slid, down the decline showed that friction affected the drop. To remove as much friction as possible and get the most accurate answer, the ball would need to be rolled down a friction-less surface. Since this isn't possible, an ultra-smooth surface that was lubricated would reduce friction as much as physically possible. We can relate what we did in this experiment to different roller coasters. During our experiment we found that the greater the angle, the greater the acceleration would be and the quicker the time it would take to reach the bottom. This is the same with roller coasters as well. When comparing to roller coasters at six flags, King Da Ka and Batman we can see that the angle of the drop affects the speed. King Da Ka had a vertical drop of 90 degrees and the speed was 128mph, while Batman had a vertical drop of 59 degrees and a speed of 50 mph. As you can see the greater the angle of the drop the greater the speed of the roller coater. Also, during this experiment and roller coasters both has friction as a force as they slide on the track. They also both have weight as a force and a normal force pushing upward from the track beneath them. This is important because it can be related to many things in life. It can also relate to a car going down a hill, the greater the incline the faster the car will accelerate. This is good to know in life so that when you are driving and you go down a steep hill you know you should apply the break because otherwise the car will accelerate very fast and will possibly go over the speed limit.
 * Conclusion:**

=Newton's 2nd Law Lab= Period 6 11/22
 * A = Lerna Girgin B = Jake Greenstein C = Rachel Knapel D = All**

What is the relationship between system mass and acceleration and net force?
 * Objective:**


 * Hypothesis:**
 * 1) A system's mass is directly proportional to net force and inversely proportional to acceleration.
 * 2) Acceleration is directly proportional to net force and inversely proportional to mass.
 * 3) Net force is directly proportional to acceleration and mass.

Place a track on a table. Put a dynamics cart down on the track. Observe the cart to see if it moves when placed at rest. This indicates that the track is uneven. If uneven, attach a metal stopping block to either side of the track as necessary to make the track perfectly even. Once the track is level, connect the photo gate timer and base and support rod, then clamp to the table. The wheel and track must be perfectly in line. Connect the photo gate timer to your laptop via the USB connector. Open data studio, and start a new experiment of Acceleration using a Linear Pulley. Attach the mass hanger to the dynamics cart with a string. Dangle the mass over the table, with the string running directly over the wheel of the photo gate timer. Now that the experiment is set up, put the 3 mass discs on the cart. Allow the cart to be along the track, and record the data. Repeat 3 more times, but each time move a mass disc off the cart and onto the pulley. Next, put all the mass discs on the hanging mass, and put all 3 mass blocks on the cart. Allow the cart to be pulled along the track, and record the data. Repeat 3 more times, but each time remove one of the mass blocks from the cart.
 * Methods & Materials:**

media type="file" key="Movie on 2011-11-29 at 13.06
 * Video:**


 * Data Table:**

Force vs. Acceleration

Mass vs. Acceleration


 * Graph:**




 * Link excel document:**


 * Sample calculations:**



1. Explain your graphs: a. If linear: What is the slope of the trendline? To what actual observed value does the slope correspond? How does it compare to this actual observed value (find the percent error between the two)? Show why the slope should be equal to this quantity. What is the meaning of the y-intercept value? The graph for force vs. acceleration was linear. We got the slope of the trendline to be 1.849, while our observed value we got as 1.87 by doing 1/m. Our total mass was .535kg and so when 1 is divided by that you get 1.87. This is very close to our experimental value. The slope of the line is supposed to be equal to the reciprocal of the mass. Even though ours is not exactly the same they are very close. After calculating our percent error between the expected and experimental data we found that we had 1.123% error. The slope should be equal to this value because slope of a linear equation is equivalent to the reciprocal of mass. The y-intercept has to do with friction. It is friction divided by the system mass, therefor if you multiply the y-intercept by the system mass you will be left with the friction. This is how you are able to find the force of friction.
 * Analysis & Results:**



b. If non-linear: What is the power on the x? What should it be? What is the coefficient in front of the x? To what actual observed value does the coefficient correspond? Find the percent error between the two. Show why this value should be equal to this quantity. The graph for Mass vs. Acceleration was non-linear. We got the power on the x to be -1.418. The theoretical value for the power on the x is -1. We were only .418 off from what it was supposed to be. The coefficient in front of the x is net force which is just hanging mass times gravity (m*9.8). On our graph we got our coefficient to be 0.0361. The theoretical value of the coefficient is .0326. Judging by just looking at the numbers it seems that our experimental value is very close to the theoretical value. When you do out the percent error for these two values you get 0.44%, which means that our experimental was very close to what it actually should have been This value should be equal to this quantity due to the equation y=Ax -b where the A is equal to net force, the same as saying mass times gravity.





2. What would friction do to your acceleration? Would you need a bigger or smaller force to create the same acceleration? Was your slope too big or too small? Can friction be a source of error in this experiment? Redo the calculation of acceleration WITH friction to show its effect. Friction would lower our acceleration. This is because it is a force acting against the way of motion and is pushing it backward cause the acceleration to be lower. You would need a larger force to create the same acceleration since the friction would be going in the opposing direction of the motion. You would need to have the original force plus the equivalent force of friction to be pushing forward so that the friction pulling the other way would cancel out with some of the force therefore making it able for the object to move at the same acceleration. An example of this is you you had a box being pushed without friction on a flat surface with a force of 10N and the acceleration is 2m/s 2. Than if that same box is being pushed along a surface that has friction, where friction has a force of 3N than in order to still have that 2m/s 2 you would need the force pushing on the box to be 3N greater than before, therefore it would be 13N. Our slope was a little smaller compared to the theoretical slope. It is nearly impossible to get rid of all friction to do this experiment, therefore friction could be a source of error, but not by that much, this is because the carts we used were mostly frictionless as they slid along the ramp.


 * Conclusion:**

The results that were obtained through this experiment, and judging from the two graphs, shows that the hypothesis made at the beginning of the lab, are true. Looking at the Force vs. Acceleration graph, the slope is positive, meaning as x increases, so does y. In this case, x is the force and y is the average acceleration. Thus, it is seen that as a force is increased on an object, the acceleration becomes bigger. An example of this can be seen within the data table; the first force is at .098 N with an acceleration of .0595 m/s/s, and the second force is at .1470 N with an acceleration of .145 m/s/s. It is a positive rate of change that occurs within the Force vs. Acceleration graph. Furthermore, the hypothesis concerning mass with acceleration was also proven correct. The line of best fit for this graph is negative meaning that mass and acceleration are inversely related, as stated in the hypothesis. The first coordinates on this graph show that at .51 kg the acceleration is .2225 m/s/s but when the mass is .6550 kg the acceleration is .156 m/s/s. These results display that the bigger the mass, the smaller the acceleration.

Using the data that we found and the expected data for this experiment we found that there was little error during this experiment. When comparing the slope on our force vs acceleration graph and the expected value, which we found by doing 1/m, we got a percent error of 1.123%, which isn't a high percent. This shows that our data was very good and close to what it actually should have been. In the Mass vs acceleration graph we got an even lower percent error. We found on our graph that the coefficient was .0361 and the expected value to be .03262, which was found by hanging mass times gravity. For this we got .44% error which is very low and again shows that our calculations were very close to what they were expected to be.

This error could come from the track not being completely level. This would cause for the cart to either accelerate faster or slower than it should have. To fix this we would need to ensure that the track is set up correctly and that the track is level and parallel to the table. Another source of error could have came from the string and the photo-gate system, they may not have been level with the track and that would have caused bad results. They all need to be perfectly in line, or the force might not transfer perfectly. Careful measuring and set up would prevent this. Finally, the tension from the mass might have been absorbed by the pulley system. The string, for example, could stretch and absorb force, reducing the force pulling the cart.

One real life example of this is in a fishing rod. A mass (a fish) is pulled along a line by a force (hand cranking wheel). In order to move the mass, tension must be applied to accelerate the fish out of water. Similar problems are present in this situation: if the fishing line stretches and absorbs the tension, the fish will be harder to pull. If the line absorbs too much force, it will snap. In order to compensate for the natural elasticity of an object being pulled, extra-strong strings are used.

=Coefficient of Friction= December 5, 2011 Part A - Rachel Knapel, Part B - Lerna Girgin, Part C - Jake Greenstein, and Part D - split up


 * __Objectives__:**
 * To measure the coefficient of static friction between surfaces
 * To measure the coefficient of kinetic friction between surfaces
 * To determine the relationship between the friction force and the normal force.


 * __Hypothesis__**:
 * 1) The coefficient of static friction will be ≥ 1 when the mass of the system is great.
 * 2) The coefficient of kinetic friction will be ≤ 0 when the mass of the system is small.
 * 3) Friction force and normal force are directly proportional to each other, meaning that if the Normal force on an object is high, so will the amount of friction force.

__**Methods and Materials**__: First, gather all of the materials. You will need a Force Meter, a USB link, a friction "cart", miscellaneous masses, a ramp, string, a laptop, and a clamp. Then, find the mass of the cart without the masses. Next, set up the lab. To do this, tie one end of the string on the hook of the force sensor and one on the cart. Place the cart on the ramp and place four blocks inside. Once data studio has been set up and ready to go, begin pulling the force sensor as close to constant speed as possible. Repeat this step for two more trials, then take away one block from the cart. When the three trials for that is over, take away one more block and pull the force sensor for the three trials. Repeat these steps by taking one block for ever three trials. When you have done three trials for just the cart, add all four blocks back plus another one. Obtain three trial results for this as well.


 * __Video__:**

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 * __Calculations__:**










 * __Free Body Diagram:__**


 * __Data__**:
 * __Chart__**:






 * __Class data__**


 * __Analysis__**:
 * Compare the slope of line with calculated m s average (% difference).

The slope of our line was extremely close to our calculated µ static average, with a % difference of only .716 %

Our µ kinetic was relatively off from the rest of the class, with a %difference of 10.4%. Our µ static, however, was extremely close to the average of the class, with a %difference of only 0.77%.
 * Compare your result with the class results.


 * __Discussion Questions__**:
 * Why does the slope of the line equal the coefficient of friction? Show this derivation.
 * The equation of a line is in the format y=mx + b. The equation of our specific line correlates to the equation f = µN. Since f is our y-axis and N is the x-axis, the coefficient of friction is the m value, or slope.
 * Look up the coefficient of friction between your material and the aluminum track. Discuss if your measured results fall within the range of theoretical values. Be sure to cite your source!
 * According to our source, the coefficient of static friction should have been between .25 and .40. This is inconsistent with our data, which yielded an average µ static of .1536. However, our coefficient of kinetic friction does fall within the source's parameters. Our µ kinetic was .1029, and the source says it can be from .10 to .30. http://www.tribology-abc.com/abc/cof.htm
 * What variables affected the magnitude of the force of friction? What variables affected the magnitude of the coefficient of friction?
 * Friction, and the coefficient of friction, were both affected by the mass of the cart. As the cart's mass increased, the force of friction, and the coefficient of friction, increased as well.
 * How does the value of coefficient of kinetic friction compare to the value for the same material’s coefficient of static friction?
 * The coefficient of kinetic friction is less than that of the coefficient of static friction. The average µ static was .1536, which was greater than the average µ kinetic of .1029. This is because while static, the friction is resisting motion, causing it to peak higher. While moving (kinetic), friction is opposing motion, but not great enough to negate it, resulting in a smaller number.

Based on the data collected upon performing this lab, some of the hypotheses were proven right. By starting out with highest mass along with the cart, 2.836g, the average coefficient achieved for static friction was .14. The lowest mass being 93g with only the cart, the average coefficient achieved for static friction was .1. The hypothesis was that the coefficient for static friction will be greater than 1 when the mass of the system is great, but this was not the case in this lab because the mass was not big enough. Had the mass been much bigger, the coefficient would have been greater than 1. The second hypothesis was proven correct because the data shows that the coefficent for kinetic friction is usually less than static friction or is ≤ 0, as was the case with the 93g system. In addition, the results also show that Normal and Friction force are directly proportional to each other because when the mass was 2.836g the normal force made the coefficient of friction greater than when the mass was 93g. When the normal force is low, so is the coefficient of friction.
 * __Conclusion__**:

Errors occurred as a result of an imperfect experiment set-up. Since the track along which the cart was pulled was totally flat (ridges), if the cart was not pulled in a perfectly straight line, the friction generated between the cart and the track could fluctuate. Also, another source of error occurs as a result of our tension sensor. Since it was pulled by hand, the string connecting the sensor and the cart may not have been perfectly horizontal. This would also skew our results. Finally, our inability to pull the cart at constant speed also affected our results.

To make sure that the next time we do this we get better results we should fix the problems we had while doing the lab this time. We would have to use a level or a rule to measure and make sure that the string is horizontal at all times when we are pulling it. The cart could be pulled by a CMV, ensuring constant speed. The carts would be set up perfectly straight along the track, to avoid any changes in friction. Another solution to fixing the problems is to making sure that before each trial we did to make sure that the graph was at zero. We could also measure with more weights to make sure that our calculations are consistent. This lab can be related to many things in life. This can relate to a train ride at an amusement park. The initial pull, causes static friction, and a greater friction than when the trains is moving at constant speed along the track, kinetic friction. The more amount of people on the ride and the greater the weight in the train the higher both the Static and Kinetic friction. This is because the force to get the train moving would be much greater if the weight of the train was greater.