Anthony,+Jimmy,+Justin,+Chloe

 Acceleration Down An Incline Lab
==Anthony Iannetta, James Ferrara, Chloe Murtagh, and Justin Tosi ==

Period 2 Due Date: December 4, 2011
** Purpose: ** ** To establish a relationship between the incline angle and acceleration. **

** Theory/Rationale: Using Newton's 2nd Law, we hypothesize the following function will equal acceleration: ** ** Procedure: ** The procedure was given, but the following video provides context for the procedure and the materials we used. media type="file" key="Chloe mur.mov" width="300" height="300"

Graph without Trendlines:
 * Data: **

Graph with Trendlines:



Using DataStudio, we were able to graph the velocity of the board going down the ramp. Taking the slope of velocity equals acceleration so we highlighted the relevant data and added a trendline to it. This is how we calculated the slope. We took 5 trials for each angle and then averaged the trials together to try and get rid of human error. This is the five trials for the 12.9 angle.



[[file:Acceleration Down an InclineV2.xls]]
Free-Body Diagrams: Up an Incline 
 * Calculations: **
 * Free-Body Diagrams: Down an Incline[[image:fbdjjim.png width="800" height="251"]] **


 * Part 2 Calculations: **



**Conclusion:** For this experiment, our purpose was to illustrate Newton’s laws in action. Based on all of his principles that we learned, we hypothesized that the acceleration would equal:
 * Graphs: **
 * [[image:sinevs.accelerationjim.png width="800" height="540"]] **

Our hypothesis was generally correct, despite 11.1% error in the experiment. The evidence that our hypothesis was correct was found in the equation of the points we graphed in sine vs. acceleration graph. The coefficient of the equation of the line represents gravity, and since ours was close to the earth’s gravity, 9.8, we can conclude that our data was somewhat consistent with our hypothesis. Our percent error for this lab was 11.1%. Error most likely came from slight imperfections in our track. Though we did our best to clean the track of dust, slight nicks in the aluminum, and other inconsistencies caused the block to slow in certain areas, creating a negative acceleration and altering our data. Also, we noticed that the block not only moved down the track, but also moved from left to right. This indicates that track was slightly tilted to one side. This would also slightly alter the acceleration. The best way to improve the data would be to obtain a track that was perfectly smooth and level. This would produce an acceleration that would better reflect the theoretical.

Discussion Questions:
1. Discuss your graph. What does the slope mean? What is the meaning of the y-intercept?

The slope is the force of gravity acting on the incline, which is equal to the mass of the wooden block moving down. The y-intercept represents the friction force opposite of the weight x force on the wooden block divided by the mass.

2. If the mass of the cart were doubled, how would the results be affected?

If the mass of the cart is doubled, the acceleration will also double. This is due to the fact that adding more mass increases the weight and thus the force parallel to the incline (weight x) increases causing the block to move faster down the ramp.

3. Consider the difference between your measured value of g and the true value of 9.80m/s2. Could friction be the cause of the observed difference? Why or why not?

Measure Value of g: 8.7236 n

The value we found is about 10 off from 9.8 due to other forces acting on the system that the lab did not account for. Friction can cause the difference because it affects the weight x force. The normal force is another force, that opposes weight y, that could affect our measured value.

** Period 2 Due Date: December 14, 2010 **

 * Purpose: **to measure the coefficient of static and kinetic friction between a piece of wood and a metal track; to determine the relationship between the friction force and normal force


 * Theory/Rationale: ** the equation, ƒ=µN, demonstrates the relationship between friction, the coeffcient of friction, and the normal force.


 * Materials: ** Friction block, Surface/glass cleaner, paper towels, 50 N force probe and computer, 1 meter long aluminum inclined track, inclinometer, 1.5 meter piece of string, masking tape, balance

** Data: Measuring Coefficient of Friction on a Flat Surface ** ** Data: Measuring Coefficient of Friction on an Incline **

One of the Graphs Using DataStudio:

To calculate the maximum static friction and the mean of the kinetic friction, we measured our data using DataStudio and analyzed it. By determining the maximum y-value we could calculate the maximum tension needed to start moving the wooden block, therefore finding the maximum static friction. Then by highlighting the points after the maximum height and then taking the mean we could figure out the average kinetic friction.

**FBD: Moving on a Flat Surface**

**FBD: Going Down an Incline** ** Calculations: Measuring Coefficient of Friction on a Flat Surface ** ** Solve for Static Frictional Constant, Friction, and Normal Force: ** ** Solve for Kinetic Frictional Constant, Friction, and Normal Force: ** ** Calculations: Measuring Coefficient of Friction on an Incline ** ** Solve for Static Frictional Constant, Friction, Normal Force: **

Solve for Kinetic Frictional Constant, Friction, Normal Force:



** Percent Difference Calculation-Static Friction ** ** Percent Difference Calculation-Kinetic Friction **

** Graphs: **

**Conclusion:** The purpose of this experiment was to find the coefficients of static and kinetic friction between wood and aluminum, and to illustrate the relationship of friction over the normal force is this coefficient. These graphs illustrate this conclusion and provide us with the coefficients. Since the slope of the line is rise over run, change in y over change in x, or friction over normal force, it also equals the coefficient of friction.

This slope for our static friction is very, very close to the stated scientific value. For our coefficient for static friction, our results differed from the classes by 2.76%, and differed from their value for the kinetic friction coefficient by 8.74%. Error could be contributed to any dust or any other particles on the aluminum track or on the wooden block, roughness in either the aluminum or the block, the block not being pulled at an absolute constant speed, or the ramp being held at slightly too high an angle. Error could be attributed to our or any group, to make all the class’ results differ.

These concepts could be applied to real life for just about an situation where somethings is being pulled. For example, to determine if a tractor full of heavy materials can be pulled down the highway, one would have to figure out the normal force and the force of friction, and also factor in the static and kinetic coefficient for friction between asphalt and rubber.


 * Discussion Questions: **

**1. Why does the slope of the line equal the coefficient of friction? Show this derivation.**

The formula for the coefficient of friction is ƒ=µN.

Since the slope the graph equal change in y (friction) over change in x (normal force), the slop equals µ, or the coefficient of friction.

**2. Look up the coefficient of friction between wood and aluminum. Discuss if your measured results fall within the range of theoretical values. Be sure to cite your source!**

According to the following table from http://hypertextbook.com/facts/2005/wood.shtml: The coefficient of friction between wood and aluminum is .22. We found ours to be .211, which only gives us about a 4% error. This is exceptionally close to the expected value.

**3. What variables affected the magnitude of the force of friction? What variables affected the magnitude of the coefficient of friction?** As illustrated by the following equation: the magnitude of friction is equal to, and therefore directly related to, the tension between the rope and the wooden block. As tension in the rope increased, as did the force of friction.

As illustrated by the following equations: the coefficient is influenced by the force of friction and the normal force. I mentioned above how the force of friction can be altered. As for the normal force, as illustrated by this equation:the normal force is influenced by the weight of the block (which is directly related to its mass). As the mass increases, if the friction were to stay the same, the coefficient of friction would decrease.

4. How does the value of coefficient of kinetic friction compare to the value for the same material’s coefficient of static friction?

µ (static) = 0.211 µ (kinetic) = 0.1712

The coefficient for static friction is greater than the coefficient for kinetic friction. This is due to the fact that it takes more normal force to move an object at rest with static friction than an object already moving with kinetic friction. If this is analyzed with the equation ƒ=µN it can be reasoned that as friction increases so does the normal force it takes to make it move. This means that the coefficient will also increase to equalize the equation. Since static friction involves more "drag" than kinetic friction, the coefficient should be more for static friction.

5. Did putting the track on an incline significantly change the coefficient of friction? Why or why not?

No simply because it did not take that much of an incline to make the wooden block move. The track was inclined at about 10º when the static friction was overcome and the block would move. This only decreases the static friction a bit because it is easier for the block to start moving and it moves on its own accord. Since the friction is directly related to the coefficient, the coefficient had to increase a little, but not change significantly.

** Period 2 Due Date: December 7, 2010 **

 * Purpose: **To find the relationship between net force and acceleration.


 * Hypothesis: **Based on Newton's Second Law, the net force and acceleration should be directly related, as per the equation F=ma. Therefore, as the net force is increased, the acceleration should also increase.


 * Materials: **Atwood’s Machine, clamp, rod, stand, 2 mass hangers, masses, photogate(s), meterstick, string


 * Procedure: **

1. Set up Atwood’s Machine and place a string through the two pulleys. Add equal mass to each side of the pulley. 2. Pull the side with the lower mass and let go, letting the heavier side accelerate down. Do not let the mass hit the pulley or the ground. 3. After each four trials add 4 grams from one side of the pulley to the other, changing the force while not changing the total mass 4. Continue the trials, while taking the average to remove possible outliers in the data.




 * Data: **

Trial Graph Without Trendline:

Trial Graph With Trendline:

We averaged the four trials together to get the actual acceleration. The reason for doing four trials is to try and eliminate any source of error during our experimentation.


 * Calculations : **




 * Graphs : **
 * [[image:avsfjimmy.png width="800" height="543"]] **


 * Follow-Up Questions: **

1. Explain your graph(s) thoroughly! 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?

Two graphs are shown, but they both demonstrate the same data, but the second one is without friction. The slope of the trendline is 1.1878. The slope is actually the mass of the one pulley gradually changing. The equation of the line reflects F=ma. Since force is y and acceleration is x, the only variable left is m which is the slope in the graph's equation. The y-intercept is the force of friction acting on the pulley system.

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 decrease the overall acceleration because it is an opposite force on the tension of the pulley. A bigger force would be needed to achieve the ideal acceleration. The slope we found was too small because the acceleration is the slope. Friction is a source of error in this experiment because it causes the acceleration to be too small.

3. Discuss the precision of your data.

Our data was tainted by the friction that is involved with the pulley, the pulley not being level, the pulley mass, and any fluctuations with the string during the pulley operating. This means that each trial is very precise in that the numbers correlate to the overall change that is supposed to be occurring, but are not accurate because of all the before mentioned factors. The basic conclusions are right and portray the information we were trying to achieve making the data very precise. The data is not accurate do to some human error and other factors that caused the numbers to be a little off from their theoretical values. As someone once said "even though it didn't happen that way it should have so we can tell it like it did." Also, we had a very high R-squared value, specifically .9992.

4. The real pulley and mass arrangement is not as simple as we assumed. In fact the pulley is not massless and frictionless means that it does require a net “torque” (a turning force) to make it rotate – this is supplied by the tension in the string. The rotational inertia of the pulley then adds an equivalent mass to the total mass being accelerated, where the equivalent mass for the pulley is approximately equal to ½ of the mass of the pulley. If the mass of each pulley is 5.6 g, could the pulley mass account for a significant potion of your error in the experiment?

We should get a lower experimental value for acceleration compared to the theoretical value we received therefore affecting with error. This is due to the fact that while calculating acceleration we did not take to account and this value would be put in the denominator. It would not be a significant amount different though.

Calculation acknowledging Friction:


** Conclusion: **

The purpose of this experiment was to prove or disprove the relationship that exists between net force and acceleration. According to Newton’s Second Law, force = mass x acceleration. In order to illustrate the direct correlation between force and acceleration, we kept the mass of the system the same, and just moved the weights from one side of the pulley to the other to alter the net force. Our results illustrated this correlation, as there was a pretty definitely linear relationship in our data. As our force increased, the acceleration of the lighter hanging weight, each point fitting very well on our linear line of fit, as illustrated by our R value of .9992.

In a perfect run, to reflect the equation defined by Newton’s Second Law, y = force, x = acceleration and the constant should reflect the mass. The mass we used was 1.06 kg, so our equation should have been y=1.06x. However, our equation was y=1.1878x, and the error in this lab (12.1%) could be attributed to a few factors. First, we assumed that friction would be nonexistent, while the friction in the pulley probably decreased the acceleration slightly. Also, we assumed that the mass of the pulley had no effect on the system, while it, too, probably slowed the acceleration slightly.

One real life application of this system is an standard elevator with counter weights. The acceleration of the elevator is controlled by the proper amount of weight hanging parallel to the elevator. Finding out this prime amount of weight can be done by using the F=ma formula.

** Purpose: **
To find the relationship between system mass, acceleration and net force.

Based on Newton's Second Law, we expect to find that the total force causing acceleration will be directly related to acceleration, and as force increases, as will acceleration. Additionally related to this, we hypothesized that acceleration will be indirectly related to the mass of the system, and as the mass is decreased, the acceleration is increased.
 * Hypothesis: **

Dynamics cart, dynamics track, photogate timer, data studio program, pulley with clamp, stopping block, mass balance, base and support rod, mass hanger, masses, mass balance, string, level
 * Materials: **

1. Set up cart and pulley, while also tying a weight to the other side of the string 2. Make the falling mass 50 grams and after every five trials transfer 10 grams from the falling mass to the cart. This changes the force without changing the total mass. 3. Using DataStudio, record the velocities using a velocity vs. time graphs. The slopes of these graphs are the acceleration. Take the average acceleration of each trial and record. 4. Make the falling object a constant mass (50 g) and add 1200 grams to the cart. Every five trials take 200 grams away from the cart. This changes the mass while not changing the force <span style="font-family: Arial,Helvetica,sans-serif;">5. Using DataStudio, record the velocities using a velocity vs. time graphs. The slopes of these graphs are the acceleration. Take the average acceleration of each trial and record.
 * Procedure: **


 * Data: One Trial on Data Studio **

Graph without trend lines**:**

Graph with trend lines**:**

We averaged the four trials together to get the actual acceleration. The reason for doing four trials is to try and eliminate any source of error during our experimentation.


 * Data: All Trials on Excel **


 * Graph: Acceleration vs. Force **

In the first part of the lab, we kept the mass in the system constant and moved the weights between the cart and the hanging mass to manipulate the force. This choice allowed us to directly compare the acceleration of the cart and the force of the hanging mass. In the first graph, we had a very good fit, with an R-squared value of almost one. Our y-intercept was not zero, but close to it. This constant symbolizes the amount of friction in our system. When I set the y-intercept equal to zero as if we were working with a frictionless system, the R-squared value decreases. Therefore, the first graph should be used and the second graph should be disregarded.
 * Explanation of Acceleration vs. Force : **


 * Graph: Acceleration vs. Mass of System **




 * Calculations : **

**See Follow Up Questions for Error Calculations.**


 * Follow-Up Questions: **

All the graphs show the legitimacy of the equation F=ma and confirm the relationships between force, mass, and acceleration. Both graphs for acceleration vs. force show the same basic information, but the second one disregards friction.
 * 1. Explanation of Graphs**

The slope is 0.5383. The slope corresponds to the mass of the cart. ( 0.554 - 0.538 )/( 0.538 ) * 100 = 2.97% error Mass corresponds to the slope of the force acceleration graph because y is force and x is acceleration. This means that the equation of the line is really F=0.538a, thus that 0.538 value must represent mass according to the equation F=ma. The y-intercept is the force of friction on the acceleration of the cart. Ideally the value should be negative because it is decreasing the acceleration of the cart as it is pulled by the pulley and moves down the track. Even though our value came out positive it is very close to zero making it relatively close to the friction involved.
 * a. Linear Graph:**
 * 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)?**
 * ( Actual - Theoretical )|/( Theoretical ) * 100 = Percent Error
 * Show why the slope should be equal to this quantity.**
 * What is the meaning of the y-intercept value?**

The power on the x is -1.2234, but should be -1. The coefficient in front of the x is 0.3992. This coefficient corresponds to the force acting on the pulley system. ( 0.4905 - .3992 )/( .3992 ) * 100 = 39.92% error This is because y represents the mass and x to the -1 represents the acceleration. In this manner the equation can be demonstrated as m=0.3992a^-1 or m=0..3992/a. The 0.3992 value is force because is in form of the equation F=ma switched around to be m=F/a.
 * b. Non-linear Graph:**
 * 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.**
 * ( Actual - Theoretical )|/( Theoretical ) * 100 = Percent Error
 * Show why this value should be equal to this quantity.**

2. Friction would slow the acceleration, because it would take more force to move the cart. To make up for friction, you would need to to add to the force, which you could do by adding weight to the hanging mass. Our slope (acceleration) was slightly smaller than expected, which could be attributed to the unaccounted for friction.

** Conclusion: **

Our hypothesis was satisfied by our results. We hypothesized that as we increased the force in the system, the acceleration would also increase proportionally. This turned out to be true, as our points lined up quite well with the line of best fit - illustrated by our excellent R value. Because there was such a consistent slope, our x and y values increase proportionally every time. We also expected to find that as we decreased the mass of the system, the acceleration would increase. This turned out to be true, as our points fit well on the drawn-in curve, as illustrated, again by the good R value. In our first experiment, our error was 2.97%. The error is obvious in the friction (the y-intercept), considering our value is positive when it should have been negative. This could be contributed to an error in our set up, as apparently there was some force acting upon our system that not only counteracted the resistant friction, but also surpassed it, ultimately pushing our cart along. Possible mistakes could have included a downward tilt in our track or in the rope between the cart and the pulley. In our second experiment, our error was about 39.92%. This is generally large error, which could be contributed to a multitude of errors on our part - including a slight slope in the track or the string pulling the cart - but also due to unaccounted for friction, or even, perhaps, air resistance met by the hanging force as it fell. To make this lab even more accurate, it would be necessary to use a cart and track that a virtually frictionless, a track that is completely unslanted and a string that is totally parallel to the track.

Inertial Mass Lab

Purpose:
== To ascertain the relationship between period and mass of an object. We will use this to find the mass of an object using its inertia, specifically by finding the periods of objects whose inertial mass we know, and comparing the periods of the unknown to those. ==

** Materials: **
Metal Weights (20g, 50g, 100g, 200g) Stopwatch Paper Towel Inertial Balance Clamp Microsoft Excel Rubik's Cube

Procedure for Known Mass:
1. Clamp inertial balance to side of table 2. Place known mass in inertial balance 3. Push inertial balance and start stopwatch at the same time 4. After 10 periods stop stopwatch and record time media type="file" key="newlabgroupvid.mov" width="429" height="429"

Procedure for Unknown Mass:
1. Place unknown cube into inertial balance 2. Push inertial balance and start stopwatch at the same time 3. After 10 periods stop stopwatch and record time 4. We have gotten the y-value (time). Plug that into the equation of the line 5. The x-value is the mass

Data:
Trials of Vibration Lab:

**Calculations: Percent Error and Percent Difference**
(Actual-Theoretical)/(Theoretical)*100 = percent error (102.09-101.46)/(101.46)*100 = .6209% error

(Actual-Class Average)/(Class Average)*100 = percent difference (102.09-104.87)/(104.87)*100 = 2.651% difference

Follow-Up Questions:
**1. Did gravitation play any part in this operation? Was this measurement process completely unrelated to the "weight" of the object?**

Gravitation did not play a role in this operation. Inertia is affected by mass and mass is affected by gravity. Since this is true, the measurement process is unrelated to the "weight" of the object. Gravity is not just acting on the object, it is also acting on the inertial balance. The gravity is a weight force so it only affects object's downward movement. The inertial balance and the mass on it are moving side to side so gravity does not have an effect on the motion.

**2. Did an increase in mass lengthen or shorten the period of motion?**

An increase in mass lengthened the period of motion.

**3. How do the accelerations of different masses compare when the platform is pulled aside and released?**

For objects with greater masses, the acceleration was less. Because the tray did not increase to speeds as great as it did for lighter weights, this made each period take longer.

**4. Do you imagine that the period would have been different if the side arms were stiffer? Would this change lengthen or shorten the period of the motion?**

We imagine, if the side arms were stiffer, the period would be the same. However, the distance that the tray traveled in each period would be less.

**5. Is there any relationship between inertial and gravitational mass of the object?**

Gravitation mass and inertial mass for an object are the same. These are two different ways to measure the mass of an object but, error aside, they will produce the same final mass because the acceleration on a falling body is always the same (-9.8).

**6. Why do we almost always use gravitation instead of inertia as a means of measuring mass of an object?**

We almost always use gravitational mass instead of inertial mass because we can get a much more accurate read of the mass using a balance, than by counting the time it takes to accomplish one period on an inertial balance. We can trust the readings of the gravitational mass because gravity is a constant (-9.8).

**7. How would the results of this experiment be changed if you did this experiment on the moon?**

Since we’re using inertial mass instead of gravitational mass, the reads should hypothetically be the same – since the readings aren’t depending on the force of gravity. The only difference would be caused by air resistance, since earth has a greater atmosphere than does the moon.

** Conclusion: **
In our hypothesis, we theorized that as the mass of the metal weights increased, the periods of the inertial balance would increase. We were correct in this hypothesis. This is shown by the data in our data table and the positive sloping line, with a slope of .0011, on our Mass vs. Vibration graph.

Surprisingly, we had very little error in this experiment, specifically .62%. This error occurred during the measuring of how long 10 periods took, as explained in our procedure. I could have stopped the timer too late or too early and this could have led to our error. Also, we secured the weights in the inertial balance with paper towels. Even though the mass of the paper towel seems negligible, it would have accounted for our less than 1% error. If I pushed the inertial balance too hard, it would cause the weight in the balance to shift, resulting in a longer period than it should be. This could have made the light weights seem slower, which would change the slope of our line and eventually our calculated mass for the Rubik's cube. Lastly, inaccuracies with the inertial balance could have led to the error. If the inertial balance was older, it would swing more freely than a newer, stiffer inertial balance. This would have led to a larger percent error.