Group5_2_ch4

Maddy Weinfeld, Amanda Fava, Ryan Hall, and Jake Aronson

=Gravity and the Laws of Motion= toc Part A: Ryan Hall Part B: Shared Part C: Maddy Weinfeld Part D: Amanda Fava NOTE: Jake Aronson was absent and made up the lab with Period 4 (see http://honorsphysicsrocks.wikispaces.com/Group6_4_ch4)

//Objectives//:
 * 1) Find the value of acceleration due to gravity.
 * 2) Determine the relationship between acceleration and incline angle.
 * 3) Use a graph to extrapolate extreme caes that cannot be measured directly in the lab.
 * 4) Determine if mass has an effect on acceleration because in free fall, the mass does not have an effect.

//Hypothesis//:
 * 1) The value of acceleration due to gravity is 9.8 m/s/s because previously determined in free-fall lab.
 * 2) The acceleration will increase as the angle increases because object is farther from horizontal and closer to free-fall.
 * 3) Mass will not effect acceleration.

//Procedure//: //Calculations//:
 * Setting up angle:
 * media type="file" key="Movie on 2011-11-16 at 08.42.mov" width="300" height="300"
 * Dropping ball down ramp:
 * media type="file" key="Movie on 2011-11-16 at 08.43.mov" width="300" height="300"
 * [[image:Screen_shot_2011-11-16_at_9.10.06_AM.png]]
 * [[image:Screen_shot_2011-11-16_at_9.10.13_AM.png]]

//Data//:
 * [[image:Gravity_Lab_Data.png width="880" height="270"]]
 * [[image:Gravity_Lab_Graph.png width="560" height="396"]]

//Analysis//: //Discussion Questions://
 * Analysis of graph set up:
 * The x axis of our graph represents sin(theta) and it goes from 0-1 because that is where the sin function lies. The y axis represents acceleration and goes from 0-10 because 10 is the maximum acceleration due to gravity.
 * The equation of our line is y = 8.2726x. The slope should be 9.8.
 * Acceleration due to gravity is 9.8 m/s2 (derived from equation below)
 * [[image:Screen_shot_2011-11-16_at_9.58.33_PM.png]]
 * Our experimental acceleration due to gravity was 8.27 m/s/s
 * Percent Error Calculations:
 * [[image:Screen_shot_2011-11-16_at_10.57.44_PM.png]]
 * [[image:Screen_shot_2011-11-17_at_8.11.24_PM.png]]
 * Class Data
 * [[image:Screen_shot_2011-11-17_at_12.08.59_PM.png]]
 * There is no clear trend between mass and acceleration and therefore, based off our class data, we can conclude that mass does not affect acceleration due to gravity.
 * [[image:Screen_shot_2011-11-17_at_9.01.33_AM.png]]
 * [[image:Screen_shot_2011-11-16_at_11.02.01_PM.png]]
 * What force is causing the ball to roll down the ramp? Is it the whole force or just part of it? If just a part, then which part is it?
 * It is the x component of the weight force. This is because the y component is balanced by the normal force and therefore just the x component of weight is causing the ball to roll down the ramp.
 * Use Newton's second law to calculate acceleration of the ball down one of your ramps. How does it compare to your calculated (average) acceleration for that incline?
 * Newton's second law gives us the equation, f=ma. When using this equation, and comparing the results to our experiment, we found that ti was very close, but not exactly the same. For the trial at 0.150 m, we found that acceleration = 0.830 m/s/s and when we used Newton's law we found that acceleration = **CALCULATE THIS.**
 * 1) Is the velocity for each ramp angle constant? How do you know?
 * No because it is accelerating as it goes down and therefore the velocity is increasing as it travels.
 * 1) Is the acceleration for each ramp angle constant? How do you know?
 * Acceleration is not constant for each of the different angles. This is because as the angle gets bigger, it gets closer to free fall, and therefore the acceleration is increasing.
 * 1) How was it possible for Galileo to determine //g,//the rate of acceleration due to gravity for a freely falling object by rolling balls down an inclined plane?
 * Galileo could have determined g by running an experiment similar to ours. If he rolled a ball down an incline plane at several different angles, he would have enough data to make a graph and find an equation of the line. Using this graph, he could extrapolate larger angles and ultimately discover that the rate of acceleration never exceeded 9.8 He also could have plugged 90 degrees into the equation of the line because this is basically a free fall.
 * 1) Dose the mass of an object affect its rate of acceleration down the ramp? Should it affect the motion of an object in free-fall in the same manner?
 * No, the mass does not affect it. This is pretty evident from our class's data because there isn't really a trend between the different masses and their g values. We also know this is true because in the equation we use to solve for g, the mass cancels itself out on both sides and is therefore irrelevant in the equation. Mass should affect the motion of an object in free fall in the same manner and we know from prior experiments that the mass does not affect the acceleration of free falling objects.

//Conclusion//:
 * After conducting the trails for multiple heights and completing the calculations, we found our hypothesis to be partially correct. The slope of our Acceleration vs. Sin(theta) graph represents the acceleration due to gravity as shown mathematically in our analysis. The theoretical acceleration due to gravity is 9.8 m/s/s, as determined during our previous free-fall lab; however, our experimental acceleration due to gravity was 8.27 m/s/s. Therefore, this acceleration is relatively off. Our graph allowed for a y-intercept (instead of setting y-intercept equal to zero), which caused our slope to be closer to 9.8 m/s/s because it allowed for most of the friction to be taken into account. The rest of the difference can be accounted for by the other sources of error, which were a good amount given our 15.61% error. Since the ball was rolling, friction is usually considered negligible, but given how small the units of measure were in this lab, the friction may have been substantial. The other forces on the ball included weight and normal which also contributed to the discrepancy in acceleration, though air resistance is negligible. Other sources of error included human reaction time with the stopwatches and lack of accuracy in height and distance measurements. The next part of our hypothesis was shown to be correct, as the acceleration and incline angle are directly related - as the angle increases, the acceleration also increases. Our smallest height of around 0.10 meters at the start of the ramp, which would subsequently have the smallest angle (theta value), also had the smallest average acceleration of 0.679 m/s/s. Our largest height of around 0.30 meters at the start of the ramp, which would subsequently have the largest angle (theta value), also had the highest average acceleration of 2.219 m/s/s. As the angle increases, it is becoming closer to 90 degrees or a straight vertical incline, which is the same as a free-fall. We have previously shown the acceleration due to gravity of a free-fall to be 9.8 m/s/s, so as the acceleration is increasing closer to 9.8 m/s/s as the angle increases closer to 90 degrees. The final part of our hypothesis was also correct, as mass does not have a significant effect on acceleration. Although the class data seems to show a slight correlation between mass and acceleration, this is actually linked to friction. The greater the mass of the ball, the more friction it causes on the ramp. This friction therefore disrupts the acceleration and can be blamed for the differences in acceleration of the balls with varying masses. When doing the mathematical calculations, mass ends up canceling out, further showing that it doesn't affect acceleration.
 * To improve our results, many of the sources of error could be eliminated or at least further minimized. The human reaction time between when the ball is actually stops/starts rolling and when the person hits the timer could be eliminated if a different method was used for measuring time. A spark timer or other timing device could be used that would indicate the exact moment when the ball starts rolling and when it reaches the bottom of the ramp. In addition, more precise measurements could be taken with a measuring tool that contained smaller units than the millimeters on the ruler/measuring tape we used. There was also a slight gap between the end of the measuring tape on the ramp and the actual end of the ramp, so this could also be precisely measured with this measuring tool to ensure the correct distance the ball is traveling down the ramp. Furthermore, more trials could be run to further ensure that our data wasn't random or outliers.

=Newton's Second Law Lab= Task A: Amanda Fava Task B: Maddy Weinfeld Task C: Jake Aronson Task D: Ryan Hall Period 2
 * Objective:**
 * Verify Newton's second law
 * Discover relationship between system, mass, acceleration and net force.
 * Predict what the graphs will look like
 * Hypothesis:**
 * The graph between net force and acceleration will be linear because as force increases, acceleration increases as well.
 * The graph between mass and acceleration will be a decreasing exponential curve.

media type="file" key="Mass1.m4v" width="300" height="300" media type="file" key="Mass2.m4v" width="300" height="300"
 * Movie for Acceleration vs. Force:**
 * Movie for Acceleration vs. Mass:**

Data and Graphs:
 * ** Theoretical Acceleration ** ||
 * 0.185 ||
 * 0.369 ||
 * 0.555 ||


 * ** Theoretical Acceleration ** ||
 * 0.555 ||
 * 0.285 ||
 * 0.192 ||
 * 0.145 ||
 * 0.116 ||

For analysis, note that we didn't use trail 5 (with the heaviest mass) in the graph as it threw off our equation. It's possible that the mass was too much and therefore caused too much friction with cart for the results to follow the pattern/be useful.


 * Calculations:**
 * Analysis:**
 * 1) Explain your graphs:
 * 2) If linear: What is the slope of the trendline? To what actual observed value does the slope correspond? How does it compare to this actual observed value (find the percent error between the two)? Show why the slope should be equal to this quantity. What is the meaning of the y-intercept value?
 * 3) The graph “Acceleration v. Net Force” is linear. The slope of the trendline on our linear graph is 1.8878, which corresponds to the actual value of 1/total mass . However, our data achieved a 1/total mass value of 1.8867. The percent error between the theoretical value and the observed value is 0.058 percent, suggesting that our results for the linear graph were successful. The actual slope should equal 1/total mass, in this case 1.8878, because in the equation a=(net force)(1/total mass) , net force is the variable “x” and 1/total mass is the coefficient, or slope. The y-intercept value represents the force of friction, which we ignored in the sample calculations.
 * 4) If non-linear: What is the power on the x? What should it be? What is the coefficient in front of the x? To what actual observed value does the coefficient correspond? Find the percent error between the two. Show why this value should be equal to this quantity.
 * 5) The graph “Acceleration v. System Mass” is non-linear. The power on the “x” on our non-linear graph is -1.535, though it should be -1.0. The coefficient in front of the “x” is 0.1817, which corresponds to the value for Net Force. The percent error between the theoretical coefficient value and the actual coefficient value is 61.81 percent, suggesting that our results were affected by sources of error. The slope should be equal to 0.294 because slope=hanging weight, which is equal to 0.294.
 * 6) 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.
 * 7) Friction would decrease our acceleration, because friction opposes the string tension from the hanging weight, so we would need a greater force to counteract the negative friction. Our slope was slightly too small on both the linear and non-linear graphs. Friction can be a source of error because it could have opposed the motion from the hanging mass. If we calculated acceleration with friction, then the equation would have been a=(hanging weight)(1/total mass)-(friction/total mass), or according to our graph, y=1.8878x-0.1033. a=(9.8)(0.010)(1/0.53)-0.1033=0.0817 m/s2.

__**Conclusion:**__ In regards to our hypotheses, we were correct with both of them. As we stated, the graph between net force and acceleration was linear since as force increases, acceleration increases as well. This is shown by when we increased the net force from 0.098N in our first trial to 0.294N in our last trial, the average acceleration rose from 0.0867m/s2 to 0.4567m/s2. Addition, as we predicted, the graph between mass and acceleration was a decreasing exponential curve, because as mass begins to increase, the acceleration exponentially decreases. This can be seen when average acceleration decreases from 0.4567m/s2 to 0.06m/s2 when we increased the mass of the cart from 0.5kg to 2.496kg.

For the acceleration vs. net force experiments, we had a 0.058% error, while for the acceleration vs. mass tests, we had a 53.5% error. In the lab, there were many possible sources of error. For example, the string connecting the cart to the hanging weight might have fully taut, not properly placed on the pulley, or might not have been parallel with the track. Also, the track could have been accidentally slanted one way or another, thus giving us an erroneous acceleration. To eliminate these errors in the future, we could use a level to ensure that the track and string are level and parallel, test the tightness of the string, and double check that the pulley is spinning properly with the string running around it. If the track and the string were in fact not level, we could use the knob on the end of the track to either higher or lower the side of the track in order to make in level.

The idea of the relationship of acceleration, net force, and system mass is very common in engineering. Construction builders must understand the workings of this relationship in order to safely build cranes and more importantly elevators. If too many people are in an elevator, it will hold to much mass to acceleration; this is why constructors must place a limit on how many passengers each elevator can safely carry. If the net force on an elevator's cable is too much, the elevator might become unsafe to ride in, and the cable might eventually snap. Therefore, builders must be very well educated in the dynamics of net force, in order to make safe products for consumers.

=Coefficient of Friction Lab= Task A: Jake Aronson Task B: Ryan Hall Task C: Amanda Fava Task D: Maddy Weinfeld

__Objectives:__ Our objectives in this lab are to 1.) measure the coefficient of static friction between surfaces, 2. measure the coefficient of kinetic friction between surfaces, 3.) determine the relationship between the friction force and the normal force.

Hypotheses: 1) The coefficient of static friction between surfaces will be between 0 and 1 because that is the normal range. 2) The coefficient will get larger as the mass increases because its related directly to the normal force, which is dependent on mass. 3) The friction force and normal force are directly related because the equation is f = (coefficient)(Normal) so as normal increases, so does friction.

__Materials:__ For this lab, we will be using a Pasco force meter, a USB link to connect it to a laptop, a friction 'cart,' miscellaneous masses, string, an aluminum track, and a clamp.

__Procedure:__ First, we will find the mass of the "friction" dynamic cart. Then, we will place the cart on the surface, and put a 500g mass in it. We will add a 15cm string to one end of the cart, and on the other end, the Pasco Force Meter. When we insert the Force Meter USB link into a laptop, we will choose DataStidio, click "New Experiment," "SETUP," check "Force-Pull Positive," and uncheck "Force Push Positive." On the display of the graph, we will change the y-axis to "Force Pull Positive," and press the zero button on the sensor once the string is left slack. Next, we will press "Start" on DataStudio, gently pull the force sensor with a constant speed parallel to the surface, which will in turn pull the block. For the graph, we will highlight the section of the straight line, click the symbol representing net force, and then select the mean as the value for tension and constant speed. Lastly, we will highlight the maximum point and record it as maximum tension, and then repeat twice for each mass, and repeat each step for larger masses. media type="file" key="coeff friction lab.m4v" width="300" height="300"

__Data and Observations:__
 * Static Friction**
 * Mass (kg) || Maximum Tension (N) || Average Maximum Tension (N) || Normal Force (N) ||
 * 0.591 || 1.1 || 1.08 || 5.79 ||
 * ^  || 1.2 ||^   ||^   ||
 * ^  || 1.2 ||^   ||^   ||
 * ^  || 0.9 ||^   ||^   ||
 * ^  || 1 ||^   ||^   ||
 * 1.089 || 2.6 || 2.04 || 10.67 ||
 * ^  || 2.2 ||^   ||^   ||
 * ^  || 1.9 ||^   ||^   ||
 * ^  || 1.7 ||^   ||^   ||
 * ^  || 1.8 ||^   ||^   ||
 * 1.339 || 2.3 || 2.5 || 13.12 ||
 * ^  || 2.7 ||^   ||^   ||
 * ^  || 2.6 ||^   ||^   ||
 * ^  || 2.7 ||^   ||^   ||
 * ^  || 2.2 ||^   ||^   ||
 * 2.339 || 4.4 || 4.3 || 22.92 ||
 * ^  || 4.5 ||^   ||^   ||
 * ^  || 4 ||^   ||^   ||
 * 3.339 || 5.6 || 5.466666667 || 32.72 ||
 * ^  || 5.6 ||^   ||^   ||
 * ^  || 5.2 ||^   ||^   ||
 * Kinetic Friction**
 * Mass (kg) || Maximum Tension at Constant Speed (N) || Average Maximum Tension at Constant Speed (N) || Normal Force (N) ||
 * 0.591 || 0.5 || 0.48 || 5.79 ||
 * ^  || 0.4 ||^   ||^   ||
 * ^  || 0.5 ||^   ||^   ||
 * ^  || 0.5 ||^   ||^   ||
 * ^  || 0.5 ||^   ||^   ||
 * 1.089 || 1 || 0.86 || 10.67 ||
 * ^  || 0.9 ||^   ||^   ||
 * ^  || 0.8 ||^   ||^   ||
 * ^  || 0.8 ||^   ||^   ||
 * ^  || 0.8 ||^   ||^   ||
 * 1.339 || 1.3 || 1.3 || 13.12 ||
 * ^  || 1.3 ||^   ||^   ||
 * ^  || 1.3 ||^   ||^   ||
 * ^  || 1.4 ||^   ||^   ||
 * ^  || 1.2 ||^   ||^   ||
 * 2.339 || 2.2 || 2.133333333 || 22.92 ||
 * ^  || 2.1 ||^   ||^   ||
 * ^  || 2.1 ||^   ||^   ||
 * 3.339 || 3.1 || 3.166666667 || 32.72 ||
 * ^  || 3 ||^   ||^   ||
 * ^  || 3.4 ||^   ||^   ||



__Analysis:__


 * The acceleration on the x is equal to 0 because the cart is moving at constant velocity, therefore there is no acceleration. Tension refers to the average tension of the trails for the specified mass (0.591 kg).
 * The acceleration on the y is equal to 0 as well because there is no motion on the y-axis, and subsequently no acceleration.
 * The class average for coefficient of static friction was 0.1722 and for kinetic friction was 0.120. Since ours was 0.176 for static and 0.0948 for kinetic, ours were pretty far off from the class data; 2.21% difference for static and 16.55% difference for kinetic. Though, still followed same general trend as class data, since our static was higher than our kinetic.

__Discussion Questions:__ 1. Why does the slope of the line equal the coefficient of friction? Show this derivation 2, Look up the coefficient of friction between your material and the aluminum track. Discuss if your measured results fall within the range of theoretical values. Be sure to cite your sources! 3. What variables affected the magnitude of the force of friction? What variables affected the magnitude of the coefficient of fricition? 4. How does the value of coefficient of kinetic friction compare to the value for the same material's coefficient of static friction?
 * [[image:Screen_shot_2011-12-07_at_10.23.11_PM.png]]
 * Because the equation of the line and the friction equation follow the same basic set-up, variables can be substituted to make y=mx+b equal to f=muN. Mu and the slope are both the only constants in the equation and are the coefficient for the x-axis variable, and can therefore be substituted for one another.
 * The coefficient of static friction between our material (plastic) and aluminum (metal) was between 0.25 and 0.4, and that for kinetic was between 0.1 and 0.3. Our static friction coefficient was lower than the theoretical range at 0.176 though not by too much; and our kinetic friction coefficient was far closer to the theoretical value range at 0.0948. Our source for these coefficients was []
 * The magnitude of the force of friction was affected by the surface the cart was on; had the surface been different, the force friction would have been different as well. The force of friction was also affected by the tension, or our pull with the string on the cart, since tension is equal to friction in this lab. The coefficient of friction and the normal force of the cart also affected the friction (f=muN). Since the normal force is equal to the weight (which is mg), the mass of the cart also affected the force of friction.
 * The coefficient of friction (mu) was also affected by the surface the cart was on. In addition, because f=muN, the normal force also affected the coefficient of friction. Since the normal force is equal to the weight in this lab, and the weight is equal to mass*gravity, the mass also affected the coefficient of friction.
 * The value of coefficient of kinetic friction was 0.0948 and the value of the coefficient of static friction was 0.176; therefore, the coefficient of static friction was larger than that of kinetic friction. The value of static friction is larger because the maximum tension is needed to move the cart from rest; since there is already a large force acting on the cart to keep it at rest, it takes a larger (specifically the maximum tension) to move it from rest. Once in motion, it is far easier to move the cart as an object's natural motion is to stay in motion; therefore the coefficient of kinetic friction is smaller.

__Conclusion:__ The lab definitely satisfied our objectives. We were able to determine the coefficients for both static friction between surfaces and the coefficient between surfaces by using data studio and then we were able to determine the relationship between the friction force and the normal force by graphing our data on excel. Our hypotheses were correct. Both coefficients were between 0 and 1, which supports what we already knew about the coefficient. We discovered that the tension force equals the coefficient of friction times normal force. Our coefficient for static friction was 0.176 and our coefficient for kinetic friction was 0.0948. We were also correct with our hypothesis that the coefficient will increase as mass increases. This makes sense because the two are directly related. We were also able to see a relationship between the tension of the string and mass; as the mass increases, so does the tension. Our last hypothesis was that the normal force and friction force would be directly related. As you can see on our graph, this hypothesis was correct, because as normal force increases, so does friction. Our percent error was 5.97% for the coefficient of static friction and 12.55% for the coefficient of kinetic friction, which are both pretty large but still fall within the 20% error range. There were several sources of error in the lab, many of which are simply human inabilities. For example, we relied on our own judgement to determine if the string was really horizontal so chances are it wasn't perfect every time. Even a slight slant in the string would change results because once it is taken off the horizontal, we need to consider the different components. To eliminate this error, we could have measured the distance from the string to the ramp and made sure it was equal throughout. This wouldn't be a perfect solution either, because we could end up moving it, but it still would at least give us a better place to start. Another error was with the speed of the cart. It was supposed to be pulled at constant speed, but again, due to the mere fact that we are humans, it was unlikely being pulled at the exact same speed the entire trial. To eliminate this error, we could have attached the string to a toy car that moves at constant speed. This lab has many real life applications. The assignment itself taught us the importance of being prepared. Having set up everything beforehand made it much easier to participate on lab day. As for the actual physics, the lab helped us visualize the concepts we have been discussing in class. Being able to see it first hand makes it much easier to understand. We could also use this lab in a literal real life situation, if we ever need to find the coefficient of friction on an object in your life, perhaps your car.