25-March- Centripetal acceleration vs angular frequency

Centripetal acceleration vs angular frequency

Purpose

To determine the relationship between centripetal acceleration and angular speed

Procedure

In this short lab we placed an accelerometer attached to a spinning disk. We then spun the diskl using different voltages and the sensor recorded how fast the disk was accelerating; the higher the voltage the higher the acceleration produced. Meanwhile, we timed a given number of rotations to figure out the period of rotation at every voltage. Using w=2pi/period we determined the angular speed at every single voltage. We then recorded the acceleration, period, and angular acceleration in a data table in LoggerPro and lastly we graphed it.

Recorded acceleration and angular speed

Graph: Acceleration vs Angular speed^2

Slope = radius = 0.13 m

Conclusion

After using a proportional fit on the graph above we see that the slope of the graph turns out to be the radius of the spinning disk. We compared the value obtained from the graph to the actual radius of the disk (obtained using a ruler) and it turned out to be the same 0.128 m or 13.8 cm. Thus we concluded that the experiment was successful as we proved that angular acceleration and angular velocity are related by the radius of the rotating object. 

a=rw^2

16-March- Modeling Friction Forces

Calculating Coefficients of Static and Kinetic Friction

Purpose

We want to calculate the Static and Kinetic frictional values for a block using different techniques

Part 1

For the first part, we placed a block of wood on a table and attached a string to it and placed a cup on the other end. We gradually filled the cup with water until the block broke free and started to move. we recorded the mass of the cup with water and repeated this 4 times while adding a block each time.

 The picture above shows how we preformed the experiment and the data below shows what was collected of the masses of both the cup and the blocks that were used to achieve maximum static friction. We found our coefficient to be 0.260.

Part 2

For the second part, we attached the block to a force sensor and used logger pro to record the amount of force needed to pull the block at constant speed across the same surface. This was repeated 4 times while adding a block after each trial

The two pictures above show the first and last trial where the blocks were hooked up to the force sensor and pulled at constant speed. 

Above is the data that was recorded and the coefficient friction we obtained was 0.313, which was higher than out static coefficient 

Part 3/4

For the final two parts, we put them together and achieved only the answer to coefficient of kinetic friction. We calculated that the static coefficient at an angle would be equal to the tangent of the angle but we never actually proved this. Instead we continue on to part 4 and used a motion sensor to find the acceleration and calculate the kinetic coefficient at various angles

The picture above shows how the experiment was performed. A hanging mass that was

 just heavy enough to accelerate the block up the ramp was used and 

the motion sensor recorded the acceleration. 

We used kinematics to solve for the kinetic coefficient of 0.234.

Our calculated value was very similar with a value of 0.227

Conclusion

I believe the methods we used to calculate the static coefficient on the table and incline were accurate but the methods to find the kinetic coefficient seemed to be all over the place and a reliable number could not be achieved. Especially when the block had to be pulled at a constant speed. This was difficult to replicate every time.

11-March-Modeling the fall of an object falling with air reistance

Modeling an object falling with air resistance

Purpose

Determining the relationship between air resistance force and speed. modeling the fall of an object including air resistance. Put on the data into logger pro and calculate the terminal velocity of your various coffee filters.

Procedure

We need to repair 5 coffee filters and a one meter stick. We also need a computer which have video capture and logger pro. We have to practice a few times to be able to use the program, plotting the point once you have the video of falling object. By using 1 meter stick and scaling the blue point on the vertical component, we are ready for the experiment. 

We still use the approach that the above handout suggests, but we will do this in the Design Technology building and to use video capture. One student will hold the coffee filter and drop them from the balcony. Another student will be ready to take the video from the stairs on the North side of the building across the balcony by using the logger pro. Student will drop 1 coffee filter for the first time and record the video. For the next time, student will drop 2 coffee filter at the same time. Make sure the the first coffee filter is always at the bottom each time. Next time will be 3,4, and 5. student need to record every single drop and save it into computer.

When we had all 5 video captures, we returned to our class room and worked and the graph. We opened the video capture by using the logger pro. We need to scale the yellow line straight vertical and horizontal component to make sure all the data will be accurate. By plotting all the blue point into on the video capture, we got the position of the coffee filter closer and closer to the ground. After that we put all the values into the position vs. time graph (Fig.2). We did linear fit for the graph and chose the some good values to calculate the slope of the graph (Fig.3).

Fig.2 Position vs. Time graph of falling object

Fig.3 Do linear fit and calculate the slope

We had to do all 5 video captures and figure out 5 slopes. These slope will be the values of k. After we get all the slopes, they are velocity of the coffee filters while falling down each time.
   

Fig.4 Graphing and finding the slope 

Fig.5 Graphing and finding the slope

  

After we get all the velocity value, we put them into a new graph which is velocity vs. force. Then we did power fit for the graph and we got a curve line F=AV^B (A is the constant, V^B is velocity m/s of the falling object raise to some power). If all the points stay close to the point, it means our experiment is good (Fig.6).

Fig.6 Power fit of the falling object

Next, we measured the mass of all 5 coffee filters and divide by 5, so we get the mass of single coffee filter. We did the calculation on the white board as shown in Fig.7 finding the force of air resistance. F=mg (m is the mass of the object, g is the gravity 9.8 m/s^2).

Fig.7 calculate the Force of air resistance

Then, we did the excel for testing all the values that we have, we use it to make predictions compare predictions with our reality (Fig.7). First, we plot in the values for k, n, m, g, and delta t as the value in (Fig.8). In this table, k is the value of A, n is the value of B (F=AV^B in Fig.7), m is the mass of coffee filter from every dropping, g is the gravity, and delta t is the changing time of the falling object. For column A in excel, we plot the value into t A9=A8+$B$5$ (t is the time interval), delta t is equal 0.01 . For column B, a is acceleration of the falling object. We plot the value into column B for B8 is 9.8 and we have F net= mg - kv^n = ma => a = g-((k/m)v^n). We plot in the value of a = g-((k/m)v^n) into B9 (Fig.11). We plot the value into column C for C9= A8*delta t. We plot the value into column D for D9=V1 + delta V (Fig.13). For column E, we plot the value into E9= (delta V)/2 (Fig.14). For column F, we plot the value into F9=X1 + delta X.

Fig.8 Excel of falling object

After putting all the values into the table, we calculated columns in excel for v and x as functions of time, determine what our model predicts for the terminal velocity of the various coffee filters. If our model is good based on the measurements, then the model should spit out values for terminal velocity that are a good match to our experiment data.

In the Fig.9, we can see that when time t=2.16s the delta velocity is almost equal 0 and delta x stays constant 1.824. We can compare to the value of the slope (v=m/s) in Fig.7 which is also equal to 1.82. Both values are very closed to each other, that means our experiment is good.

  Fig.9

Error

 There were some several mistakes while we were doing the experiment. First, the videos which we were recorded not working very well because they were leaned to one side. As we put into the logger pro, we tried to fix the vertical components, but not very well done. Next, while doing calculated for some value, we rounded up some numbers. It made the results not very accurately.

Conclusion

We have an experiment and expectation that air resistance force on a particular object depends on the object's speed, its shape, and the material it is moving through. We learned how to set up an experiment and working as a group to figure out the result based on modeling lab of falling object. We learned how to use excel as a program to predict our experiment.

09-March-Propagated uncertainty in measurements

Propagated Uncertainty 

Purpose

In this lab we had to find the propagated uncertainties for the density of three cylinder metals and for hanging masses.

Part 1.  Measuring the density of Metal Cylinders

What we used:

• metal cylinders (brass, aluminum, and bronze)
• vernier caliper
• scale

This is a vernier caliper measuring the diameter of a metal cylinder

What we did: 

1. the diameter and height of each cylinder was measured in cm
2. the weight of the cylinders are measured in grams
3. density was calculated for each metal 

Data:

This is the data we collected from measuring the metal cylinders

Calculations:

This is how we solved the equation of propagated uncertainty (dP) for density

This is the data table with the calculated numbers to get the propagated uncertainty in density

So, for the first part of the lab, we found the density and the propagated uncertainty for each metal cylinder. 

For copper: 9.08 g/cm^3 + 0.18 g/cm^3

for aluminum: 2.71 g/cm^3 + 0.06 g/cm^3

For brass: 8.41 g/cm^3 + 0.13 g/cm^3

Part 2. Determination of an unknown mass.

What we used:

• angle reader

What we did:

1. we took measurements of the lab set up for the hanging mass 
• we measured the tensions and the angles  

This is the set up for the hanging mass

Data:

This is the data that was collected from the set up of the unknown hanging mass

Calculations:

This is how we found the equations for the unknown hanging mass and the propagated uncertainty (dm) for mass

This is the propagated uncertainty for the hanging mass

We found that the hanging mass was 0.75 kg and the propagated uncertainty was 0.032 kg.
So, the hanging mass is 0.75 kg + 0.032 kg.

Conclusion

In this lab we learned how to find the propagated uncertainties by taking partial derivatives. We were able to find the propagated uncertainties of density for 3 metal cylinders and that of a hanging mass.
When we compare our calculated densities along with the propagated uncertainty, our answers come close to the actual densities of the metals.  

04-March-Non constant acceleration problem

Rocket Powered Elephant

Purpose

To use excel to determine numerically how far an elephant goes before coming to rest.

This experiment is essentially used to determine how far an elephant would move before coming to rest. The distance traveled was found by using both analytic and numerical integration.

Analytic Solution

In order to solve the problem analytically, we used the following acceleration equation (Newton's 2nd law)

We then integrated this acceleration equation  from  time=0 to time=t in order to find the change in velocity. 

We then integrated the velocity equation using the same time parameters as the previous integration in order to obtain a position function relative to time.

To solve for position we determined the initial velocity and the time at which the velocity of the elephant is zero. Plugging this values in the position function relative to time gave us the distance traveled by the elephant.

Numerical Solution

In order to solve the problem numerically,a table like the one shown below was created in order to organize the data and make use of excel's ability to take a formula and apply it to other cells within the spreadsheet.

How to Set up the columns: 

Newton's Second Law states that Force=mass*acceleration (F=ma). Thus as seen above in order to find acceleration, Force can be divided by a mass. In the scenario that was given, a rocket produced 8000N of thrust onto a total mass given as a function of time m(t)=6500kg-20kg/s*t. This equates to acceleration being -400/(325-t) (m/s^2). By using this value for acceleration as a function of time and inserting it into the excel sheet, acceleration at any given time was calculated as seen in the second column. 

Using that information, we were able to calculate the average acceleration between every time interval which was put into column 3. Change in velocity was calculated using the average acceleration within a certain interval divided by the change in time. Individual velocities at any given time were then found by taking the initial velocity of 25m/s and adding the change in velocity. The change in position between intervals was determined using a kinematics equation, taking the average of velocities within an interval and multiplying that by the time passed. Finally, position from the origin was found by adding the changes in position between intervals.

What the columns looked like:

After doing these calculations, a data table of over 800 points was produced in order to see changes in data if the time intervals changed. The table shown above had a time interval of 0.1 seconds and shows the first 2 seconds of acceleration as an example.

Below is the spreadsheet when velocity reaches 0 using 0.1 as a time interval. As shown in the highlighted region, the velocity is zero between 19.6-19.7 seconds with position being around 248.7 meters.

However, If the intervals are adjusted, you are able to determine the exact moment where velocity reaches zero more precisely. Below the intervals were adjusted to 0.05 seconds. In the highlighted region, velocity is zero between 19.65 seconds and 19.70 seconds. By making the intervals smaller, the values of time become more precise. 

Conclusion

To analyze date, its better and more efficient to do it numerically instead of analytic because of the huge amount of data than can be processed in a matter of seconds. When doing things analytic is hard to know how an object may have been traveling in between intervals of time and is also tedious because integrating is usually difficult.

When making the intervals very small, there may be a point where values start to become constant; it may be at this point the experimenter may feel that they do not need to go any smaller in precision. If solving analytically, the precision would have to be based solely on significant figures and the power of the calculator that is being used. 

02-March-2015: Free Fall Lab

Free Fall Lab

Purpose 

The purpose of this lab is not only to determine the acceleration due to gravity on a free falling object, but also to learn how to analyze data and understand the importance of experimental uncertainty. You will also practice and expand your knowledge about Excel.

Set up

The spark generator was connected to an object attached to a metal rod which allowed for 1.5 m of free falling distance. Spark tape is put parallel to the metal rod along the 1.5 m distance. The falling object was held at rest by an electromagnet before the experiment begins. When the object was released, the object creates sparks which mark the spark tape at a rate of 60 sparks/sec. The spark tape was then removed and taken for measurements.

A meter stick with +/- 0.05 m of uncertainty was used to measure the distance of each spark from the origin. This data was recorded and graphed in excel for analysis.

Part #1 Data colection

What we did was to create a table like the one below in order to make graphs that would display values for acceleration. Since the sparks were produced at a rate of 60 sparks/sec, time intervals were set at 1/60th of a second (1st column). The distance were also recorded per 1/60th of a second (2nd column). The change in distance or Delta X  was then calculated (third column). Time was then split into mid-intervals (4th column) and the speeds during these intervals was also calculated (5th column).

Part # 2 Data Graphing

Velocity was graphed using Time vs Distance in order to verify that the data made sense. This was verified as the graph showed an upwards curve due to a constant acceleration. The value of constant acceleration was then determined by taking the derivative of the y=x^2  function (deriving distance gives you velocity). Thus by taking the derivative of y=484.59 x^2, a value of 969.18 m/s^2 was found. The R^2 value was 0.9999 which showed that the curve had a direct correlation. 

Mid-interval Time vs Mid-interval Velocity was then graphed which gave us acceleration. The acceleration was given by taking the slope of the graph and was determined to be 970.03 cm/s^2 or 9.70 m/s^2. This value was approximately .11 m/s^2 slower than the accepted gravitational acceleration value. The two graphs gave similar accelerations of 9.70 m/s^2 which is -1.12% from the actual value of acceleration. 

Part #3 Data Analysis (how we obtain results from the graphs above)

1. Show that for constant acceleration the velocity in the middle of a time interval is the same as the average velocity for that time interval.

           First you need to look at your Velocity vs Time graph and select a Mid-interval time from your x-axis and                draw a doted line upwards to the trend line and from that point you need to draw a horizontal doted line to            the the y-axis to obtain the corresponding value for velocity.Then you simply solve for velocity using this                values. This is the velocity in the middle of that time interval.  Now to obtain the average velocity for that                time interval you need to find average velocity.  As you can see for constant acceleration, the velocity in                the middle of a time interval is the same as the average velocity for that time interval.

    2. Describe how you can get the acceleration due to gravity from your velocity/time graph.          
        Compare your result with the accepted value.

        To obtain acceleration from a Velocity vs. Time graph you need to find the slope of the line.
        or you can take the derivative of  the velocity function. Comparing your result with the accepted value of g             can be done in two ways

                     Absolute difference = (experimental value - accepted value)

                                                      = (9.7 m/s^2 - 9.8 m/s^2)

                                                      = -0.11 m/s^2

                           

                     Relative Difference =  (experimental value - accepted value)/ accepted value x 100%

                                                      = (9.7 m/s^2 - 9.8 m/s^2)/ 9.8 m/s^2 x 100%

                                                      = -1.12%

    3. Describe how you can get the acceleration due to gravity from your Position vs. Time graph.
        Compare your result with the accepted value.

        From an X vs T graph you obtain a function of velocity. If you then integrate the velocity fuction, you get a             fuction for acceleration. You then compare your result using absolute difference and relative difference.

Part # 4 Analazing the class data for g

The values of gravitational acceleration were taken from the class and a class average was computed. This average was determined to be 9.53 m/s^2 and standard deviation was used to determine a range. The standard deviation had a value of 12.27 m/s^2 thus giving us the +/- value for our results.

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    1.  What pattern is there in the values of our values of g?                                                                                               As expected, the distance traveled by the falling mass increases every period of time. We can  also                        observe that the velocity of the object constantly increases as the mass falls.

    2.  How does our class average value compare with the accepted value of g?

           Absolute difference = (experimental value - accepted value)

                                        = (9.53 m/s^2 - 9.8 m/s^2)

                                        = -0.27 m/s^2

                           

           Relative Difference =  (experimental value - accepted value)/ accepted value x 100%

                                       = (9.56 m/s^2 - 9.8 m/s^2)/ 9.8 m/s^2 x 100%

                                       = -2.75%

        

    3.  What pattern if any is there in the class values of g?    

         As expected there is no pattern. The data is randomly distributed around the average value of the class                value of g

    4.  What might account with any difference between the average value of your measurements and                              those of the class?

          Friction from air resistance could not have been zero because the experiment was not performed in a close           container of something similar. The accuracy of the electromagnet was not very reliable because as we                 noticed on our Velocity vs. Time graph there were two dots  that were completely off from the trend-line.

    5.  Write a paragraph summarizing the point of this part of the lab. What were the key ideas? What were                    you supposed to get out of it?

         

        The point of this part of the lab was to calculate the average g value for the class. From this value we                     calculated the standard deviation of each group's g value. Thus the purpose of this part of the lab was to               analyze experimental data and evaluate experimental uncertainty. Because all of our measurements (For             example, the distance between each dot on the  paper strip) were uncertain by nature, our results were also         uncertain. And so we calculated the error in each of our values and compared it to the accepted value of g           by obtaining the Standard Deviation of the mean.

Error 

The values that we found and the class average were both lower than the actual value. This may be due to friction along the metal rod as the object was in free fall. This friction could have slowed down the object and not allowed for full acceleration. The class average was 2.75% lower than the actual value compared to our experiment's 1.12%. This may be due to inconsistency in the equipment and the nature of the data recording as there were many different people taking data from different strips. This could add up to human error that may have affected the average. 

Conclusion

The experiment was a success as there was only a range of about 3% error between the data. The group acceleration was determined to be 9.70 m/s^2 while the class average was 9.53 m/s^2. These values were compared to the actual value of 9.81 m/s^2. The values were determined using excel and taking the derivative of the position vs time graph or the slope of the velocity vs time graph.