March 22: Lab #7 Modeling Friction Forces

Lab #7 

Modeling Friction Forces

Ricardo Gonzalez and Josue Luna

3-22-17

Using our understanding of friction forces and how they work, we will be conducting five experiments. Each experiment will be used to model either the coefficient of static or the coefficient of kinetic friction.

1. Introduction
Our understanding of friction is all the same. Not all surfaces are smooth therefore the two surfaces generate heat. When looking at a motion, the maximum amount of force needed to barely get an object to start moving from rest is called static friction. The amount of force required to keep an object moving at a constant velocity is called kinetic friction. Both static and kinetic friction are parallel to the motion of the object. Also. the friction force is opposite to the direction of motion, however friction may also cause motion.
We will be conducting multiple experiments. By measuring or calculating the maximum amount of force to barely get a block to begin moving we can find the coefficient of static friction. Similarly, by measuring the amount of force needed to move an object with a constant velocity, therefore no acceleration is present, we can calculate the coefficient of kinetic friction.

2. Procedure
Part 1
To measure the maximum static friction force, we will be measuring the total amount of force require to barely get a block to start moving by utilizing an ideal pulley and a hanging mass that we can measure. We placed the felt side of the block on the tables surface. Next we clamped a pulley on the corner of the table. A string was attached to the block, placed over the pulley and then tied to a hanging mass of 5 grams. We began adding 5 grams to the hanging mass until the force due to gravity of the hanging mass would begin to move the block. We repeated the process by placing 200 grams on top of the block and once again added increments of 5 grams to that hanging mass until the block barely began to move. Two more trials were conducted by adding 200 grams each time and adding increments of mass to the hanging mass.

Here is a two-dimensional representation of our apparatus.

Here is a photo of our experiment for Part 1

Part 2
To measure the kinetic friction of the same block, we will move the block with a constant force and therefore moving the block at constant velocity. We first began by retrieving a laptop to utilize the logger pro software and a force sensor. We connected to the force sensor to the laptop, set the force sensor range to 10-N, held the force sensor vertically to "Zero" the force sensor on the logger pro software and placed the force sensor horizontally on our table surface to "Zero" the sensor. We tied a string from the block to the force sensor. Starting with just the mass of the block, we placed the felt-side of the block facing the ceiling. We hit "Collect" on the logger pro software and began pulling the block using the force sensor at constant velocity.

 

We stored the latest run, we placed 200 grams on top of the block and repeated the process by pulling the block at constant velocity. 

We then repeated the process with 400 grams and 600 grams placed on top of the block. 

Part 3
To calculate the static friction from a sloped surface, we began by placing the block at an end of the board and placing our phones to measure the angle. As the board was raised, at a certain angle created by the board with respect to the table, the block just barely began to slide down the board.

 Part 4
To find the kinetic friction from sliding a block down an incline, we raised the wooden board at an incline large enough so that we know the block will accelerate down the incline. Using the motion sensor, we can measure  the acceleration of the block down the incline and also measure the angle of the inclined surface using our phones. As the block moves down the incline, an opposing force of due to kinetic friction is decelerating the motion and can be calculated.

Part 5
To predict the acceleration of a two-mass system, we can use the coefficient of kinetic friction we found in part 4 to theoretically find a value of acceleration for the system and then compare that to the experimental value measured using a motion sensor.

3. Measured Data/Results

Part 1

The coefficient of static friction = m/M

- Mass of block = 0.189 kg 

hanging mass = 0.075 kg

-Mass (Total) of block = 0.389 kg 

hanging mass = 0.125 kg

- Mass (Total) of block = 0.589 kg

hanging mass = 0.225 kg 

-Mass (Total) of block = 0.789 kg

hanging mass = 0.39 kg 

Notice, we graphed the ratio of the masses in terms of grams. Whether in grams or kilograms, the ratio, or the slope, is unaffected. 

Part 2

The Kinetic Friction Force = Coefficient of Kinetic Friction x Normal Force

Therefore, the Coefficient of Kinetic Friction = Kinetic Friction Force/Normal force

Here we measured the forces resulted from pulling the block. 

Run 1 = mass of just the block with the mean force applied. 
Run 2 = mass of block + 200 grams with the mean force applied
Run 3 = mass of block + 400 grams with the mean force applied
Run 4 = mass of block + 600 grams with the mean force applied

 

Below is a graph of the mean force vs the Force from kinetic friction. 

Part 3 

By drawing the forces on the system, we can find the sum of the forces in the x direction, the y direction, and then simplify for the coefficient of static friction between the block and the wooden board at the maximum angle.

Part 4

Here is a graph for our motions velocity vs time 

Coefficient of kinetic friction 0.323 

Part 5

The acceleration = 4.79 m/s^2

4. Analysis In Part 1, we plotted the ratio of the masses. Just at the moment the block barely starts to begin moving, the force of the tension is equal to the weight of the big M (our hanging mass) and therefore the maximum static friction is equal to the weight of the hanging mass. When we solve for the coefficient of static friction, we get a ratio of the blocks mass over the hanging mass. If we think of the ratio as a slope (rise/run), then our rise (y-axis) is the mass of the block and our run (x-axis) is the mass of the hanging mass. When all points are plotted, the slope of the line gives us the coefficient of static friction between the block and the wooded board. Th coefficient of static friction from the slope was 0.435.

In Part 2, we found the mean force needed to move the block across the surface of the wooden board at constant velocity. Since there is no acceleration, then the force needed to move the object at constant velocity is equal to the kinetic friction force opposing the motion. When we solve for the coefficient of kinetic friction, we get a ratio of the Kinetic Friction Force and Normal force. When graphed, the slope of the line gives us the coefficient of kinetic for the wooden block onto the surface of the wooden board. The coefficient of kinetic friction was 0.232 which is reasonable since the coefficient of static friction should be larger that that of the coefficient of kinetic friction tat we found in part 1 to be 0.435.

In part 3, we set up a sum of forces in the x and y direction and solved for the coefficient of static friction between the block and the surface of the wooden board. This is only achievable when we utilize the maximum angle the block can be raised at an incline just before the block begins to accelerate down the incline. When we solved for the coefficient of static friction , we got 0.445 which is just about the same as the coefficient of static friction we calculated in part 1 which was 0.435.

In part 4, we raised the block on an incline at an angle we know the block would accelerate down the incline. When we found the acceleration of the block using a motion sensor, we were able to a sum of forces that accelerated the block and solved for the coefficient of kinetic friction. Surprisingly, the coefficient of kinetic friction was 0.323, different from the number we calculated in part two which was 0.232.

In part 5, we can theoretically calculate the acceleration of a block with a two mass system, if we add a large enough mass to accelerate the block from rest. Using the coefficient pf kinetic friction found in part 4, we theoretically found the acceleration to be 4.79 m/s^2. We did not manage to conduct the experiment and so we do not have a real life value for the acceleration to compare too.

5. Conclusion
In conclusion, we can use our current knowledge of forces to calculate both the coefficient of static and kinetic frictions. Not so simple is to be able to interpret, the ratio of masses as it is not so obviously seen. Through the calculations, we can conclude that the values for kinetic friction were within reasonable values between parts 2 and 4. However, the coeffiecients of static friction between parts 1 and 3 there was a differnce of nearly 40 percent between the two values calculated. A reason for this is simply becasue in part 1, the coefficient of static friction relied on a ratio of masses as opposed to in part 3 where the coefficient of static friction was reliant on the angle itself.

March 15 Lab # 5 Trajectories

Lab #5 

Trajectories

Ricardo Gonzalez and Josue Luna

3-15-17

The objective of this experiment is to have an understanding of projectile motion to predict the impact point of a ball (a marble in our case) on an inclined plane.

1. Introduction
We know when an object is launched at an initial velocity off of table at height (Δy), the horizontal velocity stays constant and gravity is the only force affecting an object. During the time that the object falls, the object has traveled a horizontal distance along with a vertical distance at a given time. We have set up a small experiment were we would like to know how far a marble travels at the moment the marble strikes a board of length (d) placed against the table at angle (θ). Before we measure through experimental trials what distance down the board the marble strikes, we can set up an equation using our known knowledge of kinematics to theoretically give us a distance. Having theoretically calculated the distance, we would then execute the experiment and measure the distance the marble would strike the board.

2. Procedure
We first began by gathering the needed materials and tools for the experiment, such as; Two Aluminum "V-Channels," a small marble, two wooden blocks, a string, a wooden board, a ring stand, a clamp, a white blank piece of paper, and a piece of carbon paper. We set up the apparatus in such a way that one v-channel would accelerate the marble down at an incline and then horizontally. The end of the horizontal v-channel was aligned to the same overhang length as the table and can then be used to simulate an object moving off a table with an initial velocity.

 We released the marble from a repeatable point and found an area where the marble landed. On that area we first taped a blank white piece of paper on the floor and a piece of carbon paper directly above the blank piece of paper. We then released the marble from the repeatable point three times. Those three times the marble strikes the carbon paper. Interestingly, the marble created a large black dot on the white piece of paper due to the carbon paper.

This is a simplified way of visualizing our apparatus. 

 Using the sting that was taped onto the edge of the table, we measured the horizontal distance the marble traveled from the edge of the table, the string, to the center of the large black dot on the white piece of paper. With the measured height and horizontal distance, we can calculate the initial velocity the marble experienced.

A simple visualization of our apparatus with the board leaning on the corner of the table for a certain angle. 

Next we placed a wooden plank with one end aligned to the end of the table and the other end on the floor. We placed two large masses in front of the board in order to stabilize the board from falling down. creating a sloping angle that we measured with our phones. 

Next we decided to let the marble once again roll down the v channel from the repeated point to find the area of the board where the marble would strike the board. At the point where the marble struck the board, we taped down a blank white piece of paper to the board and a piece of carbon piece of paper directly above the white paper. 

The carbon paper once again helped us find the distance the marble had struck the board with the distance down the board starting at the point where the board and the edge of the table meet. 

3. Measured Data

Height to the v-channel (Δy) = 0.931 +/- 0.001 m

The horizontal distance traveled by the marble to the floor (x) = 0.63 +/- 0.02 m

The angle of the wooden board (θ) = 48° +/- 1°[In respect to the floor]

4. Results 

We  first began with our basic equations of kinematics. We set Δx in terms of time and substituted the value of t into our equation for Δy. We substituted dsinθ for our Δy.

We next solved for d.
d in our case is the distance down the board  

We found the equation used to find the theoretical value of d. 

5. Analysis 

We found the theoretical value of d by first calculating the initial velocity. 

On the bottom of the photo above we have listed the distances for both the experimental measurement and the theoretical value. The theoretical value we have derived an equation for suggest the marble would hit the board at 0.707 meters. Also, the experimental measurement we measured through experimentation gave us a value of 0.775 meters. The difference between the two values is roughly about 0.06 meters, or, 6 centimeters. The percent error from our values is 9.6%. That error was the accumulation of the uncertainties for the values we had calculated for the height, the distance the marble struck the ground, and our measurement of the angle of the board. To expand, our equation for the theoretical value of d is dependent on our initial velocity and angle, both of which have a range of error. 

6. Conclusion

When we calculated the theoretical value for d, we did not take into account of any other forces onto the marble during its flight time other than gravity. Although air resistance is present, given the small surface area of the marble, we completely ignored the air resistance force that would not have affected our theoretical value. When measuring the experimental value, we used a meter stick to measure the distance from the top of the board to the black dot created below the carbon paper on the blank white piece of paper, which was not perfect. A range of uncertainty should be accounted for. Technology has advanced so much in recent years, one of which is the ability to measure angles with our phones. The measurement we took was not accurate to the nearest whole number and thus our value of uncertainty of one degree was relatively large in comparison to our measurements, in respect to a percentage of error. 

March 13: Lab #4 Modeling the Fall of an Object with air Resistance

Lab #4 

Modeling the Fall of an Object with air Resistance

Ricardo Gonzalez

Josue Luna

3-13-17

In this experiment we want to prove that air resistance is a factor of all free falling objects, regardless of mass. 

1. Introduction

 When we have looked at free fall motions in the past, we assumed the motion was ideal and the only force accelerating the object was gravity. We know that that is not always the case. Air resistance is an affecting factor in most cases due to the objects shape, the material it is moving through, and as the objects speed. Likewise, as an object speeds up, the force of air resistance must increase. The equation for air resistance is
F(resistance) = kv^n , v is the velocity, where k is a constant value due to the shape of the object and n is an exponential value that the velocity takes.

2. Procedure

For this experiment, we wanted to track the motion of an object during free fall and graph its components of motion.  We used the logger pro software to capture a video of the motion of a single coffee filter. We scaled the distance on the camera with the use of scaled reference points from the height. The distance from one hash to another is exactly 1 meter. By adding the scale and placing points on the video after each selected frame per second, we are able to graph the velocity over time. We did the same procedure for 2, 3, 4, and 5 coffee filters. We created an excel file to find the components of the motion numerically.

3. Data

mass of 50 coffee filters = 43.9 +/- 0.1 g

mass of a single coffee filter = 0.89 +/- 0.002 g

The above photo shows the position vs time graph of 1 coffee filter. Slope = 1.154 m/s

The above photo shows the position vs time graph of 2 coffee filter. Slope = 1.912 m/s

 The above photo shows the position vs time graph of 3 coffee filter. Slope = 2.027 m/s

 The above photo shows the position vs time graph of 4 coffee filter. Slope = 2.308 m/s

 The above photo shows the position vs time graph of 5 coffee filter. Slope = 2.471 m/s

4. Results
The above photo shows the slope of the the weight (y-axis) vs velocities (x-axis).

The photo above and below show our numerical data of 1 filter.

The photo above and below show our numerical data of 5 filters. 

5. Analysis
As we know, the formula of air resistance accounts for two unknowns, k and n. When we graphed the velocities of each group of filters vs the weight (in Newtons) of those filters, we can apply a curve fit. From the curve fit, we can see that the curve is exponential, as the speed increases, so does the force of air resistance. What we mean by this, is that when an object falls, at a certain point, the force of gravity is going to equal the force of air resistance, therefore the object is no longer accelerating and in now at its terminal velocity. A larger mass would be needed with the same object in order for the object to once again accelerate and reach a higher terminal velocity which is evident on our graph. The equation of the line from the curve fit is Ax^B. A in this equation is our constant K value and B is our exponential value of N from our equation of air resistance.
By our values measured and calculated, we created an excel file to numerically show the motion of the coffee filters. I have provided two photos of the excel file that show we have set the motion of the coffee filters with the values of k and n as constants. By looking at the file, we can observe that the terminal velocity of the 1 coffee filter reaches 1.38 m/s, however when we found the velocity using the logger pro software, the graph measured the terminal velocity of 1.154 m/s. Also from the excel file, we can see for the five filters, that the filters approach a terminal velocity of 2.492 m/s while the slope of our position time graph for five filters shows that our terminal velocity is 2.471 m/s. These clear deviations in values are due to the uncertainty from our curve fit. There is a range of values for both our k value and n value. Since these uncertainty values were not taken into account in our numerical representation of the motion, the value found from our logger pro software would have been the better choice for this experiment.

6. Conclusion
In summation, the power law of Air resistance holds true as we proved that constants k and n are derived from our own interpretation of the objects motion as it experiences free-fall. With a certain shape of an object and a certain weight (N), there is a limit the speed that object is able to accelerate too since the force of air resistance is going to increase as speed increases until that force of air resistance is equal to the force of gravity pushing it down. An obvious deviation in our data between the slopes of the position time graphs of each group of filters and the numerical value of the corresponding group was due to percent uncertainty found from our curve fit we graphed.

March 8: Lab #3 Non-Constant Acceleration

Lab #3

Non-Constant Acceleration

Ricardo Gonzalez

Josue Luna

3-8-17

Our goal is to find how far an object will travel with a non-constant acceleration in the opposite direction of the initial motion. 

1. Introduction 

So far many of our experiments have assumed acceleration is constant, however in real life cases, that is not always the case. When lets say a rocket is launched from NASA with a constant thrust, the mass of the rocket changes with time as the rocket fuel burns. In this case, the force of thrust is unchanged while the mass decreases therefore acceleration is much larger. 

2. Procedure

What we have is an elephant moving down an inclined hill toward a flat horizontal surface. At the moment the elephant is moving across the horizontal surface, the elephant has an initial velocity of 25 m/s. Notable, the elephants mass is 5,000 kg and a rocket of 1500 kg is strapped onto the elephant with the direction of thrust opposite to the motion of the elephant. As fuel is burned from the rocket, the rockets mass begins to decrease given by a function. We want to calculate how long must the hill be in order for the elephants velocity to approach zero and begins to move in the opposite direction.

3. Data

Initial velocity of the elephant = 25 m/s

mass of elephant = 5000 kg

mass of the rocket = 1500 kg

Constant Rocket thrust = 8000 N

the function m(t)= 1500 kg - 20 kg/s*t

Newtons first law Fnet = ma

4. Results

5. Analysis
We have the means to find the value of x, our maximum distance that the elephant will travel with the given information in two ways. Both analytically and numerically. Analytically, we can use Newton's second law to find our function for acceleration by setting a in terms of t (time). By taking the integral of our acceleration function with limits (0,t) , we are able to find our change in velocity. What we have is a function in terms of( time with an initial velocity constant. Once again, we can take the integral of our velocity function [v(t)] with limits (0,t) to give us our position function in terms of time. When we set our velocity function equal to zero, we are able to find how long the elephant took to come to rest.The time we calculated was 19.69 seconds. Lastly, we are able to substitute our value of time into our position function and find our total distance the elephant traveled so that its velocity was 0. We can see that the value of x was calculated as 248.7 meters.
From a numerical approach, we are able to set an excel file in such a way so that we are able to find the precise value of x as we set our time interval to be a much smaller value. As shown in the photo above we have set: the first column as the time interval, the second column as the acceleration, the third column as the average acceleration, the forth column as the change in velocity, the fifth column as the velocity, the sixth column as the average velocity, the seven column as the change in distance, and finally our eighth column as the distance. When we set our time interval of one second, we can see that at the 19th second, the position changed from a positive value to a negative value. This tells us that between the 19th second and the 20th second,the elephant changed direction. As we changed our time interval to a tenth of a second, we now can see that between the 19.7 and 19.8 second, our direction of motion changed. While we did not use a smaller time interval, we would see that the value of x would begin to change in such a small amount of distance relative to the total distance traveled that any further time interval changes are needed.
If we were to change the mass of the elephant to 5500 kg, the fuel burn rate to 40 kg/s*t, and the thrust force to 15000 N, we can see that our values change. our new value of x is roughly 12 meters between 12.7 and 12.8 seconds.

6. Conclusion
When accelerations are constant, finding positions and velocities at given times are much simpler. When an acceleration is not constant, we must find a function for the acceleration and find its components of velocity and position by taking a few integrals. Although useful, at times the functions and or integrals may be much more complicated that anticipated. When such a case arises, we can use numerical approaches to find the same results with out the headache. 

March 1: Lab #2 Free Fall Lab

Lab #2
 Free Fall Lab

Determination of g and some statistics for analyzing data

Ricardo Gonzalez

Josue Luna

3-1-17

Lab Part 1

This lab experiment was conducted in order to prove that objects under free-fall will fall at the speed of gravity, 9.8 m/s^2.
Lab Part 2
Secondly, we are able to measure the percent error in our value of gravity due to uncertainty.

1. Introduction
Lab Part 1
We have a rod of 1.86 m tall with a magnet holding an object from falling 1.5 meters. At the moment the magnetism is broken, the object falls with no other forces other than gravity in the downward direction. As the object falls, the position is recorded onto a spark-sensitive tape during equal time intervals. By measuring the distance within a time interval, we are able to find the acceleration through a position time graph and velocity-time graph.
Lab Part 2
When we look at data, it is inevitable to have variation between data, as nothing is ever perfect. We can calculate the difference between our data by finding standard deviation which would give us the range in which our data fluctuates.

2. Procedure 
Lab Part 1
We were handed a spark-sensitive tape that was had the distance traveled marked for each time interval. We creating an excel file and on the first column we input the time after every time interval. Notice, the frequency used to mark the distance onto the spark sensitive tape was 60 Hz, therefore, a mark was made for every 1/60th of a second. On the second column, we placed all distance measurements for each time interval on the tape. For the third column, we measured the change in distance, Delta X. Under column D, we found the Mid-Interval Time. Lastly for the final column, we measured the Mid-Interval Speed. By selecting The Mid-Interval Time and Mid-Interval Speed, columns C and D, we created a scatter graph, giving us a Velocity-Time graph. Furthermore, we created a Position-time graph by utilizing the Time and Distance columns, columns A and B.
Lab Part 2
We created a second Excel file. On this excel file we gathered all the data that will be used to find the average standard deviation. he specific data we re looking at is the value of gravity that each group experimentally found. We numbered the first column from 1-10 for each group. Under the second column we placed the corresponding experimental-value of gravity from each group and found the average. For the third column, we subtracted the average value of gravity to each individual experimental-value of gravity for each group and then took the sum. Too find the deviation, we squared each result from the difference taken and once again, we took the average. Finally, we took the square root of the deviation, giving us the average of the squared deviations.

3. Data
Lab Part 1

Lab Part 2

(Please note that we used cm/s^s as opposed to m/s^s. )

4. Results

Lab Part 1

As we can see from the graphs below that we have increasing functions for our Position-Time graph and Velocity Time graph. Looking at the Position-Time graph, if we took the slope of the line, we can get the Velocity. Now if we took the derivative of the function for the Position-Time graph, we would be given the function of the Velocity-Time graph. When analyzing the Velocity-Time graph, if we took the slope of the line, we can find the acceleration. Likewise, we can also take the derivative of the Velocity-Time function which would also give us the Acceleration. I would also like to mention, if we found the area under the curve of the Velocity-Time graph, we would get the total displacement. Likewise, If we took the integral of the Velocity-Time function, we would be given the Position-time Function. Simply by looking at the function of the velocity, we can see if we took the integral, mentally in our heads without any real effort, we would get the number 9.5 m/s^s. That number is our acceleration due to gravity. Our acceleration is a constant value as we can see, there is no longer any variables in our function after taking the derivative of the Velocity function. Notably, our acceleration calculated was not exactly 980 cm/s^2 (9.8 m/s^2) as we initially believed due to errors and uncertainty. 

Lab Part 2

From the results we have gathered, we can see that 1 standard deviation is equal to 1 sigma. Our 1 sigma, in this experiment is equal to 6.1 cm/s^2 therefore giving us a range of values acceptable to our calculated value of gravity. Our range is from 955.4-967.6 cm/s^2, that is that at least 68% of our data were within 1 standard deviation of our value of gravity. Secondly, going further away from our first standard deviation from our average to the range of 949.3-973.7 gives us a second standard deviation from the average, in which at least 95% of our data was within 2 standard deviations.We can see a pattern in the data gathered for the value of gravity, in such that not one group was  further away from two standard deviations of our average. As we compare our average value of gravity from all group, we can see that we are still far from the value of gravity, as much as 31 cm/s^2 and as low as 6.3 cm/s^2. 

5. Conclusion

Having analyzed the data, we have to conclude that there must have been another factor that was not taken in account for when conducting this experiment. I say this because as we can see, random error was a clear reason for having deviations in the calculated value of g. These deviations could have been made when measuring the distance after each time interval on the spark-sensitive tape. Uncertainty between measurements of the same distance is completely possible which is why some groups were able to within two standard deviations from the average value of gravity. A systematic error, such as air resistance could have affected the acceleration of the object as it fell. Also, the time the object took to begin falling in response to the electromagnetism disengaging could have, to a certain extent, affected the first time interval, inevitable affecting the following intervals. 

The point of this lab was not only to learn how to calculate the acceleration do to gravity, assuming an ideal case where no other forces are applied to the experiment, but also to be able to use tools like Excel to do the Calculations for us. Key concepts of this lab were that not all experiments are perfect, in such that there will always be at least some variation between data. However, we are able to find an average and thus a standard deviation from our average. By finding a standard deviation, we are able to utilize a range of acceptable values within a number of deviations.