Steven,+Ryan,+Navin,+Aaron

**Class:** Honors Physics Period 2 **Date:** Monday, December 20, 2010 **Purpose:** To find how the acceleration of an object moving down an incline is related to the angle of the incline.
 * __ Lab: Acceleration Down An Incline __**
 * __﻿ ﻿ __****__﻿﻿__Group:** Steve, Ryan, Aaron, Navin


 * Hypothesis: **The acceleration of an object is dependent on the angle of the incline. As the angle of the incline increases, so will the acceleration. Based off of prior knowledge of Newton's Laws, acceleration can be found by using the below equations:

Mass Sliding Up Incline due to Hanging Mass: Mass Sliding Down Incline:


 * Materials:** wooden block, aluminum track, ring stand, clamp, meter stick, 2 photogates, picket fence, tape, photogate stands, pulley, string, masses, mass hanger

media type="file" key="Movie 11.mov" width="330" height="330"
 * Procedure:** The procedure was supplied in the lab handout. However, the video below displays a data run, showing how we performed each trial and the materials that were used.

__Part A:__ The following graphs were derived from data studio. They display the 5 sets of trials used to produce an average acceleration. Next to each of the following is the corresponding angle measurement for θ. By using a velocity vs. time graph, the slope is m/s/s or m/s^2, which is equivalent to acceleration. 15º: 16º: 17º: 18º: 19º: The following data tables were from each of the five trials of each of the different weights that we used. The first one corresponds with the first graph (see below), and the second corresponds with the second graph below. Better Data:
 * Data Tables and Data Studio:**

__Part B:__

__Part A:__
 * Graphs with Explanations:**
 * __First Results of Accelerating at Varying Inclines__**



Our first results were very far off from what was anticipated. Our R^2 value was .59 (which is much more incorrect than normal) and our slope was supposed to be equal to the force of gravity, but ours was more than triple that value. We came in later in order to remeasure the acceleration of the block.


 * __Better Results for Acceleration at Varying Inclines__**

This graph is a collection of accelerations measured at varying angles for a block that slides down a surface. These accelerations were recorded by a photo eye seeing a lens that was placed on the block. The resulting slope should be the value of gravity on the block, and the difference is the cause of error. Our results were consistent, which can be seen because our R^2 value is .9696. Our results are greater than expected, but this was because of error.

Spreadsheet Document: __Class Results:__ || || --- || %Difference=(|Experimental-Average|/Average) *100 %Difference=(|11.223-10.443|/10.443) *100 %Difference=7.47%
 * Percent Difference:**
 * ~ Name ||~ Slope ||~ y-internet ||~ R2 ||~ slope with y-int = 0 ||
 * = Rachel ||= 10.398 ||= -1.9849 ||= 1 ||= 3.99 ||
 * = Evan ||= 10.476 ||= -1.8539 ||= 0.9976 ||= 4.03 ||
 * = Nicole ||= 9.0393 ||= -1.9593 ||= 0.978 ||= 3.72 ||
 * = Justin ||= 8.7236 ||= -1.7436 ||= 0.9782 ||= 2.31 ||
 * = Andrew ||= 13.406 ||= -3.6733 ||= 0.9839 ||= 3.38 ||
 * = Ryan ||= 11.223 ||= -2.5289 ||= 0.9696 ||= 2.63 ||
 * = Emily ||= 10.089 ||= -2.0541 ||= 0.9987 ||= 2.23 ||
 * = Jessica ||= 9.7156 ||= -1.8744 ||= 0.982 ||= 3.68 ||
 * = Jae ||= 10.393 ||= -2.6813 ||= 0.9891 ||= 4.36 ||
 * = Deanna ||= 9.8699 ||= -2.264 ||= 0.9888 ||= 3.01 ||
 * = Chris H ||= 10.089 ||= -2.0515 ||= 0.989 ||= 4.3 ||
 * = Eric ||= 10.0376 ||= -2.3396 ||= 0.9951 ||= 4.21 ||
 * = Sam ||= 12.299 ||= -3.0632 ||= 0.9935 ||= 4.15 ||
 * **AVG.** || **10.443** ||

__Part B:__


 * Analysis:**

1. Make a graph of acceleration versus sinθ, using Excel. //See above graphs.//

2. Find the coefficient of friction between your incline and the block using the equation of of your trendline.

(From Graph with Better Results) y-int=-2.5289 y-int=µgcosθ 2.5289=µ(9.8)(cos(15)) µ=.267

3. Calculate the percent error between the slope and g(earth). //Show this calculation.//

% error = |Actual value- Theoretical value| / Theoretical value X 100 % error = |11.223- 9.8| / 9.8 X 100 % error = 14.52% error

4. Compare the value of the coefficient of friction between your incline and the block to that from last week's lab. The coefficient of friction between the incline and the block of wood this week was 0.267 while last week's was 0.1501<span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; font-size: 12px; line-height: 17px;">. Both labs had friction between the same two materials– wood and metal. Despite this, however, there was a difference between the two coefficients. There is only one explanation for this– error. Error,especially human error, is the only cause for this type of change.


 * Calculations:**

__Pre-Lab:__ __Part A:__ Pre-lab calculations were used along with help from excel.

__Part B:__ Mass of B with µ taken from graph=172.5 g with µ taken from average acceleration=154 g


 * Discussion Questions:**

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


 * The slope of the graph is the force of gravity on the system which is 9.8 m/s^2 but we got 11.223 m/s^. The y-intercept is the friction force between the object and the surface.**
 * Note: See Data and Graphs**

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


 * If the mass of the cart were doubled, then the results of acceleration would be much higher. This is because more weight is pushing down and forcing the cart to go on the x-axis. We saw this when we first tried this experiment with 300g as opposed to 200g.**

3. Consider the difference between your measured value of g and the true value of 9.80m/s^2. Could friction be the cause of the observed difference. Why or why not?


 * In our case, we found that our value of g was higher than 9.8m/s^2 at 11.223m/s^2. This being the case, it is difficult to say whether friction played a cause or not because the error on our part is so high. Our results do not seem to indicate so, but one would theorize that since friction changes based upon acceleration, and gravity is dependent on acceleration, then gravity would decrease due to friction. This is not the case because of the error in this lab.**

4. How were your results in Part B? Why was the expectation that your results be within 2% considered to be reasonable when in other labs we allow much larger margins of error?


 * Error Analysis/ Conclusion:** Before this experiment, we hypothesized that the acceleration of an object would be dependent on the angle of the incline. As the angle of the incline increases, so will the acceleration. Based off of prior knowledge of Newton's Laws, we believed that acceleration could be found from the following equations:

A) Mass Sliding Down Incline:

B) Mass Sliding Up Incline due to Hanging Mass: For part A, our hypothesis was generally correct despite having a 14.52% error. Once we collected our data and graphed it, we got an equation for the line on our graph. This line's equation was y=11.223x-2.5289. Because the slope was theoretically 9.8, which is value of earth's gravity, there was a little bit of error. We can conclude that, since our slope was larger than it should have been, there was an extra force pushing down on the wood block. This force most likely came from our group giving the wood block a slight push down the incline. Error also came from the track itself. We noticed that, as we performed trials, the wood block moved from side to side as it moved down the track. This is a result of the track being slightly tilted to one side. Because of this and possible additional forces acting on the system, our acceleration was greater than the theoretical value.

-**__ Lab: Coefficient of Friction __** **Group:** Steve, Ryan, Aaron, Navin **Class:** Honors Physics Period 2 **Date:** Monday, December 13, 2010

// 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. //
 * // Purpose: //**

// The coefficient for static and kinetic friction can be found by measuring different movements of two surfaces. We can find Normal Force through weight, and friction force through Tension, and divide them. This will give us the coefficients friction, all based around Newton's Second Law (F=ma). //// The relationship between friction force and the normal force is linear and proportional, as one increases, so does the other. // **M****aterials:** Force meter, USB link, wooden block, masses, string, aluminum track, clamp, laptop with Excel **Data Tables:**Mean Tension at Constant Speed:Maximum TensionMinimum Angle to Break Static Friction between block and aluminum:Maximum Angle before breaking static friction: **Data Studio Graphs:** This first data set represents the movement of the wood against the track with .5kg on it, for a total of .652kg. This was then dragged across the track by a string. The subsequent data sets are with increased weight. Notice the peaks at the beginning of each data set. This is the static friction needed to begin movement. The greatest amount of energy is needed to start the movement.This data set has 1.152kg. It can be observed that the force needed to pull has jumped up at least one Newton. This change continues as we increase weight. This data set has 1.652kg on it.This data set is 2.152kg. We took the mean of the data set to find the kinetic friction for each weight. Since kinetic friction is the friction throughout the entire movement, it makes sense that we took the mean. Last, we had a weight of 2.652kg on the wood. **Analysis:** 1. From measured data in Part A, use Microsoft Excel to plot a graph of Static Friction vs. Normal Force. Add a second line for the kinetic friction data, on the SAME graph. Set the y-intercepts to zero, and show the equations of the lines with the regression coefficient.
 * // Hypothesis: //**

2. Compare the slope of line with calculated µs average (% difference). The average static µ of the class was 0.2069, which, compared to our derived value of 0.1797, as a 13.15% difference. % difference= |experimental-average|/average x 100 Fs=|.1797-.2069|/.2069 x 100 Fs=13.15% ** Averages of the class: **

*Please note that these averages were recalculated because we derived new values for static and kinetic friction once we corrected a mistake.
 * Group || Static Friction || Kinetic Friction ||
 * Ours || 0.1797 || 0.1501 ||
 * 1 || 0.1985 || 0.1792 ||
 * 2 || 0.2199 || 0.2008 ||
 * 3 || 0.2397 || 0.2131 ||
 * 4 || 0.2070 || 0.1793 ||
 * 5 || 0.2251 || 0.1769 ||
 * 6 || 0.1783 || 0.1590 ||
 * Avg || 0.2069 || .1798 ||

3. Compare your values of µ found in Part A with those found in Part B. The coefficient neu found in Part A will be very similar to neu found in Part B because the change in angle for Part B is very small. The small change of 10 degrees will cause the block to move on its own because gravity now helps it move. The value of neu, as a result for static and kinetic friction, will be less than in part A when the track was level.




 * Discussion Questions:**

After drawing a free body diagram of the wooden block, there are the values of normal force and weight on the y axis and the values of tension and friction on the x axis. Because acceleration in both cases is 0, "ma" cancels out and leaves normal force equal to weight and tension equal to friction. Because we recorded the values of weight and tension, these values can be used to show normal force and friction. Once plugged into the graph, the slope, which is rise over run or change in y over change in x, equals friction force over normal force. This also equals µ. The derivation is shown below:
 * 1. Why does the slope of the line equal the coefficient of friction? Show this derivation. **

m=∆y/∆x m=f/N

f=µN f/N=µ=m

[] According to this website, the coefficient of static friction between wood and any clean metal (aluminum in our case) is between .20 and .60. The static friction from our trials was .1797 which does not fall into the range. **3. What variables affected the magnitude of the force of friction? What variables affected the magnitude of the coefficient of friction?**The force of friction was effected by the tension pulling in the opposite direction. We found that these two forces are equal, so if one increases, the other would as well. We also found that as mass increased, and subsequently as weight increased, there would be more friction since there would be greater contact between the two surfaces. The coefficient of friction is dependent on the Friction Force/Normal Force. As one changes, so does the coefficient of friction. Thus, things like tension and weight would also effect the coefficient of friction. **4. How does the value of coefficient of kinetic friction compare to the value for the same material’s coefficient of static friction?**The coefficient of static friction will always be higher than that of kinetic friction, and this was the case in our experiment. This is because, the force is takes to get an object moving (static) is much greater than when the object is already moving (kinetic). **5. Did putting the track on an incline significantly change the coefficient of friction? Why or why not?** Putting the track on an incline did not affect the coefficient of friction by much because we did not have to change the angle of the track by a significant amount in order to break the static friction. We only had to raise the track to an angle of about 10 degrees before the static friction was broken, so as a result the coefficient for friction only changed by a small amount.
 * 2. Look up the coefficient of friction between wood and aluminum. Discuss if your measured results fall within the range of theoretical values.**

The objective of this lab was to measure the coefficients of static and kinetic friction between two surfaces, and to observe the relationship between friction force and Normal force. We went about measuring these coefficients by actually creating friction between two surfaces: aluminum and wood. Using a sensor to test tension, we could graph our results multiple times for different weights and observe the difference as we went up. Using the maximum point on the tension graph (see "Data Studio"), we found the static friction; while the kinetic friction was found using the mean value of each graph. This gave us our values, and we generally found that as weight increased tension, which is equivalent to friction force, increased by one Newton. So overall, the data suggests our hypothesis was correct. In fact, the way we found the coefficients was more or less exactly how we found them. We used the tension and weight, which then equaled the friction and Normal, graphed them, divided f/N, and used that slope for the static and kinetic friction. This further fuels our hypothesis that friction and Normal force are linear. This is clear from our graph because as one increased, as did the other.
 * Conclusion:**

Error in this lab could have come from many places. In part A, it could have come from the decreasing strength of the rope, which was hanging on by a tiny thread towards the end. The lack of strength in the rope could have decreased tension and thus friction. Any foreign dirt on the track would also have generated extra friction and corrupted the results. Last, not pulling the sensor evenly would lead to different results each time and make it impossible to have a straight line in data studio. Error in part B would mostly be human error, but if there was again dirt particles on the track, more friction would be added.

= __ ﻿Lab: Atwood's Machine __ =

Group: **Steve, Ryan, Aaron, Navin** Class: **Honors Physics Period 2** Date: **Monday, December 6, 2010**

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

// Materials: //
 * // Stand, rod, clamp, Atwood's Machine, 2 mass hangers, masses, laptop with photogate, string, masking tape. //**

Procedure:
 * (initial setup)**
 * 1. Tighten rod into the stand.**
 * 2. Clamp pulley onto top section of rod.**
 * 3. Attach pulley output wire to photogate timer. Connect photogate timer to laptop via USB.**

4. Place string through the pulley and tie a mass hanger to each side.
 * [[image:procedphoto-Raj3.jpg width="320" height="240"]]


 * (lab procedure)**
 * 5. Place an equal mass to each mass hanger. Tape masses to hangers with masking tape to prevent from falling off.**

6. Take mass (2g) from one hanger and move to other hanger to perform trials.
 * [[image:procedphoto-Raj2.jpg width="320" height="240"]]
 * [[image:procedphoto-Raj2.jpg width="320" height="240"]]


 * [[image:procedphoto-Raj.jpg width="320" height="240"]]


 * 7. Perform each trial 5 times.**
 * 8. Repeat steps 6 and 7 five times for a total of 25 trials.**


 * media type="file" key="procedmovie-Raj.mov" width="297" height="297"

Data/ Graphs: [[image:iwoa[fhsdf[oiha.png]] //We then took those averages and multiplied them by the total mass, which was constant at .320 kg, in order to find the total force.// [[image:sagvkbhljndsad.png]] [[image:xdtcfyvgiubohiajnsd.png]]
 * //Using data studio, we were able to record the acceleration by taking the slope of the velocity. We were then able to formulate the following graphs made by data studio.//**
 * //A: 0.162, B: 0.158//**
 * //A: 0.164 kg, B: 0.156// //kg//**
 * //A: 0.166// //kg////, B: 0.154// //kg//**
 * //A: 0.168// //kg////, B: 0.152// //kg//**
 * //A: 0.170// //kg////, B: 0.150// //kg//**
 * //After collecting data from each of the 5 mass variations, we recordex each trial into excel and found the average acceleration.//**
 * //From this data, a graph was fabricated to show Force vs. Acceleration.//**
 * The slope of this graph is .32, which coincidentally is the same value as the total weight of our system during the lab. Because the slope is force over acceleration, there is a very relevant explanation to why these two values are the same. Using the equation, force=mass X acceleration, it can be inferred that mass equals force divided by acceleration. Because the slope of the trend-line and our total mass are the same, there is essentially no error. If the weight of the pulley and string were added to our experimental value of mass, the slope may have been different. However, these two masses were ignored in this lab. The y-intercept of our graph is zero because if there was zero acceleration, there would have to be zero force. //All work and equations for this explanation are shown below.//**


 * F=ma **
 * F/a=m **


 * F=ma **
 * F=m x 0 **
 * F=0 **

Error:


 * % error = |Actual value- Theoretical value| / Theoretical value X 100 **
 * % error = |0.32- 0.32| / 0.32 X 100 **
 * % error = 0% **


 * This lab lacks any error as far as we can calculate because the actual weight of our total mass was 0.32kg, and our theoretical weight, which we calculated using the slope of the graph, was 0.32kg.**

Questions and Analysis:


 * <span style="font-family: Arial,Helvetica,sans-serif;">1. **//Note:// **//number one seemed to flow better when left in the description of data. It made it easier to follow along. Refer to the data for the answers to the questions of number one. Thank you.//**


 * <span style="font-family: Arial,Helvetica,sans-serif;">2. Friction causes a decrease in our acceleration because it resists the movement of the string and the pulley. As a result, our acceleration is slightly too small, but not enough to make a significant difference in our results, as the pulley had very little friction. Because of the friction, it would require a greater force to create the same acceleration was we want to find. Our slope was too small. Friction is a source of error in this experiment because it was not included in the calculations, even though it was present in the experiment. **


 * <span style="font-family: Arial,Helvetica,sans-serif;">3. Our data is quite precise, in that, it is the same no matter how much we repeat the trials. There was maybe a .01 difference in each of our trials, leading one to draw the conclusion that all our data repeated nicely and was precise. Looking back, there were a few outliers, that while are not too far off, but are not very close comparatively. There are logical explanations for the minute lack of precision on certain trials; the most prevalent of which would be the weights flinging about wildly as they accelerated both up and down. Going down the list, nothing stands out as not precise, and some answers are even more accurate than others. Overall, the precision of our data was a success and wholly repeatable.**


 * <span style="font-family: Arial,Helvetica,sans-serif;">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? **

Conclusion:


 * The purpose behind this lab was to find the relationship between net force and acceleration. As we went through our procedure, the data made it clear that net force and acceleration are directly related. As is presented in our data, when the net force increases, so does the acceleration and vice versa. This explains why our graph is linear. This leads us to conclude that the relationship between net force and acceleration is a direct, proportional relationship.**


 * As was seen in the error analysis, this lab had 0% error. Does this necessarily mean Steven, Ryan, Navin, and Aaron are the greatest scientists ever who deserve A+'s... sadly no. The theoretical value is highly approximated and only taken at that decimal place, meaning it can indeed be slightly different. Now, we also did not take in to account friction. Seeing as the theoretical is the same as the experimental, we know that friction is nearly non-existent, but it is still present and a more exact answer would have been found if we had taken it into account.**

= __ Lab: Newton's Second Law: __ =

Group: **Steve, Ryan, Aaron, Navin** Class: **Honors Physics Period 2** Date: **Monday, November 29, 2010**

//Purpose:// **//Find the relationship between system mass, acceleration, and net force.//**

//Materials:// ** The materials that we used were a dynamics cart, track, Photogate timer, Data studio application, super pulley with clamp, string, mass hanger, and an assortment of masses. **

Procedure:
 * 1. Organize ﻿materials and setup track.**

2. Setup data studio for experiment.
 * Place track on table or flat surface. Make sure the track is level to prevent error.
 * Clamp down track at the end of the table. Attached to the clamp should be the super pulley.
 * Tie one end of the string to the cart and the other to a mass hanger.
 * Place car on track and line up the string with the pulley mechanism.
 * Attach the pulley output wire to the photogate timer. Via USB, attach the photogate timer to the computer.
 * Note: Below are pictures of the correct setup.
 * [[image:q3wz54x6ec57r6v8t7.jpg width="448" height="336"]]
 * [[image:ex7rtcuyivuobinp.jpg width="448" height="336"]]
 * [[image:rdcfv9ebyoudfipnow.jpg width="448" height="336"]]


 * 3. Perform Trials**

4. Vary the amount of weight on the cart and hanger mass. Make sure the total mass of the cart and hanger stays consistent.
 * With the help of others (one operating data studio, another holding the cart, and another making sure the cart does not hit the pulley), perform trials using various weights.
 * Note: You should use the same amount of masses each time, only varying the mass on the cart by offsetting it with the mass of the mass hanger.
 * To perform trials, pull cart back and release. Simultaneously, click start on data studio. Make sure one person catches the cart before it hits the pulley.
 * Repeat these steps 3-5 times to ensure accurate results.
 * Once the trial is over, make sure you stop the data collection on data studio.
 * Find the velocity vs. time graph and select consecutive points that are consistent.
 * Select fit, then linear.
 * A box will pop up showing multiple values. Make sure you record the value of M (slope). This is the acceleration of the cart.
 * media type="file" key="iuiuhucucoxhc.mov" width="300" height="300"
 * [[image:wz4x6e5c7r6tv87bu9.jpg width="384" height="288"]]

5. Repeat the same process. This time, keep the hanging mass the same and change the mass on the cart. The goal of this is step is to find the mass that lets the cart barely move. The values below include the weight of the cart itself.
 * Keep a total mass of 25 grams. Any kind of combination can be used. Below are the masses we used: (Keep in mind that the cart value does not include the cart itself, which is 500 grams)
 * Cart: 0, Hanger: 25
 * Cart: 2, Hanger: 23
 * Cart: 10, Hanger: 15
 * Cart: 20, Hanger: 5


 * Cart: 2000, Hanger:1005
 * Cart: 3400, Hanger: 9

Data/Graphs:


 * //Our first data set shows the relationship between acceleration and force.//**




 * Above is the table which contained all of our values for different placements of weights onto the hanging mass. In all, the total mass stayed the same, as it should, but the force changed when we moved weights from one side to the other. This is because force is calculated with the equation "F=ma," where F= force, m=mass, and a=acceleration due to gravity. This factors into the larger equation for the rope-pulley system:**


 * a=** //((Hanging Mass)(9.8))// **/((Hanging Mass) + (Cart Mass))**




 * The above graph is a linear fit that represents the change in force based on how acceleration changed in the equation. This means that as acceleration increases, the amount of force exerted by the hanging mass //MUST// increase. The relationship between these two was found by keeping the total mass constant, and adding weights onto the hanging side. This changed the hanging mass, and thus the force value, and finally as represented in the graph, the acceleration. The slope of the best fit line is the force/acceleration, which equals the total mass of the system. This can be derived from F=ma... in this case m=F/a. However, it can observed that the theoretical value of the mass of the system does not equal the actual value.**

% **error = (|0.525- 0.518|** / **0.518)** X **100**
 * % error = (|Actual value- Theoretical value| / Theoretical value) **X **100**
 * % error = 1.35% **


 * The percent error for the theoretical weight, and the actual weight is 1.35%. In actuality, the .525 should be the slope to make the F=ma equation true in the form of y=mx+b; where y=F and x=a. The y-intercept value is the force when there is no acceleration. In the experiment, there is no acceleration only when an object is at rest. Thus, the y-intercept value represented the force exerted when the object is supposedly at rest for a few seconds. In this case, it is not very little and can be attributed to human jolts of the hand. **


 * //The next data set shows the relationship between the hanging mass and acceleration.// **




 * The table above has the hanging mass in column one and the acceleration in column four. The equation a=F/m ties the hanging mass to the acceleration. In our experiment, the rope-pulley equation is:**


 * a= **//((Hanging Mass)(9.8))// **/((Hanging Mass) + (Cart Mass))**


 * The hanging mass value determines the force, which then determines the acceleration.**




 * The relationship shown above is linear. It shows how acceleration changes based on the hanging mass's weight; as the hanging mass increases so does the acceleration. This makes sense both in the experiment and the equation. First, if we add more weight to the hanging mass, it is going to fall down faster and thus, the acceleration of the entire system will increase. We witnessed this quite painfully when we put too much weight onto the pulley. This relationship appears in our equation because, using the equation a=** //((Hanging Mass)(9.8))// **/((Hanging Mass) + (Cart Mass)), we find that by changing the hanging mass, the force changes. This can cause total mass to change and result in an increase of decrease in acceleration.**


 * //The final relationship was that between the hanging mass and the force.//**




 * The table shows in column one the multiple hanging masses we tested with. Column five shows the force resulting from mass** X **acceleration. This is based on the equation F=ma. Our relationship for the mass vs force was always the same, and remained constant throughout all trials.**


 * This is a linear relationship between the hanging mass and the force. This means that as we increased the weight onto the side of the hanging mass, the amount of force that hanging mass had during our trials increased, and this was a steady increase. In order to do this, we kept the total system mass constant, like previously, and added weight from the cart onto the hanging mass. This would change the force because force is calculated through F=ma, where F=force, m=hanging mass, and a=acceleration due to gravity. In the end, each weight change had its own force and the larger the weight, the more force exerted by the hanging mass. The slope of the best fit line is force/mass, which in equation terms is the acceleration based of F=ma... or in this case a=F/m. In this case, the acceleration is equal to the acceleration due to gravity that has acted throughout the experiment and is correct. There is no error in this relationship, as it should be because the acceleration should always be 9.8 in the F=ma equation. There is not y-intercept in this best fit line.**


 * This relationship tells us that as the hanging mass changes, so does the force. Using this information, we can infer that if the hanging mass were left the same, then the force would be the same as well. The force relates to the larger equation for the rope-pulley system:**


 * a= **//((Hanging Mass)(9.8))// **/((Hanging Mass) + (Cart Mass))**


 * Based on that data presented, it is clear that if we change the force, the entire system changes, if we leave it as it is, the entire system is left as is. So, the relationship between the total mass and the force would also be linear assuming we changed the weight of the hanging mass. **

Analysis Guide Questions**:**


 * 1.** //Note:// **//number one seemed to flow better when left in the description of data. It made it easier to follow along. Refer to the data for the answers to the questions of number one. Thank you.//**


 * 2. Friction between the cart and the track would result in a decrease in speed, and a decrease in acceleration for the cart. As the cart moves along the track, friction will resist that movement and cause the cart to slow. As a result, it would require a bigger force pulling on the cart to generate the same acceleration. According to our calculations we got a slope that was too small as a result of friction because slope in the graph is our acceleration. Friction is a source of error in this experiment because we did calculate the change of acceleration it caused and did not take friction into account when we did our calculations. Friction in Newtons can be seen as the y-intercept on our graph (0.05 N) in newtons.**

Percent Error:

% **error = |0.5066- 0.518|** / **0.5066** X **100**
 * % error = |Actual value- Theoretical value| / Theoretical value** X **100**
 * % error = 2.3%**

Error Analysis:


 * Error in our calculations occurred when we did not take friction into account. Friction between the cart and the track caused the cart to go slower and we did not calculate the difference caused by friction. As a result, our calculated acceleration was less than expected. Another source of error was that we were not able to keep the track and the pulley at a steady height. The clamp keeping the track on the desk created a decline for the cart and caused the cart to move faster. We were not able to keep the string and the track at a level height, and gravity was able to affect the cart's movement.**

Conclusion:
 * The purpose of this lab was to find the relationship between system mass, acceleration, and net force based on Newton’s Second Law. Newton’s Second Law states that F = ma and that the force is directly proportional to the mass and acceleration of the object. It also says that mass and acceleration are inversely proportional. We ran tests for both force vs. acceleration and mass vs. acceleration. The results from our experiment proved Newton’s Second Law to be true. When the force was increased, the acceleration of the cart increased proportionally with it since the mass decreased on the cart. The error for our trials was caused by the friction between the cart and the track since we did not take it into account. The acceleration we calculated was less than what found out from our experiments because we did not calculate how friction affected our experiment. If we had calculated the difference caused by friction, our actual acceleration would be closer to the theoretical acceleration since friction would have slowed down the cart. As a result, our percent error was 2.3%.**

= __ Lab: Inertial Mass __ =

Group: **Steven, Ryan, Aaron, Navin** Class: **Honors Physics Period 2** Date: **Monday, November 22, 2010**

//Purpose:// **//Find the mass of an object only using its inertia.//**

Hypothesis: **We believe that as an object increases in mass, so will the time it takes to do ten vibrations, as well as one period of vibration because "heavier" objects are more difficult to move. Using this understanding, we can find that the relationship between mass and inertia are proportional.**

=Materials:= Inertial balance, clamp, stopwatch, known masses, paper towel, Rubik’s cube (unknown mass)

=Procedure:= 1. Clamp the inertial balance tray to table


 * 2. Place paper towel in the tray so the mass does not move**
 * 3. Place 200g mass in tray on paper towel**
 * 4. Push the balance tray and start the stopwatch**
 * 5. Count ten slides and stop the stopwatch**
 * 6. Complete this 3 times for each mass**
 * 7. Complete trials for 6 different masses**
 * 8. Place the Rubik’s cube (unknown mass) in the balance tray**
 * 9. Complete steps 3 through 6 for the Rubik’s cube**

=Experimental Set-Up:=


 * This was our set-up during the experiment. The inertial balance was clamped onto the table and a paper towel was placed inside the tray to reduce movement of the masses, which would be placed on top of the paper towel.**

=Data Collection:= Trials:




 * We performed three trials for each of our seven masses, each time getting different, and decreasing times as expected. We then found the average of all three trials per mass and the time it took to complete a single period of vibration. The mass and period of vibration would then be transformed into a graph to determine an equation and what the mass of the unknown mass is.**
 * Graphs:**
 * The graph above shows how long a period of vibration took for each of the different weights we used. It is clear that the periods increased as the mass increased; thus supporting our hypothesis.**


 * Calculations to Find Unknown Mass:**




 * From the graph, we found the equation to find mass of an object based off of its period of vibration. In order to find the mass in grams of the Rubik's Cube, we input the time of vibration into the equation of the line shown above. From this we derived 106grams exactly as the mass of the Rubik's Cube.**

media type="file" key="st,nr,ac,rl 100g wieght.mov" width="300" height="300" media type="file" key="rl,st,ac,nr unknown weight.mov" width="300" height="300"
 * Videos:**
 * Above is a trial video of our data collection for the 100 g weight.**
 * The above video is a trial of our data collection for the unknown weight of the Rubik's Cube.**

=Error Analysis= Percent Error:




 * Percent Difference: The class average was 104.44g, deriving from the values of 92.09, 120.83, 99,3, 109.5, 104.3, 102.09, and 106.**

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


 * Despite gravitation being present, it did not actually play a part in this operation because the side-to-side movement was dependent more on the mass as apposed to the gravitational pull on the objects. Gravitation kept the object from floating away, but the actual make up of the object, its mass, is what determined the length of periods. The process was unrelated to the weight because the gravitational pull that constitutes the "weight" is a constant, whereas the mass can change is what changes the "weight" of an object.**

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?
 * Each value that we tested displayed a different acceleration. Compared to other weights, acceleration would be higher if the weight was lighter, and vice versa.**

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?
 * The period of motion would have indeed been different had the side arms been stiffer. The change would have caused the period of motion to be shorter because the stiff side arms would have resisted to touch and not bounced as far back and forth. When the side arms are stiff, nothing is able to move as much.**

5. Is there any relationship between inertial and gravitational mass of the object?
 * It can be seen that inertia and gravitational mass have a common relationship because in the experiment, as the "weight" increased**, **the resistance in movement for the tray increased. This is seen because there was an increase in inertia on the tray, and it reduced speed.**

6. Why do we almost always use gravitation instead of inertia as a means of measuring mass of an object?
 * We almost always use gravitation instead of inertia to measure mass because it is much easier to find the weight than to find the inertia and weight is used more often.**

7. How would the results of this experiment be changed if you did this experiment on the moon?** The results of this experiment would be the same on the moon despite the difference in gravity because we have already established that gravity does not play a part in the periods of motion. The mass does not change on the moon, it is still made of the same "stuff" as on earth. Thus, the motion will go the same distance on the Earth and moon.

=Conclusion:= Our error for this lab was 4.53%. The main source of error came from human error involving the stopwatch. Once the inertial balance swayed back and forth 10 times, a stopwatch operated by human hands would be stopped. Obviously, the time would be a little bit off if the time was stopped a little bit before or after the 10 movements were finished. The same thing could have happened for the beginning of the trial. The time may have been started a little bit before or after the balance was pushed into motion. Looking over our data, our hypothesis seems to have been supported. Evidence suggests that objects of greater mass take more time on their periods of vibration due to their increased mass.