Emily,Elena,+Andrew


 * Acceleration Down An Incline Lab**
 * Due Date: 12/21/10**

Purpose: How does the acceleration of an object down an incline depend on the angle of the incline?

Hypothesis: The relationship between the sin of the angle of the incline and the acceleration of the object down that incline should theoretically equal g. Due to friction we estimate that the value of g will be slightly greater because with friction g=a/(sinx-µcosx) compared to a system without friction g=a/sinx. This is because g is inversely related to (sinx-µcosx). If the value of (sinx-µcosx) decreases then the value of g increases.

Materials: Wooden Block, Aluminum Track, ring stand, clamp, Meter stick, 2 Photogates, picket fence, tape, photogate stands, pulley, string, masses, mass hanger.

Procedure:

1. Attach a small clamp to the track and place it on a ring stand rod. Set up the track with one end raised higher than the other. Carefully measure the exact height. 2. Tape picket fence to block so that the small black lines are at the top. Be sure that no tape is on the bottom of the block. Find the mass of the block/fence. You may want to tape a 200 or 500-g mass to the block to give it more mass if it is too light. 3. Measure the thickness of one of the small black lines on the picket fence. BE PRECISE! 4. Set up the height of the Photgate Timer so that the small black lines will run through the laser. This can be anywhere along the length of the track, as long as the block is moving all the way through. 5. Open “Data Studio” and select “Photogate – Picket Fence”. Click on Settings, “Constants”, and enter the measurement you made in step 2 under “Band Spacing”. 6. Move the block to the end of the track. 7. Release the block and record the acceleration through the gate. Repeat this measurement several times. Record all the values in your Data Table. 8. Change and re-measure the height, and repeat the acceleration measurements. Do this for at least 5 different heights.
 * PART A: Acceleration Down an incline**



Data: In our data table above, we doubled the acceleration we received from our experiments because we only measured half of what we should have measured on the "picket fence." We only measured from the distance in between two edges; however, we were supposed to measure from one edge to the next edge. By measuring in between, we cut the distance in half, cutting the acceleration in half. In order to account for this, we doubled the acceleration from 12-20 degrees, and for 20-26 degrees, we inputted the correct information into data studio.

Class Data Table:



Data Studio Graphs:

For 12˚:



For 14˚:



For 16˚:



For 18˚:



For 20˚:



For 22˚:



For 24˚:



For 26˚:



In data studio, we took the slope of the velocity graphs to get the acceleration, which we then recording in Excel and averaged them for each trial in order to construct a graph that will give us an approximation for gravity.

Excel Graphs: Our graphs come t=from the equation: a = g*sin(theta) - (f/m)



Because of the inaccuracy of our measurements, we attained this graph. Though the slope has doubled (since we doubled the acceleration), the slope (gravity) is not close to the theoretical value of g. In order to correct this, we tried doing more trials, and took out our first two data points, giving us a graph like this:



Though we are now higher than the value of g, our R^2 value is better, indicating more accurate results. Our distance from the theoretical value of "g" may result from the friction/mass force (the y-intercept), for it is relatively large compared to other groups' frictional values.



Explanation of Issue:

In this lab, we measured the incorrect distance on the picket fence, which marred our data. Instead of measuring from the leading edge of one band to the leading band of the next, we measured the distance in between them, effectively cutting the acceleration of the block in half. By doubling the distance, we double the acceleration, and doubling the acceleration doubles the slope of the line on our sin x vs. acceleration graph, the slope being g. By making this correction, we decreased our error.

Analysis/Calculations:

Pre-Lab Derivations:








 * Calculation for Hanging Mass Needed to Accelerate a 0.193 kg mass up 0.5 meters in 1.5 seconds:





ERROR CALCULATIONS/DISCUSSION:

As evidenced by our data table, the friction decreases as the angle increases, showing that friction affects acceleration much more when the angle islow. This accounts for our large percent error in the beginning (12˚) and our decrease in error by the time we reached 26˚.
 * For Friction:**


 * For Theoretical vs Experimental Acceleration:**

We compared the theoretical value of acceleration with the value of acceleration we attained from our experiments in order to see how far off we were.


 * Overall Error Calculation (our value for "g" versus the theoretical value):**




 * Percent Difference Between Our Results and the Class Average:**





Discussion Questions: 1.Our graph is sin of the angle versus the acceleration of the block. We are measuring how the angle affects the acceleration. The slope of our graph represents gravity and on the equation it is the coefficient of x. And the y intercept represents the friction/mass force between the block and the track on the equation.

2. If the mass of the cart was doubled the acceleration should not change at all, for acceleration is independent of mass, a fact that we proved in our pre-lab derivations (sin x - µcos x = a). The masses simply cancel.

3. Friction should be the observed difference because friction opposes the weight on the x axis. In a frictionless world, the formula for the forces on the x axis would be:

wx=ma.

mgsin x = ma (masses cancel)

g sin x = a

g = a/ sin x

However, with friction, the equation changes.

wx – f = ma.

mgsin x-f = ma

f = µN

f = µmgcos x

mgsin x – µmgcos x = ma. (masses cancel)

g(sin x – µ cos x) = a

a/ (sin x – µ cos x) = g.

Because there is friction (µ cos x), the value of acceleration should decrease as (sin x - µ cos x) decreases because acceleration and (sin x - µ cos x) are directly related. Likewise, acceleration should decrease as friction increases. We expect that the force of gravity was inhibited by friction on the x-axis, thereby lowering it; however, our results indicate that gravity was higher than the theoretical value, which could be attributed to the random speeding up of our cart. Theoretically, though, friction should combat the force of gravity on the x-axis, thereby decreasing the value of "g." But theoretically, the value of g should never change, for as friction increases, acceleration will decrease enough in order to maintain that 9.81 value of "g," for g is constant.

Error Analysis/Conclusion:

One source of error is that the block was not moving very well. Although we cleaned off both the track and the block, when we ran our trials there were still parts of the track when the block would slow down and then speed up. We are not sure why this happened because even when we used different parts of the track, the block would still slow down and then speed up. This would impact our data because since the velocity was changing that means the acceleration was changing. The change in velocity also caused a change in acceleration, for when the velocity starts to decrease, we know that acceleration actually becomes negative, which contributed our error. Also, for some trials the block was not moving in a straight line; instead, it moved on a slight slant, which could have impacted our results. As seen in calculations above our results were conflicting. Our initial measurements may have played a part in it. We measured the length of the black block and used that length (.005m) in data studio, however we were supposed to measure the length of the black and white blocks, which would have given us a distance of .010 m. For the final trials when we adjusted the length and we got lower percent errors. The results we obtained from our lab supported our hypothesis. Prior to the lab we hypothesized that the slope of our line (acceleration divided by sin of the angle) should be 9.81, which is equivalent to the force of gravity. Our slope was 13.46 m/s^2. Although this is greater than the force of gravity our number accounts for the friction as well as any error in our lab. Because we found a relatively large amount of friction in our lab, it makes sense that we found a greater value for g. When we went to analyze the graphs we came across several problems. As seen in the graphs above our first graph was clearly in correct. As discussed above we found the root of our problem was in the measurements we put into data studio because we only used half the distance, which also gave us only half of the acceleration. When we redid several trials we were much more successful. We found that as the angle increased the friction decreased, which means that friction affects acceleration more on smaller angles. Although, we ran into some trouble completing this lab, our final result still proves our hypothesis correct; the acceleration divided by sin of the angle would equal g (and our experiment also accounted for the friction in the system). Our value for g is higher than the theoretical, perhaps stemming from the way that we let the car go through the force meter. If we had pushed it even slightly, then the acceleration would have increased, increasing the value of g. Even though our results were not uniform, we can deduce from them the relationship between the angle of an incline and the acceleration of an object down that incline. Our results indicate qualitatively that as the angle increases, the acceleration increases, which satisfies our purpose qualitatively.


 * The Coefficient of Friction Lab**
 * Due Date: 12/14/10**


 * Andrew Miller, Elena Solis, Emily Van Malden**

Purpose:
 * To measure the coefficient of static friction between surfaces
 * To measure the coefficient of kinetic friction between surfaces
 * To determine the relationship between the friction force and the normal force, and how the normal force divided by the weight gives the coefficient of friction
 * To measure the relationship between friction and the normal force between it, and an angle of incline

Hypothesis: In this experiment, we are trying to determine the equations for static and kinetic coefficients of friction on flat surfaces and inclines. We are also trying to determine the relationship between normal force (weight) and friction force (tension) and determine how these two factors create the coefficient of friction. By Netwon's Second Law (∑F = ma), we know that the Tension force equals the friction force, because acceleration is (theoretically) 0 because we will move the block at constant speed. By finding friction this way, we will then plug this value into the equation f = un, where u is the coefficient of friction and n is the normal force acting on the object (weight). By knowing the mass and gravity, we can calculate the normal force, and then divide f by n to attain the coefficient of static or kinetic friction.

Materials: Force Meter USB Link Wooden Block Masses String Aluminum Track Clamp Excel Protractor

Procedure:

Measuring Coefficient of Friction on a Flat Surface:

1. Mass the wooden block. 2. Clamp the surface board to the table top. 3. Place the block on the surface and put 500-g on top of it. 4. Tie a short (15 cm) string to the block at one end, and to the force meter on the other. 5. Plug the force meter into your computer. Choose Data Studio, and “Create Experiment”. A force-time graph will automatically open. 6. Go to SETUP and check **//Force – Pull Positive//** and uncheck **//Force – Push Positive//**. Then on the graph display, click the y-axis label to change the name to **//Force- Pull Positive//**. 7. Leaving the string slack, press the button “ZERO” on the sensor. 8. Press START on Data Studio, and gently pull the block with the force sensor. a. Be sure to pull with a very slow constant speed once it starts to move. b. Hold the string parallel to the board. 9. You should get a graph like the one here: 10. Highlight the straight line part and click **S**. Record the MEAN as the value for Tension at Constant Speed. 11. Highlight the maximum point and record that value as the Maximum Tension. 12. Repeat twice more with the same mass. 13. Repeat Steps 8 – 12 adding more mass each time (best to make large changes).

Measuring Coefficient of Friction on an Incline: 4. Place the track on an incline by clamping it to a ring stand. Make the angle just slightly less than the angle measured in Step 3. The block should NOT slide down on its own. When you nudge the block just slightly, it should continue down the ramp at constant speed. Have each member of your group conduct this step 2 times, recording each angle measurement, then take the average.
 * 1) Attach a protractor to the track, using the nut-screw assembly that slides onto the track.
 * 2) Secure a 200-g mass to the block with masking tape, and place at the raised end of the track.
 * 3) Slowly lift the end of the track until the block just begins to slide down the aluminum surface. Record the angle at which this occurs. Have each member of your group conduct this step 2 times, recording each angle measurement, then take the average.


 * DATA** ([[file:frictionlabbb.xls]])

Data Tables: We recorded the data from our flat level experiment (Part A) in a table. We knew that acceleration should theoretically be zero because the block was either motionless or moving at a constant speed. To calculate the coefficient of static friction, we found the average MAX tension of the string pulling the block (which was the same as friction because acceleration is zero) and divided that by the respective weight of the block. We knew that the highest point of tension would lead us to the static coefficient because it was that final friction force we had to overcome to get the block to start moving. To calculate the coefficient of kinetic friction we used the average tension (the unmarked tension columns) divided by the respective weight. We knew that we could use the average tension (found using the straight part of a below DataStudio graph) because that was the force we had to continually overcome while moving forward.)
 * __Part A:__**

__**Part B:**__ We recorded the data from Part B, our experiment where we used various inclines and recorded how they affected the coefficient of kinetic or static friction. The coefficients of friction found in part A relate to friction of an object on a flat level surface, whereas in Part B the coefficients of friction relate to an object on an incline. Objects on an incline steep enough, will slide without any apparent push. But what’s really pushing is its weight, which is a vector and as such can be broken up into components parallel to and perpendicular to the inclined surface. Thus, whenever an incline is present, an object’s weight will be acting as a force, however small, pulling it down the slant. And if that parallel component is greater then the friction force holding it back the object will move. So it makes sense that on e average the coefficient of friction on an incline is less then that for an object on a flat surface, for on an incline the object has a fraction of its weight pushing it along as well, making friction easier to overcome. Objects on flat level surfaces on the other hand, like in Part A, have nothing initially to help them to move, and so the friction they have to overcome will be greater. The bigger the coefficient of friction, the greater the resulting friction, and vica versa.

GRAPHS: We then graphed the changing weight of the block with the amount of kinetic/static friction created when we tried to pull it. There was a direct relationship between them; as the weight of the block was increased, so was the friction. This makes sense for the greater the pressure of one surface on another, the greater the forces by and between them are, so the more friction. In addition, it makes sense that our static friction data is bigger then our kinetic friction, because the block already has motion behind it when dealing with kinetic friciton. For static, the block is completely motionless; it has to start from rest. __To Find Tension or Force of Friction With DataStudio:__ We used DataStudio and a PASPORT sensor to find out what our friction force equals. Because acceleration was zero, the tension of the string on the block was equal to the friction against the block. We looked at the highest point on the graph to see what static friction was, for once the tension we exerted overcame that static friction, the block started to move. We then tried to find the straightest part of the graph and find the mean of that section for that would give us our kinetic friction or the force we had to overcome continually in order to keep the block moving. We then used these friction forces to find the coefficient of friction.
 * __Part A:__**

500 g mass + block:

1000 g + block: 1500 g + block: 2000 g + block:


 * __Part B:__**



Error Calculation Compared to Actual Coefficient of Friction Values between Aluminum and Wood: (Abs(|Theoretical- Actual)/ Theoretical)X 100 =Percent error

Static Friction: (abs(.2200-.2579)/ .2200 )X 100 = 17.22% error

Kinetic Friction:

Percent Difference Compared to Class Average:



Percent Difference between calculated u and the slope of our graph:

Static Friction: abs(.2616-.2579)/(.2616+.2579)/2 X 100 =1.5764% difference

Kinetic Friction: abs(.2115-.2103)/(.2115+.2103)/2 X100 =.5690 % difference



Discussion Questions:

1. Sum F=ma F-T=ma At constant speed so a=0 F=T F=uN T=uN

U=T/N N=mg U=T/mg

Kinetic Friction: Tension (N)/ Weight(mg) = u According to our graph we got a the coefficient of Kinetic friction as .2103

Static Friction Tension (N)/ Weight (mg)= u

According to our graph we got a the coefficient of Static friction as .2597 Static Friction is greater than Kinetic Fiction because it takes more force to make the block start moving; it takes less force to keep the block in motion (kinetic friction).

2. The coefficient of friction for static friction between aluminum and wood is .22 according to [].As seen in our experiment this value is close to the value we obtained and as seen in the calculations above it matches our theoretical.


 * 3** . The weight of the wood and the tension in the rope pulling the wood would affect the magnitude of the force of friction. The greater tension in the rope, the greater the force of friction would de as we proved in our lab. Also a greater weight would cause a greater force of friction on the system. The variable on coefficient of friction would be the angle because then he weight is broken down into it x and y components (see calculations).

4. The value of kinetic friction was always less than the static friction. This is because it takes more force to get an object to begin moving than it does to keep an object in motion. Static friction measures the maximum friction a force has to overcome to move so it makes sense that it is greater than the kinetic friction.


 * 5.** Putting it on an incline decreased the coefficient of friction because the weight was broken up into its x and y components. Also it requires less force to move an object on an incline because gravity is acting more strongly on it.

Error Analysis/Conclusion (Refer to calculations above): The error in our lab most likely occurred during our data collection. For the first part when we had to drag the wood across the track, if we zeroed the force meter when there was tension in the string this would cause us to get a lower reading for the tension because the meter would have zeroed the tension that was already existing. This alone would not have impacted our results greatly because we ran several trials to get the most accurate readings. Another source of error could be from the track. If there were any dust particles on the track it would change our results. This makes sense because our reading was higher than the predicted amount of friction because there was friction between the aluminum and the dust on the wood block. Also if the string was not level or if the block was not pulled at constant speed our results would have been affected. For the second part of the lab the error resulted from our imprecise protractor. The measurements we got from the protractor were not exact because we took the angle of the flat part so that might not give an accurate reading for the angle. In this lab we were trying to determine the static and kinetic coefficients of friction for aluminum and wood. We also tested the effects of an incline on the coefficient of friction. In our lab we used Newton’s Second Law to establish that the tension acting on the system is equal to the friction acting on that system (see derivation above). We proved that tension does equal friction. On our graphs the slope of the line is equal to the coefficient of friction. For kinetic friction we got a value of .2103 and for static friction we got a value of .2597. Our class average was .2170(static) and .1876 (kinetic). We also found online that the coefficient of friction is .22 so our results were fairly accurate. We also found that by putting the ramp on an incline the coefficient of friction decreases because the weight is broken down into its x and y components. The results from our lab report our hypothesis allowing us to conclude that the tension is equal to the force of friction of a system.


 * Atwood's Machine Lab:**
 * Due Date: 12/7/10**


 * Andrew Miller, Elena Solis, Emily Van Malden**

Purpose: To investigate the relationship between net force and acceleration

Hypothesis: Net force and acceleration have a direct relationship. If the acceleration increases then the net force will also increase.(Based on Newton's Second law, see conclusion).

Materials: Atwood’s machine, clamp, ring stand, rod, 2 hanging masses, masses, string, tape, photogate, Excel, Data Studio

Procedure:
 * 1) Set up ring stand, screw in until knob is tight around metal rod.
 * 2) Place clamp onto top, and then clamp pulley to the top of the metal rod.
 * 3) Attach pulley to clamp.
 * 1) Connect Pulley to USB port of the photogate.
 * 2) Attach hanging weights to each end of the string and thread string through the pulley.
 * 3) Tape (mass of tape is negligible) masses on each hanging weight so that both of the masses are .100 kg and the total mass (assuming the mass of the string and pulley are zero and including the mass of the weight of the hanger which is 5 g)) is .200 kg.
 * 4) Perform one trial with the same mass of both sides.
 * 5) Then we take masses (in 5 gram intervals) off weight A and put them on mass B so that the total mass is constant at .200 kg for the entire experiment. For our calculations mass A will always be lighter than mass B.
 * 6) To make sure we get an accurate acceleration reading we will put mass B (the heavier mass) higher so that it has more time until it reaches the ground.
 * 7) Start Data Studio when the person holding the weights lets go.
 * 8) Make sure to analyze the part of the graph before or after the two weights hit each other.
 * 9) Repeat 6 times and record observations.
 * 10) Make a graph of results using Excel and Data Studio.
 * 11) Compare last week’s result to this week’s.

Observation Table:


 * Trial || Mass A || Mass B || Observation ||
 * 1 || .100 kg || .100 kg || No movement ||
 * 2 || .095 kg || .105 kg || Mass B moved downward which and Mass A moved upwards. The trial was over when mass B hit the ground. ||
 * 3 || .090 kg || .110 kg || The same results as trial 2 occurred only they happened faster than the pervious trials. ||
 * 4 || .085 kg || .115 kg || The same results as trials 2,3 occurred only they happened faster than the pervious trials. ||
 * 5 || .080 kg || .120 kg || The same results as trials 2,3,4 occurred only they happened faster than the pervious trials. ||
 * 6 || .075 kg || .125 kg || The same results as trials 2,3,4,5 occurred only they happened faster than the pervious trials. ||
 * The total mass is always constant at .200 kg

Data:



We did multiple trial runs to calculate the velocity of the pulley (the velocity is the same for both the mass going upwards and downwards--one is just negative depending on how you draw your coordinate plane). Then, we took the slope of the velocity graph in order to get the acceleration (which is also the same-- it is just positive or negative depending on the coordinate plane). We then used the acceleration as well as the mass (0.2) in order to find the actual force.


 * Net Force vs. Acceleration Table ||  ||   ||   ||   ||   ||
 * Trial # || Mass A (kg) || Mass B (kg) || Total mass (kg) || Theoretical Acceleration (m/s^2) || Actual Aceleration (m/s^2) || Total Theoretical Force (N) || Total Actual Force (N) ||
 * 1 || 0.1 || 0.1 || 0.2 || 0 || 0 || 0 || 0 ||
 * 2 || 0.095 || 0.105 || 0.2 || 0.4905 || 0.398 || 0.0981 || 0.0796 ||
 * 3 || 0.09 || 0.11 || 0.2 || 0.981 || 0.849 || 0.1962 || 0.1698 ||
 * 4 || 0.085 || 0.115 || 0.2 || 1.4715 || 1.34 || 0.2943 || 0.268 ||
 * 5 || 0.08 || 0.12 || 0.2 || 1.962 || 1.79 || 0.3924 || 0.358 ||
 * 6 || 0.075 || 0.125 || 0.2 || 2.4525 || 2.26 || 0.4905 || 0.452 ||
 * 6 || 0.075 || 0.125 || 0.2 || 2.4525 || 2.26 || 0.4905 || 0.452 ||



The slope of the line is equal to force divided by acceleration, which is the mass in kg. 0.2 kg is exactly what the total mass of the system is, proving the accuracy of our experiments. The y-intercept value is the value of the friction on the rope. Due to the fact that it is incredibly close to 0, we can assume that it didn't play a huge role in our experiment because the rope was basically frictionless.







Error Calculations:

//(abs( Actual-Theoretical)/Theoretical) X 100//

(abs(.389-.4905)/.4905)X 100 //=20.69 % error//
 * =** Percent Error

Average Error for this experiment= 9.95% error
 * Trial || Percent Error ||
 * 1 || 0% ||
 * 2 || 20.69 % ||
 * 3 || 13.46% ||
 * 4 || 8.94% ||
 * 5 || 8.77% ||
 * 6 || 7.85% ||


 * DISCUSSION QUESTIONS**

1. see graphs for explanations.

2. What would friction do to your acceleration? Would you need a bigger or smaller force to create the same acceleration? Was your slope too big or too small? Can friction be a source of error in this experiment? Redo the calculation of acceleration WITH friction to show its effect. If enough friction was present in our experiment, it would slow down how fast the string was moved over the pulley and thus slow down how quickly the falling heavier mass could accelerate downward. In order to create the same acceleration with friction as we would have with a system without friction, we would need to use a heavier force that would make the net force (W2 – f) equal to the force of the falling mass without friction (which would just be W1). In our graph, slope is the total mass of our equation. The Mass of an object stays constants despite any outside influences. So the prescence of friction, would not affect the total mass in a system, it would only affect the acceleration and force (our x and y value). So our slope, our coefficient of x, is accurate. Friction can be a source of error in this experiment if one mixes their experimental data with their theoretical data. Our experimental data consists of the total mass (which is not affected by friction) and the acceleration taken from the pulley machine (which was collected under real-life conditions and naturally takes friction into account finding the mass’s velocity and acceleration). However our theoretical acceleration was collected using the equation: g(Mb-Ma)/(Ma+Mb) or basically F/m. It does not take friction in account. To see how accurate our theoretical value was to real life conditions we can do the following calculations, plugging in the value of friction from our trendline or (-8 x 10**-17** ):

Theoretical Acceleration: F**ideal** = (Ma+Mb) a **ideal** g(Mb-Ma) = (Ma+Mb) a**ideal** 9.81(.105-.095) = (.2) a**ideal** .4905 m/s/s = a**ideal** F**ideal** - F **fric** = (m1 + m2) a **experimental**

g(Mb-Ma) - F**fric** = (Ma+Mb) a **experimental** 9.81(.105-.095) – (-8 x 10**-17** ) = (.2) a **experimental** .09810000000000008 = (.2) a**experimental**

.4905 m/s/s = a**experimental**

So the effects, of friction while present, would be so minimal that they should not affect our theoretical acceleration that much.

3. Discuss the precision of your data.

In this experiment, our experimental data was pretty precise. The mass was easy to collect for we used premarked weights that allowed us to record the mass easily. The weight of the pulley was negligible, for as light as it was already, only half of it would have been calculated into our experimental equations. And the pulley system we used was designed to record real-life situations. So even though the presence of friction was minimal x 10 to the negative million, the effects of friction were recorded anyway and used in our collections. When graphing our data, we had a linear graph with a R2 value of one. This makes sense because Newton’s Second law is a linear equation and as one variable) acceleration moves, the function should change consistently.

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?

The mass that would be added to our equations ((1/2) *.0056 kg or .0028) would slightly edit our acceleration. Because it would be adding to the total mass of the system, it would be put at the bottom of Atwood’s machine’s equation a = (Mb – Ma)g/(Ma + Ma + **(1/2)Mpulley** ). Because the same value would be divided by a bigger denominator, the outcome should be a lower value. Due to the lack of effect accounting for friction had on our acceleration calculations, leaving out the mass of the pulley most likely accounted for most of our error and for why our theoretical value was bigger then our experimental one.

Error Analysis/Conclusion:

As seen in the calculations above, we had a little less than 10 percent error. The error occurred because of our calculations and the actual performance of our lab. In the equations we used (see above) we did not account for friction. Though there would not have been enough friction to drastically change our results, the friction did contribute to our error. Our theoretical accelerations were slower than our actual accelerations because we did not account for friction. If we accounted for friction then we would have had slower actual accelerations, which would be closer to our theoretical values. Also, we did not include the mass of the pulley, which would have changed our results. This week’s gizmo showed us that an increase in the mass of the pulley would decrease velocity and therefore acceleration. Therefore, if we included the mass of the pulley our actual accelerations would be slower and closer to our theoretical accelerations. Some of our error may have also been from when we released the hanging masses. The moved in a circular pattern and not straight down which would influence our results. Also the weights collided so if we weren’t careful in analyzing our graphs we may have included the part of the graph where they collided which would change our results; although we were pretty accurate with our graphs so the error would have been minimally affected by this. The majority of our error came from not accounting for friction and the mass of the pulley and the error might have been minimally influenced by the error during the trials.

In this experiment we were investigating the relationship between net force and acceleration. We hypothesized that if the acceleration increased the net force of the system would also increase because they are directly related. We based tour hypothesis on Newton’s Second, Force= mass(acceleration). Since force and acceleration are on opposite sides of the equation we can conclude that they are directly related. Our results prove our hypothesis correct and verify Newton’s equation. As seen in the tables above as we increased the mass of weight B, the force of the system also increased. Because both the acceleration and the force were increasing we can conclude that the acceleration and the net force are directly related.

//**Due Date: 12/01/10**//
 * Newton's Second Law Lab**

// **Andrew Miller, Elena Solis, and Emily Van Malden** //

//Purpose: To find and confirm the relationships between mass, acceleration, and net force.//

//Hypothesis: We are proving that the relationships between net force, and mass/acceleration are direct relationships and will have a linear graph, while the relationship between acceleration and mass will be indirect, and have a power curve.//

//Materials://

//Dynamics Cart with Mass// //Dynamics Cart// //Track// //Photogate timer// //Data studio// //Microsoft Excel// //Photogate Port// //Super Pulley with Clamp// //Base and Support rod// //String, Mass hanger and mass set// //Wooden or metal stopping block// //Mass balance// //Level//

//Procedure to Test the Relationship between Force and Acceleration://

//Procedure to Test the Relationship between Mass and Acceleration://
 * 1) Clamp track to table.
 * 2) Attach pulley wire photogate port.
 * 3) Attach photogate port to computer.
 * 4) Thread pulley string through the cart (505g) carefully and then through the actual pulley wheel.
 * 5) Thread the same string through the hanging mass (5g). The hanging mass can now pull down the cart, accelerating it.
 * 6) Place 25g of mass units onto the cart—the total mass is 535 g.
 * 7) Measure acceleration using data studio.
 * 8) Do not add any mass units—keep the mass constant, and place a 5g mass unit onto hanging mass.
 * 9) Measure acceleration by taking the slope of the velocity graph on data studio using a line of best fit.
 * 10) Keep adding mass to the hanging mass, and taking away the SAME mass from the cart in order to measure acceleration with different hanging masses and cart masses, while keeping the total mass the same.
 * 11) Start the cart from the same measurement from the pulley (for example, we placed the front of the cart at 70 cm from the pulley each time).
 * 12) Record results in table form in Microsoft Excel and graph. Create line of best fit.

//Procedure to Test the Relationship between Force and Mass://
 * 1) Clamp track to table.
 * 2) Attach pulley wire photogate port.
 * 3) Attach photogate port to computer.
 * 4) Thread pulley string through the cart (505g) carefully and then through the actual pulley wheel.
 * 5) Thread the same string through the hanging mass (5g). The hanging mass can now pull down the cart, accelerating it.
 * 6) Place 2 g on the hanging mass, which is enough to accelerate the cart.
 * 7) From here, add mass units to the cart at constant intervals of 50g (We have made sure that the hanging mass can pull the cart’s mass).
 * 8) Find acceleration using data studio.
 * 9) As before, take the slope of the velocity graph using a best-fit line on data studio to find the most accurate acceleration.
 * 10) Calculate percent difference between your average force and your highest and lowest force to see how constant your force really was.


 * 1) Clamp track to table.
 * 2) Attach pulley wire photogate port.
 * 3) Attach photogate port to computer.
 * 4) Thread pulley string through the cart (505g) carefully and then through the actual pulley wheel.
 * 5) Thread the same string through the hanging mass (5g). The hanging mass can now pull down the cart, accelerating it.
 * 6) In order to keep acceleration the same, establish a ratio of the mass of the cart to the mass of the hanging mass. Our ratio was 8:1 (the cart was eight times the mass of the hanging mass).
 * 7) Add mass units to keep the ratio true for different trials.
 * 8) Use data studio to figure out what the constant acceleration is and to verify that it is constant.
 * 9) Calculate the force using the changing masses and the constant acceleration.

//**DATA:**// // Newton’s second law states that F = m * a. In our experiment, we has three different experiments, in which we kept each variable constant. We created table and graphs for each experiment and saw the relationship between each variable. //

// Our excel tables were very helpful in keeping track of each of our variables. // // The mass of the hanging object and the cart were easy to find using a mass balance and objects with known masses. We added them together to find the total mass. // // To find a purely theoretical estimate of what a certain variable should be, we used the equation: // // a = (g * mass of hanging object)/(total mass) // // and plugged in whatever constants we knew for each trial to find the one missing one. // // The force was found using Newton’s second law F = m * a. // // The data studio graphs helped us check that all of our trials were similar in nature (same slope/acceleration/shape) but were especially helpful when it came to calculating the actual acceleration of each trial. Using the velocity graph, we found a linear line of best fit and looked at the slope of that line. The rate of change of velocity is acceleration so by looking at the slope of velocity we found our actual acceleration. // // Finally, the excel graphs were very helpful to see if we were able to find overall accurate relationships between each of the variables. If the shape, slope, y-intercept and trendline of the graph matched our data and theories, then we knew we were on track. //

//**ACCELERATION VS FORCE:**// // Acceleration vs Force //

// The slope of the trendline (the change in Force over the change in the acceleration) data is .535 which is the same as our total mass (mass of the hanging weight and the cart added together). This is the exact same value as the mass we kept constant in our tables. And so we have a 0% error. //

// If we consider Newton’s Second Law, we can see why the slope of our trendline is the same as our constant total mass. // // F = m * a // // F = .535 Kg * a // // Since Force is our y –value and acceleration our x-value, we can replace “F” with y and “a” with x. So we get: // // y = .535 x // // In our trendline, the term -7E-17 refers to friction’s impact on the cart’s force. It is so minimal, close to zero, that for the purposes of this experiment, its effects can be ignored. //

// The y-intercept value is the y value when x is zero or in this case, the Force when acceleration is zero. If the acceleration of the system is zero, then the force being exerted on the cart is almost nonexistent. //

//**MASS VS ACCELERATION:**// // Mass vs Acceleration // // The slope of this trendline (the rate of change of acceleration over the rate of change of mass) is .0375, which is not obviously similar to any of either our mass, acceleration, or our force, which should be constant. If we use Newton’s second law, we will be able to see which idea the coefficient was referring to. Our force was kept pretty constant so we can replace that in the equation. // // F = m * a // // .0463 = m * a // // Since mass is the x-value and acceleration is the y-value, we can replace them in the equation and solve for y. // // .0463 = x * y // // x * y = .0463 // // y = .0463 (1/x) // // y = .0463 (x**-**1) // // From this equation we can see that the coefficient in the trendline should match the constant force we calculated in our table, while the power x is raised to should be negative 1, not the -1.392 our trendline has. //

// The constant force we calculated in our table and the coefficient in the trendline do not match. In our table we had an (actual) constant Force of .0463 Newtons while the (theoretical) coefficient of the trendline is .0375. Our percent error is shown below, as well as the explanation for why the trendline and our actual data is so off (the latter is explained in the conclusion). //

//**MASS VS FORCE:**//

//Here, in order to keep the acceleration constant, we created a ratio of the mass of the cart to the hanging mass. This ratio was 8:1 respectively; therefore, we added weights to both the hanging mass and the cart in order to keep the ratio the same in each trial, thereby keeping the acceleration the same, as shown by our data.//

// Mass vs Force // // The slope of our trendline (the rate of change of mass over the rate of change of force) is 1.01, which is the same as our calculated acceleration, which was kept constant. In fact, the slope or coefficient of the trendline is the exact same value as the acceleration we found in our the data studio graph, so our percent error would be 0%. //

// If we take another look at Newton’s second law, we may see why that coefficient is the same as our acceleration. // // F = m * a // // F = 1.01 (m/s^2) * a // // Since mass is the x-value and Force, the y, we can replace them with their respective variables and see why they match. // // y = 1.01 * x // // The slope intercept value would be the y – value when x is negative, or rather the Force when mass is zero. In our experiment, a mass of zero would mean there is neither a cart nor hanging mass to work with, so it would make sense that the resulting force/weight is zero. //

//Sample Calculations://

//Acceleration vs. Force://

//F=ma//

//F-force causing acceleration// //m-mass being accelerated// //a- acceleration//

//F=.535 kg(a)// //F=.535 kg ( .0593 m/s2)// //F= .032 N//

//We are solving for force. The mass is constant at .535 kg and then to solve for force we would plug in a value from our table for acceleration. By multiplying the mass by the acceleration we will find the force.//

//Mass vs. Force://

//F=ma// //F=.5681 kg(1.01 m/s2)// //F=.574 N//

//This time we are changing the mass in constant ratio. The mass on the pulley is 1/8 the mass on the cart. This makes the acceleration constant at about 1.01 m/s2 by plugging in masses and the acceleration we solve for force.//

//Mass. vs. Acceleration://

//F=ma// //F=.612 kg (.0724)// //F=.0443 N//

//We kept the force the same for all the trials. We plugged in mass and acceleration to solve for force. As seen in the charts, the force is the same (or very close to the same).//

//Theoretical Acceleration Sample Calculation:// //a = (m2g)/(m1 + m2), where m1 = mass of cart and m2 = hanging mass and g = 9.81 m/s^2// //a = (0.01)(9.81)/(0.525+0.01)// //a = .1834 m/s^2//



Calculations for Percent Error:





Analysis Guide:

2.Any friction present in the lab would have caused the acceleration to decrease, which would have caused the force to decrease. Thus, it would have required a larger mass in order to get the acceleration and force that we received from our "frictionless" trials.


 * Conclusion:**

The purpose of our lab was to qualitatively prove Newton’s Second Law true. His second law says that F=ma, an equation that shows that the force of an object is directly proportional to both the mass of the object and the object's acceleration. This equation also says the acceleration is inversely proportional to its mass. In our lab our data proves the aforementioned hypotheses, for as we increased the total mass, the acceleration decreased; as we increased the mass, the force increased; as we increased the acceleration, the force increased. Although the computer-calculated acceleration did not match the theoretical acceleration calculation, this is not important. Our data shows a trend that supports the preceding relationships, even though the accelerations do not match. Our percent error would be large if we compared our two different accelerations; however, that didn't affect our main purpose of the lab, which was to qualitatively verify the equation using calculations. If the calculations show the trend as they do, then they are perfect to satisfy our purpose. As seen in the tables above, our calculations do prove the equation true because the relationships were what we expected.

For two of our graphs, we received an error of 0%, showing the accuracy of our trials. For the other graph (mass vs. acceleration), our error was rather large because it was extremely hard to keep the force precisely constant. Even though the difference between our average and highest and lowest was relatively small (.002), it is clear that this difference actually has a large effect on the slope and the exponent. In the future, we will do more trials to receive a more accurate average force, and be more precise in keeping the ratio between the cart mass and the hanging mass the same in each trial. The hanging mass was 1/8 of the cart mass. When we calculated what the hanging mass should be using the cart mass in order to keep the ratio constant, we received decimal answers; however, we could only add whole number weights to the hanging mass. Because of this, we could not ensure that the force stayed completely constant. In the future, we would use more precise weights to account for this.

Friction did not factor into this lab, for the track that we used for the cart was close enough to being frictionless that the friction is negligible. This shows why our y-intercept in our acceleration vs. force graph is so close to 0.


 * Inertia Lab 11/23/10:**
 * Andrew Miller, Elena Solis, and Emily Van Malden**
 * Period 2**

Purpose: To find the mass of an object only using its inertia.

Hypothesis: We are investigating the relationship between mass and the period of vibration, which we believe to be direct. We are tying to figure out the mass of an unknown object using its inertia, a number that we will calculate by using objects with known masses and calculating their periods. We will then graph them, and plot the period of the unknown object as a function of its mass, and we will see where the two lines intersect, and this will be the mass.

Materials:

3 Timers Weights of Varying Mass (5g, 10g, 50g, 100g, 200g, 300g, 500g) Inertial Balance Paper Towel Clamp Microsoft Excel

Procedure to Find Period Using Objects with Known Masses:


 * 1) Clamp the inertial balance to the table.
 * 2) Put a paper towel onto balance in order to prevent unnecessary movement of the weight.
 * 3) Put a weight with a known mass onto the inertial balance.
 * 4) Pull the balance towards you and release (do not pull hard enough that the weight moves—the weight must remain still.
 * 5) Start time: when the balance is released.
 * 6) Count 10 periods (one period = movement forward, and then backwards, so displacement for one period = 0).
 * 7) Take average of the three times for one trial in order to account for error, then repeat twice.
 * 8) Average all three trials, then divide this by 10 to get the time for one period for a certain mass.
 * 9) Create graph of results on Microsoft excel.

Procedure to Find Mass of Unknown Rubix Cube:


 * 1) Using clamped inertial balance with paper towel on it, place rubix cube onto balance.
 * 2) Pull balance towards you and release (do not pull hard enough that the cube moves—it must remain still).
 * 3) Start time: when the balance is released
 * 4) Count 10 periods (one period = movement forward, and then backwards, so displacement for one period = 0)
 * 5) Take the average of the three times that you receive for one trial.
 * 6) Repeat twice more, then average the three trials.
 * 7) Divide by 10 to get the time for one period.
 * 8) Graph this as a horizontal line to see where it intersects with the best-fit line of your other graph (verify your results with a graphing calculator).
 * 9) Verify your results algebraically.

Below is a bird's-eye view of our setup:




 * Data:**

Table to Find Period Length with Known Masses and Period Data for Unknown Mass:


 * Mass (g) || Trial 1 (s) || Trial 2 (s) || Trial 3 (s) || 10 period Average (s) || 1 Period Average (s) || 1 Period Average for 9 total Periods (s) ||
 * 5 || 3.14 || 3.12 || 3.39 || 3.217 || 0.3217 || 0.3574 ||
 * 10 || 3.49 || 3.71 || 3.37 || 3.523 || 0.3523 || 0.3915 ||
 * 50 || 4.16 || 3.97 || 3.93 || 4.020 || 0.4020 || 0.4467 ||
 * 100 || 4.49 || 4.54 || 4.69 || 4.573 || 0.4573 || 0.5081 ||
 * 200 || 5.83 || 5.73 || 5.78 || 5.780 || 0.5780 || 0.6422 ||
 * 300 || 7.15 || 7.09 || 7.11 || 7.117 || 0.7117 || 0.7907 ||
 * 500 || 9.55 || 9.51 || 9.52 || 9.527 || 0.9527 || 1.0585 ||
 * Rubix || 4.65 || 4.63 || 4.64 || 4.640 || 0.4640 || 0.5156 ||


 * Graphs:**



This is our graph if we only timed 9 periods, not 10 for our known masses.



We calculated the point of intersection both graphically and analytically dividing by 9 periods instead of 10 (in case we counted incorrectly and did not include the first period in our trials); however, the result was that the mass of the rubix cube was 68.21 g, a number that we found using the same analytical calculations and graphical analysis as above. Therefore, we concluded that we did in fact count correctly.



As seen in the table, we performed three trials for each mass and averaged them together to find the average time in seconds it took the mass to complete 10 periods. Then we divided that average time by 10, in order the find the average time for just one period. Here is an example for the 5 gram Mass: Based on our graph of mass vs. vibrations, we were able to come up with an equation of best fit for our data. .0012x + .3317 = y
 * CALCULATIONS:[[file:Calculations Used to Find the Length of a Mass’s Period.doc]]**
 * Calculations Used to Find the Length of a Mass’s Period:**
 * Calculations Used to Find that Mass of the Unknown Object:**

where x is the mass in grams and y is the time in seconds of one vibration or period.

We found the Rubix Cube’s average period length (using the method shown above) to be .464 seconds. We plugged it into the equation for our data as y and solved for the cube’s mass in grams.

In order to check to see if our calculations were accurate, my group turned our data into two equations. The data of the known masses was .0012x + .3317 = y, while the unknown Rubix cube became a straight horizontal line, y = .464. Similar to the graph above, we plugged these equations into our calculator and solved for their intersection. The algebra would be the save as above, we just let the calculator above.
 * Calculations Used to Verify that Mass of the Unknown Object:**


 * Error Calculations:**





**Discussion Questions:**
 * 1) Gravity plays a part in the weight of a object because weight is mass multiplied by gravity so gravity is a part of the objects’ weight. However, gravity does not play a part of the mass of the objects because gravity is at a constant 9.8 m/s2. The weight is unrelated in this experiment because gravity has the same effect on all the objects so the object’s weight does not affect the relationship between the time for one period and the mass of the object.
 * 2) An increase in mass lengthened the period of motion because period of motion and the mass are directly related. If the mass is increased it will move slower because if the mass increases that means the inertia also increases, making the period of motion longer.

3. When the object is smaller the period is faster because the object has less inertia so it takes less force to move that object. Therefore the object can reach a faster speed more quickly because it has a smaller mass (and a faster acceleration). The heavier the object, the bigger the inertia and the more force must be used to get that object to move. Therefore, if we place the same amount of force on the bigger mass, as we did on the smaller one, it wouldn’t move as fast and will have a much slower acceleration.

4.If the side arms were stiffer the period of motion would be shorter. The period of motion would be short because the arms would not be able to move as far so the time each period takes will be shorter.

5. The inertial mass and gravitational mass of an object are the same. They are the same because gravity affects both inertia and the mass. We know this because of Newton’s second law which says the force of an object= mass(acceleration). Therefore the inertial mass and gravitational mass are the same because gravity is the same on Earth (9.8 m/s2) so they two are the same.

6. We use the gravitation instead of the inertia as a means of measuring the mass of an object because it’s easier and more accurate. Gravity is constant so its more simple to calculate the weight by using a scale. Gravitational mass is easier to use because it is weight, so its common knowledge.

7.The results of this experiment would not change if we did this experiment on the moon. The results would be the same because the mass of the object would not change even though the gravity on the moon is one sixth of Earth’s gravity. The weight of the object would change on the moon because of the difference in gravity but the mass of each object would be the same. So all the objects would be equally and proportionally affected by the moon’s gravity so the results of our experiment should be the same.


 * Conclusion:**

As seen in the calculations above we had 8.72 percent error. The error from our lab was a result of discrepancies between he equipment and human error. The flexibility of the arms of the inertial balance affects the length of the object’s period of motion. If the arms are more flexible the vibrations will be greater so the time each period takes will be longer. Since we found the mass of the Rubix Cube to be 110.25g and the actual mass was 101.41gwe can conclude that some of our error resulted from having more flexible arms of our inertial balance. However, the flexibility of the inertial balance would not have created almost 10 percent of error. Therefore, we know that human error played a part in our error. The human error most likely occurred during the timing of the period. Even though we had three timers going for each trial and then averaged them for each trial and then averaged the three trials so find the time of the period of each mass we tried, we were not perfect with our timing. As humans, there is no way that we could have started and stopped the stopwatch at the exact moment it started and stopped. We thought the error may have occurred from the counting. We though that we may have started counting at one instead of zero. But we divided the times for our known masses by 9 instead of the original 10 to account for that but the error was even worse (see graph above). Our rubix cube time was perfect, for we attained the same results each time; therefore, we concluded that the period length of the rubix cube was correct, and did not divide that by 9. Also, our time for the rubix cube period was within milliseconds away from other groups, which affirmed our thoughts that our rubix cube time was correct. Therefore, we know that the error form our lab was a combination of human error and the flexibility of our inertial balance. In this we proved our hypothesis correct. We hypothesized that the relationship between the mass of the object and the period of the motion is direct. We now know that this is true because when the mass of the object increased, so did the time of the period of motion. The object that was 500 grams had the longest period of motion and the lightest object, which was 2 grams had the shortest period of motion. Therefore, proving our hypothesis correct. We then used this information that we graphed to determine the weight of the unknown mass of the object. The actual mass was 101.41 g and we found it to be 110.25 grams, though we had a fairly substantial percent error, we know that it was due to human error and our balance’s flexibility and not an error in our procedure.