Hallowell,+Litmanov,+Vanderberg

=LAB: ACCELERATION DOWN AN INCLINE=

Due: 1/3/11
OBJECTIVE/ PURPOSE:

To find out how the acceleration of an object down an incline depends on the angle of the incline.

HYPOTHESIS:

The acceleration of the object //will// depend on the angle of the incline. The larger the angle, the higher the acceleration and vice versa. This hypothesis is made based on the derived equation a = gSin**θ** - **μ**gCos**θ**. Acceleration and theta or, the incline angle, have a direct relationship and so, if theta increases, so will acceleration.

PHOTO OF SET-UP:
 * PART A**

EXAMPLE FROM DATASTUDIO:


 * This is a sample set of trials from DataStudio for Part A. This was our data for when our angle was 20 degrees. In this sample, there are two boxes that give the information for the linear equation of each trial. We used the slope to give us the acceleration for each trial. Although, as we explain below, when plugging in the settings for the picket fence, we only measured half of what we needed to, therefore, we recorded what the slope of each line was and multiplied it by 2 to give us the acceleration of each trial.

DATA TABLE:


 * After getting all of our data, we then learned that we had typed in the wrong measurement for the picket fence on the DataStudio. This caused us to originally record accelerations that were half of what we wanted. Once we found the average of the trials for each angle, we then multiplied the averages by two. It was these accelerations that we used to create our graph.

CLASS DATA:

GRAPH:

ANALYSIS: In order to calculate the coefficient of friction between the incline and the block, we need to use the equation a = g(sin(angle)) – (ug(cos(angle))). We plug in one of the angles we used to collect our data, as well as the acceleration calculated for that angle. For our calculation below, we used the trail in which we had 15 degree angle. For “g”, we use the coefficient in front of the “x” from our equation on our graph. This coefficient represents weight divided by mass, which gives us our “g”. The work for this calculation is shown below. For the gravity of the earth, we used the value, 9.81 m/s^2. Again, we used the slope from the equation from our graph above. The percent error was 2.84%, which shows that we did a pretty good job in making sure we were precise and accurate with our data collecting. The work for this calculation is shown below. In order to compare the coefficient of frictions from this week’s lab and last week’s lab, we calculated the percent difference between the two. For last week’s lab, we used the class average from Part A. The value was equal to .1847. The value for this week’s lab was solved for earlier. The work for this calculation is shown below.

CALCULATIONS:

Theoretical Acceleration for 20 degree incline: In order to solve for the theoretical acceleration of a certain trial, we created a free body diagram for the situation in Part A. We decided to find the theoretical acceleration for when we had a 20-degree incline. For our theoretical coefficient of kinetic friction, we used the class average from Part A of last week’s lab. That value was .1847. The rest of the calculation is shown below.


 * PART B**

DATA TABLE:

PERCENT ERROR:


 * The goal of Part B was to get less than 2% error. After performing five trials and finding the average, we were able to see that we had accomplished our goal.

CALCULATIONS: Our job was to find out the acceleration needed to travel .5 m in 1.5 seconds, starting from rest. The work for this calculation is shown below.

Once we found the acceleration needed, we needed to find an equation that would allow us to find the mass of the hanging weight needed to travel the acceleration. The force-body diagram for this situation is shown below:

We then derived a formula by using only variables. We would then be able to plug in our values into this equation.

Once we derived our formula, we were then able to plug in our values. We knew the mass of "m1" was .367 kg. We knew "g" was equal to 9.8. We decided to use an angle of 20 degrees. We also decided to use the coefficient of friction of .1847. This value came from the class average of Part A from last week's lab. The work for this calculation is shown below.

DISCUSSION QUESTIONS:

1. By deriving the formula for acceleration down an incline, we get the equation a = gSin**θ** - **μ**gCos**θ**. The equation on our graph follows the format y = mx + b. This format is similar to the equation we derived. Therefore, the slope of our graph is equal to g, the force of gravity, and the y-intercept is equal to the coefficient of friction times gravity times the cosine of theta, or the friction force.

2. If the mass of the block were doubled, the results would **not** be affected. This is because the force of gravity is constant and the force of gravity acting upon the block will always be the same regardless of its mass, and the acceleration of the block will remain the same, no matter the mass.

3. Our measured value of g is higher than the true value of g. Friction is most likely the cause of this difference because if we were to assume that the track was frictionless, and set the y-intercept of our equation (which is the value of the friction force), the measured value of g would decrease to below the true value of g. Thus, we can deduce that friction is the cause of our measured value being higher than the true value.

4. Our results were within the 2% error required; we had 1.7% error. The 2% error expectation was considered reasonable because we used a derived formula to calculate the mass required. Due to this fact, the mass we calculated should have been the mass needed to achieve the result. If we got more than 2% error, then one of our calculations had to be wrong.

CONCLUSION: In our experiment, the results we obtained coincided with our hypothesis. As we raised the angle of the slope, the acceleration increased. This shows the direct relationship between the angle and acceleration. For example, with a sine angle of .25882, the block accelerated at .534 m/s^2, while at a sine angle of .30902, the block accelerated at 1.024 m/s^2. This example further displays the direct relationship of the angle and acceleration.

For the error in our lab, it was possible that there was human error. If we did not let go of the wooden block perfectly, there could have been a slight force applied causing the block to move left or right. To help prevent this, the group member with the steadiest hands should have been the one to release the block. Another source of error could be from the track itself. Our block would often grind against the side of the track, causing friction to be applied. To prevent this, we could have tried to keep the block centered and not use data recorded when this happened.

A real world application for this would be when driving. A way to save on gas and help the environment would be to not accelerate while going downhill, instead letting gravity do the accelerating for you. Since you know that going down a slope causes acceleration, your knowledge of physics not only helps you save money on gas, it will also limit carbon dioxide emissions. The reverse is also true, however. When going up a steeply inclined hill, more gas must be applied to accelerate your car.

= = =LAB: COEFFICIENT OF FRICTION=

Due: 12/13/10
Excel Sheet:

PURPOSE: >
 * 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: The coefficient of static friction will be greater than that of kinetic friction. This is because friction is higher when an object is still rather than in motion. Since the force of friction and the coefficient of friction are directly related, as shown in in the equation //f//=µN, the larger the force of friction is, the larger coefficient of friction is.The relationship between the friction force and normal force will be direct because of the equation //f//=µN.

PART A
PHOTO OF SET-UP:

DATA: Maximum Tension

Mean Tension at Constant Speed Note: The average maximum tension in the first table and the average tension in the second table that we calculated were both equal to the friction on static/kinetic friction seen below in the graph.

GRAPH: ANALYSIS/ CALCULATIONS: In our graph above, there are two equations next to each of the trendlines. Both equations come from the friction formula, f=uN. The "y" in the equation represents the amount of friction. The "x" in the experiment represents the normal force. That means that the numbers we see next to the "x"s in both equations are the coefficients of friction. We calculated the percent difference between our data and the class's data below.

Static Friction

Kinetic Friction

PART B
PHOTO OF SET-UP:



DATA: Static Friction

Kinetic Friction

ANALYSIS/ CALCULATIONS: In the calculation below, we describe how we found out the coefficients of friction for both scenarios using angles. Once we determined our coefficients of friction, we determined the percent difference between our values from Part A and our values from Part B. They are shown below as well.







DISCUSSION QUESTIONS: 1. Why does the slope of the line equal the coefficient of friction? Show this derivation. The slope of the line equals the coefficient of friction because of the equation //f//=µN. //f// is the force of friction, µ is the coefficient of friction and N is normal force. We graphed our results using friction force on the y-axis and normal force on the x-axis. By using the equation y=mx, we deduced that the slope, m, is the coefficient of friction since //f// is y and N is x.

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! The theoretical coefficient of friction between wood and any clean metal is between .2 to .6. The calculated value of our static friction is .2199 and our kinetic friction is .2008, so both values fall in the theoretical range. @http://www.carbidedepot.com/formulas-frictioncoefficient.htm

3. What variables affected the magnitude of the force of friction? What variables affected the magnitude of the coefficient of friction? The tension from pulling on the wooden block affected friction, changing it from static to kinetic. The weight, which is the same as normal force, affected the force of friction because they are directly related. The normal force also affects the coefficient of friction because they are inversely related.

4. How does the value of coefficient of kinetic friction compare to the value for the same material’s coefficient of static friction? The value of the coefficient of kinetic friction is slightly lower than that of the static friction. This is because the force of friction is directly related to the coefficient of friction. Static friction is always larger than kinetic friction, so the coefficient of static friction is higher than that of kinetic friction.

5. Did putting the track on an incline significantly change the coefficient of friction? Why or why not? No, the incline did not affect it significantly. This is because the weight and the acceleration remained the same in both experiments. The weight is the same because the same materials were used. The acceleration was 0 because the system was either not moving or moving at constant speed.

CONCLUSION: After performing our lab, we can conclude that our hypothesis was correct. Originally, we believed that the coefficient of static friction would be greater than the coefficient of kinetic friction. After performing parts A and B, we were able to see that those original ideas were correct. The graph that we created from the data in Part A shows that the coefficient of static friction was .2199. This number was greater the coefficient of kinetic friction of .2008. In Part B, after we performed our calculations using trigonometry, we saw that the coefficient of static friction was .2378 and the coefficient of kinetic friction was .1831. Again, these results proved that our hypothesis was correct. In our hypothesis, we also stated that the friction force and normal force were directly related due to the equation f=uN. The graph from Part A of our lab proved that our hypothesis was correct again. In the graph, you can see that the trendlines for the data are diagonal lines, ascending across the graph. Because the normal force was shown on the “x” axis and friction force was shown on the “y” axis, we can see that due to the diagonal lines, when one force is increasing, the other is increasing as well. This proves that our hypothesis was correct. After calculating our percent difference, it is easy to see that some error occurred in both parts of the lab. In Part A, one of the main sources of error was our ability to move the sensor at a constant speed. Our group found it very difficult to get consistent results because it was very challenging to move the sensor perfectly every time. We also found that the block would sometimes stick when we would try to slide it, which also could have caused some inaccurate results. In Part B, the main source of error was our inability to get a precise angle for our data. As we gradually picked up the ramp and increased the angle, the ball on the protractor used to measure the angle would shake slightly. This made it hard to get an exact measurement. Another source of error came from our protractor and its very small intervals between each angle. It was nearly impossible to tell the difference between each individual angle on the device. This made for many estimates on the angle, which definitely could have caused some error in the lab. In order to improve this lab and minimize the error, we believe it would have been very effective if we could have used some sort of machine to move the sensor at a constant speed. One device that comes to mind would have been the Constant Motion Vehicle we used in an earlier lab this year. That would have given us much more consistent results in Part A. The results for Part B could have been improved if we had some sort of digital protractor that gave us angles to the tenth or hundredth place. This piece of technology would have ensured that we got consistent results every time rather than using our best guess to get our measurements.

=LAB: NEWTON'S SECOND LAW=

Due: 12/6/10**


OBJECTIVE: What is the relationship between system mass, acceleration, and net force?

HYPOTHESIS: When comparing the system mass, acceleration, and net force using Newton's Second Law of Motion, expressed in the equation of f=ma, the relationship between net force and acceleration is direct and the relationship between mass and acceleration is indirect.

PROCEDURE: > with Clamp, Base and Support rod, String, Mass hanger and mass set, Wooden or metal stopping block, Mass balance, and a level
 * 1) The materials used in this lab were a Dynamics Cart with Mass, Dynamics Cart, track, Photogate timer, Data studio, Super Pulley
 * 1) Place ramp near the edge of the table and attach the pulley to the ramp with a clamp.
 * 2) Attach a piece of string to the cart and make the string parallel to the ramp.
 * 3) Attach a mass to the end of the string.

Part A:
 * 1) We attached 15g to the end of the string and 25g was placed on top of the cart. For this experiment, system mass was kept constant at 40g to find the relationship between acceleration and force.
 * 2) We ran each trial three times using the photogate and Data Studio to record the information. We used velocity time graphs and found the slope of each line, which is the acceleration. These were averaged together and used in the Excel spreadsheet.
 * 3) This was repeated a total of five times, each time taking 5g from the cart and adding it to the end of the string until there was no weight left on the cart.

Part B:
 * 1) For this experiment, the mass at the end of the string was kept constant at 55g to find the relationship between mass and acceleration.
 * 2) We placed 1600g on the cart and recorded three trials using Data Studio. These were again averaged and used in the Excel spreadsheet.
 * 3) This was repeated eight times, each time removing 200g from the cart until there was nothing left.

DATA TABLES:

Mass Constant

Force Constant



GRAPH (Mass Constant):

GRAPH (Force Constant):



ANALYSIS QUESTIONS: 1a. After creating our graphs, our mass constant graph was linear. The slope of the trendline for our graph came out to be .5033. This value corresponds to the total mass of the system when performing the experiment in kilograms. The actual observed value of our experiment should have been .538 kg. The percent error is shown below. The slope should be equal to our total mass because of the equation f=ma. On our graph, we have force shown on the “y” axis and acceleration shown on the “x” axis. Therefore, in the equation for the trendline, the “y” represents force and the “x” represents acceleration. That means that the slope represents the total mass of the system. The y-intercept value shows how much force is used in the trial when there is no acceleration. In order to find the y-intercept, you plug in 0 for “x”, which means you are using 0 acceleration. For our graph, the y-intercept is .0915. Ideally, the y-intercept would be 0, which shows that we had a little error in our experiment.



1b. For our non-linear graph, force constant, the power on the “x” was -1.2424. The theoretical value for this power should have been -1. The coefficient in front of the “x” for our equation was .371. The theoretical value for this coefficient should have been .539 because that was the force that was kept constant throughout all of our trials for this part of the experiment. The percent error for these two values is shown below. The coefficient should be equal to the constant force used in the experiment because of the equation f=ma. When you divide both sides by “m”, you get the equation a=f/m. This is the equation that should be used on the force constant graph. The acceleration is on the “y” axis and the mass is on the “x” axis. In the equation, acceleration is represented by “y” and mass is represented by “x”. This means that, if no error were present, the constant force would represented by the coefficient.



2. If friction was involved in our experiment, then our acceleration would go down. In order to create the same acceleration with friction, you would need the force to be increased. This is because friction decreases the net force, which then means that the acceleration decreases as well. Our slope was too small, which shows that friction can be used as a source of error in our experiment. Below shows the difference that friction makes in our mass constant. The friction used will be .0915 because that was the value that was left on the equation, which showed that we had error in our experiment.



CONCLUSION: Overall, after completing both parts of our experiment, we can conclude that our hypothesis was correct. We were able to see in our “mass constant” table and graph that when force increased, acceleration increased as well. This showed the direct relationship between new force and acceleration, which can be seen in the equation, f=ma. We were also able to see in our “force constant” table and graph that when mass was decreased, acceleration was increased. This showed the indirect relationship between total mass and acceleration, which can also be seen in the equation, f=ma.

After completing both experiments, we were able to see that we experienced some error during the process. For the first experiment, we had only 6.44% error, which showed that very slight changes could have allowed us to be almost perfect. Our second experiment had 31.17% error. In order to avoid the error, we could have possibly cleaned the track a little better to avoid any friction. We also could have made to set set-up the pulley and the string more precisely in order to avoid any possible error. The only way to improve on this experiment and decrease the bad variables would have been to slow down and collect the data more precisely. Overall, I feel that our experiment went smoothly, answered the objective, and supported our hypothesis.

=LAB: INERTIAL MASS=

Due: 11/22/10


OBJECTIVE: Find the mass of an object only using its inertia.

HYPOTHESIS: We believe that the time of one period of motion will increase as the mass of the object on the balance increases. We will use our calibrations in order to find the mass of the Rubik's Cube.

PROCEDURE: >
 * 1) Gather equipment (inertial balance, known masses, clamp, stopwatch, paper towel)
 * 2) Clamp inertial balance to table
 * 3) Place known mass on inertial balance
 * 4) Secure weight in place with a paper towel
 * 1) Pull inertial balance to make it move back and forth
 * 2) Use a stopwatch to record 20 periods of motion
 * 3) Divide this time by 20 to get the time of one period of motion
 * 4) Once three trials are done for a mass, average the time of the three periods of motion together
 * 5) Repeat steps 3-8 for each of the different masses
 * 6) We then performed the same steps on a Rubik's cube in order to get the time of one period of motion.

PICTURE OF SET-UP:

This picture shows our set up before we began to perform the trials.

DATA TABLE:

GRAPH:



CALCULATIONS: We used the equation from the graph above in order to find the mass of the Rubik's Cube. We plugged in the time we observed for the Rubik's Cube into the equation for "y".



FOLLOW-UP QUESTIONS:
 * 1) Yes, gravitation played a part in this operation. Gravity dictates the "weight" of the known mass which causes the time to either be faster or slower.
 * 2) An increase in the mass lengthened the period of time.
 * 3) The heavier the weight was, the slower the acceleration and the lighter the weight was, the faster the acceleration was.
 * 4) Yes, the time would have been different if the arms were stiffer. The times would have been faster because the arms would have less of a range of motion.
 * 5) The relationship between gravitational and inertial mass is that they are equal, so the more gravitational mass there is, the more inertial mass there is.
 * 6) It is easier to find gravitational mass rather than inertial mass and gravitational mass can always be found no matter what gravity is present the gravitational acceleration on each object will be the same.
 * 7) The results would be changed because there is less gravity and no air resistance, so the periods of time would be shorter.

CONCLUSION: After completing the lab, we can conclude that our hypothesis was correct. The time for one period of motion was increased as the mass of the object on the balance increased. For example, as seen in our data table, it took .312 seconds per period for a 10-gram object, whereas it took .329 seconds per period for a 20-gram object. This evidence shows that weight and gravity definitely had an effect on the time of a period of motion. We were also able to find a mass of the Rubik’s cube using only inertia and our data, which shows that we succeeded in completing our objective. Throughout the experiment, there were many instances when error could have occurred. First, there was likely error when starting and stopping the stopwatch when calibrating the balance. It is very tough to start the timer at the exact moment when you release the balance and stop it when it has finished. Second, the objects with less mass had a much shorter period, which could have produced another form of error. We could have miscounted the amount of periods that occurred because it was happening so fast. This would have definitely changed our data. This showed in our data with inconsistent changes in time for the objects with less mass. In order to fix this error, we would have had to use very precise timing devices with laser sensors in order to get perfect data. This would allow us to know the perfect amount of periods as well as the precise time of the trial. In the future, we would need better measuring devices to eliminate the error.