Working with Spreadsheets

Purpose/Introduction: The purpose of this experiment was to get familiar with electronic spreadsheets (i.e. Microsoft Excel) by using them in some simple applications. Some applications used were adding, multiplying, and dividing cells and displaying them in a separate cell. The first step to the experiment was to open a Microsoft Excel spreadsheet that calculated the function,

                f(x) = Asin(Bx) + C

                whereas:                   A = 5

                                                B = 3

                                and          C = π/3

Procedure: The procedure was to place these three variables, A, B, and C, on the right side of the spreadsheet and consider them constants. Then, we were to place a column for “x” and a separate column for “f(x).” The “x” column was to start at 0 radians and continue to 10 radians, in increments of .1 radians. The trick was to NOT enter the x values by hand, rather, use the copy feature to complete this action. After completing this task, we did the same but with the f(x) values and that was it for Step 1 of the spreadsheet part of the lab.

Next we placed the data into Graphical Analysis and received a sine graph. We fitted our curve to the same equation and our results were as follows:

A = 5.00

B = 3.00

C = 1.05

We can say we expected these results because the equation from the spreadsheet was exactly the equation from the Graphical Analysis, therefore, for every x value, the f(x) value was expected to be the same in both programs.

After this part of the lab, we opened another spreadsheet and did the exact same thing with a different equation with different constant variables. The equation was:

Whereas:            initial velocity = 50 m/s

                                Initial position = 1000 m

                                Change of time = 0.2 s

                                And g = -9.8m/s^2

Again, we were to input x values from 0 to 10, but this time in increments of .2 and not by hand. We continued by using the copy feature to complete both the x and f(x) values.

Data:

When we received our f(x) values, we copied and pasted them into Graphical Analysis; consequently, we received a concave down parabolic equation. We fitted our curve to the equation, y = Ax^2 + Bx + C and our results were as follows:

A = -4.90

B = 50.0

C = 1000

We can say we expected these results because the equation from the spreadsheet was exactly the equation from the Graphical Analysis, therefore, for every x value, the f(x) value was expected to be the same in both programs. Also, A is equal to exactly half of the acceleration of gravity. B is equal to the initial velocity and C is equal to the initial position.

Conclusion: Some of the graphs were off, but after re-fitting the equations, the Graphical Analysis program gave us the desired values. The results that the program gave us we some absurd value that could not be true; therefore, we knew we had to re fit the curve. Also, there was some confusion on the second part with the acceleration of gravity, but if you read the problem, the object must be going up and then falling. 

Vector Addition of Forces

Purpose/Introduction: The purpose of this lab was to study vector addition using graphs and also using components. We also used a circular table to confirm our results. To begin, we were given three different vector magnitudes and their angles. The vectors that were given to us are represented in this table.

Vector	Magnitude (grams)	Angle (Degrees)
A	200	0
B	100	55
C	200	135

We decided that instead of drawing a vector with a magnitude of 200 cm, we would scale the magnitudes to 1 cm = 50 g. Therefore, the magnitudes to draw were 4 cm (A), 2 cm (B), and 4 cm (C). The next procedure was to draw a vector diagram of the three vectors and find the resultant vector using a protractor and straight edge.

Vector diagram of vectors A, B, and C. Vector D is A + B + C.

The resultant force from the vector diagram was D. The magnitude of D was approximately 4.8 centimeters or about 240 grams. This means that in order to balance the circular force table, you would need to put approximately 240 grams on the fourth pulley.

The second part of the lab was to find the resultant vector (D) by using components. We graphed the vectors according to the scale and the angle given. We used sine and cosine to determine the opposite and adjacent sides. 

Data:
î represents the x coordinate

ĵ represents the y coordinate

cosθ= î /hypotenuse

sinθ= ĵ / hypotenuse

Vector	î substituted	î	ĵ substituted	ĵ
A	4cos0 = î	4	4sin0 = ĵ	0
B	2cos55 = î	1.147	2sin55 = ĵ	1.638
C	4cos45 = î	-2.828	4sin45 = ĵ	2.828

We calculated Vector D to have a magnitude of 251 grams at 62.5°. The way we found that out is that we added the three vectors and received two components, î and ĵ. We took the arctangent of (ĵ/ î) and received the angle of Vector D. In order to find the magnitude of D, we added î² and ĵ² and took the square root of the sum. We received an answer of 251, which was consequently our magnitude.

The next step was to confirm our results by setting up a force table and arranging the pulleys so that they represent our given vectors at the given angles. After placing the three pulleys on the table, the ring in the center was NOT at equilibrium. We placed a fourth pulley at 242.5° (180° + 62.5°) with 250 grams and the ring equaled out.

Circular force table that shows Vectors A, B, C, and D (resultant vector)

The next and final step was to confirm our results using a university website. We confirmed that the sum of our vectors was approximately 252 grams and about 62.5°.

Vector program from http://phet.colorado.edu/en/simulation/vector-addition

Conclusion: This lab showed that vectors can be calculated in many ways. Today’s lab demonstrated two ways to calculate the sum of more than one vector. The first way is by a graph. Although this method is less accurate than components, it is still effective. The reason why some calculations are off is because the measuring devices (rulers, protractors, ect.) are not always accurate and one is basing the measurement of the resultant on the measuring device. This is one source of error of the graphing method. The other method has a more accurate measurement. The component method uses trigonometry to calculate the angle and magnitude of the resultant vector. This is far more accurate than the graphical method because one is using angles and magnitudes rather than a ruler to measure a vector. There is hardly any error with this method. With the website that we were supposed to check our resultant vector, the reason that was off is because the program only allowed certain magnitudes and angles; consequently, one is not able to put in the correct magnitudes and angles or the given vectors. Using the component method will ultimately give one the most accurate resultant vector.

Acceleration of Gravity on an Inclined Plane

Introduction: The purpose of this lab was to study the acceleration of gravity by studying the motion of a cart on an incline. Also, the gain further experience using the computer for data collection and analysis. In this lab, we used the computer to collect position vs. time data for a cart accelerating on an inclined track. The effect of friction was eliminated by comparing the acceleration of the cart when moving up and down the track. We averaged the slightly increased acceleration (when going up) with the decreased acceleration (going down) to obtain the acceleration that depends only on force of gravity. If we let g be the acceleration when the cart moves up and down the track, then we still have to consider the angle of the track. Therefore, we can use 

where the a’s are the accelerations of the up and down incline. We measured the acceleration by looking at the slope of the v vs. t curve for the cart.

Procedures: The first step to this lab was to assemble the track and measure the angle of the incline. The way our group did it was that we took side A and subtracted side B and then divided it by side C. This gave us the sinθ. After multiplying both sides by acrsin, we got the angle of our incline track. Assuming side C and side D continued until interception, the angle made by these two sides is θ. 

This picture illustrates the above

As we placed the motion detector on the top of the track, we were able to see the velocity vs. time graph. When we pushed the cart up the track, the velocity went from negative to positive. When the velocity was equal to zero, that meant the cart reached its peak and stopped at the top. This video shows the process of pushing the cart up, and then letting it come down the track.

The slope of the interval from the negative to zero velocity was the acceleration as the object was going up the track. On the other hand, the interval from zero to the positive velocity is the when the cart is coming back down the track. The slope represents the acceleration of gravity as the cart is going down the track. 

This graph illustrates one trial at the slope of 1.74° 

Data:

 For the calculations, each trial had the same equation. The equation was as follows:

As for the work, the different trials for both θ is equal to 1.74 and 3.637.

This is the work for the when θ is equal to 1.74

This is the work for the when θ is equal to 3.637

Conclusion: We repeated the experiment with two other angles and it seemed as we increased the angle, the percentage error decreased. Therefore, we concluded that with a steeper incline, a more accurate acceleration of gravity is produced. We also figured that one source of error was because of the accuracy of the meter stick. If we had a stick with smaller measurements, the answer could have been a little closer. Also, the answer could have been thrown off if the table we had the track on was uneven or unbalanced. Although these sources of error could not have been avoided, our data shows that g was off no more that -13%. One thing that can make this experiment better is experimenting with different object with different masses. For example, you can roll a ball up the track; this may or may not affect the results. We learned that gravity is always acting on the cart. The moment when someone pushed the cart, the only force was gravity of the world does not change because you pushed the cart.

Acceleration of Gravity

Purpose/Intro: The purpose of this lab was to determine the acceleration of gravity for a falling object, and also to use the computer as a data collector in order to gain experience.In this experiment, we used the computer as a data collector to collect some position (x) vs. time (t) for a rubber ball that we threw into the air.  Previous knowledge told us that velocity of an object is equal to the slope of the x vs. t curve. Therefore, we changed the graph to v vs. t (which calculates the slope of x vs. t) and graphed the slope. We used both graphs (x vs. t and v vs. t) to determine the acceleration of gravity of the ball.

After we tested the experiment multiple times, we calculated percentage error of each trial, which is equal to

We found that the velocity of falling object has a slope of 4.9 and therefore, the acceleration ( the derivative of the velocity) is 9.8. The graph of the position graph should be a parabola because the ball goes away from the motion detector (+ slope of parabola), reaches its vertex (zero slope), and then come back down toward the motion detector(- slope). Also, the slope of the graph is negative, therefore, the ball is falling or is in free fall.

This graph shows the position vs. time graph. The slope of of this graph is the velocity of the falling object.

This graph is velocity vs. time graph. The slope of this line is the acceleration of the falling object, consequently, the acceleration of gravity.

Data:

                               
Conclusion: In the end, we did accomplish our purposes by determining the acceleration of gravity. Also, we did practice using the computer as a data collector. Therefore, the purposes were accomplished. Also, we realized that we received different percentage errors for the different trials. One of the reasons why the experiment had errors is because there were multiple “throwers.” This could have affected the path the ball could have taken. Also, we believe the hand that threw the ball was recorded with the motion detector. Finally, we believe that air resistance could have been a source of error because the height the ball was thrown could have received more air resistance as opposed to a ball thrown at a shorter height.