3/18 Modeling Friction Forces

Purpose: To determine static and kinetic frictional force using different experiments. 

Procedure:

PART ONE: Static Friction

FIGURE (1)

For the first part of the lab, we were determining static friction between a block and the tabletop. We did this by attaching a pulley to the table attaching a string to the block and a cup. We hung the cup off the edge of the table while the block sat on top of the table. We then proceeded to add water to the cup until the block started to move just a little. That meant that the weight of the cup and therefore tension in the string was great enough to overcome the static friction. ( f_static≤µ_sN) By obtaining the mass of the cup we were able to calculate the maximum force one can apply before overcoming static friction (FIGURE (1)). We did this process four times with four different masses and we recorded the mass of the cup with water each time. We then calculated the normal force and frictional force created a Friction v. Normal graph (FIGURE (2)). 

CORRECTION IN FIGURE (2): The mass of the blocks was entered incorrectly. The data should read from top to bottom: 131, 252, 359, 485. 

FIGURE (2)

The slope of this graph gives us the coefficient of static friction: µ_s=.3475 ±.01287

PART TWO: Kinetic Friction

For the second part of the lab, we were determining the kinetic frictional force between the blocks and the tabletop. This time, we had the block sitting on top of the table with a string attached to it. We set up a force probe and tied it to the other end of the string. We then pulled the force probe horizontally while collecting data in LoggerPro (FIGURE (3)). We did this three additional times with larger block masses each time. 

FIGURE (3)

After we finished collecting or data we plotted it on a graph again. (FIGURE (4)) The slope on this graph again gives us the coefficient of static friction: µ_s=.2712 ± .009813. We then know that the coefficient for kinetic friction is anything greater than that. (f_kinetic=µ_kN). 

FIGURE (4)

PART THREE

For the third part of the lab, we placed a block on a horizontal piece of plywood and lifted one end (the one closest to where the block sat) until the block started to slip. We then measure the angle where that occurred. The calculations to find the coefficient of friction are shown in FIGURE (5).

FIGURE (5)

PART FOUR

In the fourth part of the lab we did a similar procedure as part three but we measured speed with a motion detector. From measuring speed, we can calculate the kinetic friction between the surfaces by obtaining acceleration from our graph (FIGURE (6)). We know that the acceleration would be the slope of the velocity graph. Therefore, the acceleration is .7825 m/s^2.

FIGURE (6)

Once we have the acceleration we can use the angle and acceleration to calculate the coefficient of kinetic friction. The steps to find µ_k are in FIGURE (7).

FIGURE (7)

PART 5

In the first part of the fifth part of this lab, we were to derive and equation for acceleration using the coefficient of kinetic friction from part four. The work for that is shown in FIGURE (8):

FIGURE (8)

FIGURE (9)

The second part of the experiment was to calculate the mass that would be required to overcome static friction and move the block, if the unknown mass was attached to a string and to the block and then hooked up to a pulley (FIGURE (9)). After we found the unknown mass we could plug the into our equation for acceleration and get an estimate for acceleration as well. The work for this is shown in FIGURE (10).

FIGURE (10)

When we tried the experiment with our calculated mass (we had to use 40g), the block moved a little bit but it was not a steady movement. We got a steady movement at 50g, which probably means that the precise mass is somewhere between 40g and 50g. We also set up a motion sensor so we could obtain a velocity v. time graph as well as the acceleration. (FIGURE (11))

FIGURE (11)

The graph shows that the acceleration of the block was .573m/s^2. This value is not very to the one we calculated. Errors that may have attributed to this include errors in initial measurements such as the angle measured in part three. This would have led to an error in out calculation of the coefficient of friction and an even bigger error in part 5 calculations. When we were doing the experiment, our motion sensor was also acting very strange. It would take measurements differently each time, and our initial test looked completely different from our final test (shown).

Conclusion:
Overall, the purpose of this lab was fulfilled. We demonstrated everything that we had learned in class about friction and most of our calculations and experimental results were accurate. There were many possible errors in this lab since we did five different experiments. The possible errors may have included: (part (1)) we could have added to much water to the cup. This would obviously lead to a great mass of the cup and therefore a greater frictional force. When we were adding drops to the cup we were not adding drop by drop by drop, because that would take a long time and we did not have that much time in lab; (part (2)) When we were pulling the force sensor we could have pulled it at a slight angle instead of parallel to the table. The force sensor also could have been calculated inaccurately and we would lose accuracy. Also, it it highly doubtful that we were able to pull the blocks at a constant speed and the same speed every time we took new data.; (part (3)) As previously stated, an error in the part could have been the angle we calculated. We could have read the measurement incorrectly or we could have stopped at an angle greater than when the block first started to slip. This would give us a greater frictional value and coefficient of friction.; (part(5)) we could have recreated the 20 degree angle inaccurately which would give the block greater acceleration and therefore a different coefficient of friction.; (part(5)) As previously mentioned, our previous measurements could have given us inaccuracy in this part of the lab. Also, there was the problems with our motion sensor.

3/4 Propagated Uncertainty in Measurements

Purpose: To determine unknown values (such as density or mass) and uncertainties, and then calculate propagated uncertainty of the unknown values.

PART ONE

FIGURE (1)

Procedure:

For the first part of the lab we were given three cylinders and we were to measure the height, diameter, and mass of each one (FIGURE (2)). We measured the height and diameter of each cylinder using a caliper (FIGURE (1)).

Once we found the height, diameter, and mass we could find the volume and density (FIGURE (2)).

FIGURE (2)

After that, it was time to find the propagated uncertainty. First we needed to obtain an equation for density in terms of mass, height, and diameter. The equation we came up with is :

 

As shown in FIGURE (3), we then set up the equation to obtain uncertainty for density using partial derivatives and the uncertainties in mass, diameter, and height. 

FIGURE (3)

Once we obtain the partial derivatives we were able to start plugging in our measured values of mass, diameter, and height (FIGURE (4)).

FIGURE (4)

Conclusion:

For the first cylinder, we calculated a density value of 2.9114 g/cm^3  ± .038. The actual value of Aluminum is 2.7 g/cm^3, which is not within our calculation of uncertainty. This tells us that we made an error somewhere, most likely in one of the measurements. 

For the second cylinder, we calculated a density value of 9.187 g/cm^3 ± .181. The actual value of Copper is 8.96 g/cm^3, which again is not within our calculation of uncertainty. Again, this probably means we made an error in a measurement,

For the third cylinder, we calculated a density value of 7.78 g/cm^3 ± .1375. The actual value of Iron is 7.87 g/cm^3, which is within our calculation of uncertainty. Yay!

PART TWO

Procedure:

For the second part of the lab we were determining an unknown mass and the propagated uncertainty of that mass. The mass we were calculating was hanging from to spring scales at different angels (FIGURE (1) & (2)).We measured the force of each spring scale and the angle. We did this with two different masses. 

FIGURE (1)

FIGURE (2)

  After we took all of the measurements, we set up our equations (FIGURE (3)).

FIGURE (3)

This gave us the equation to find the mass and also our equation to derive so we could find our uncertainty. All that was left was plugging in our data (FIGURE (4) & (5)):

FIGURE (4)

FIGURE (5)

Conclusion:
It looks like our data makes sense. The uncertainties are not that high and each mass is a reasonable number. Although, there is still room for error. When we were taking our data some of the measurements were hard for us to read, which leads to inaccuracy and therefore systematic errors.We also used different types/brands of spring scales. This could mean that the scales are calibrated differently and would also give us inaccuracy on our measurements.  Of course there also could have been random errors, due to the equipment we were using. In order to check and make sure that our data makes sense, we could compare it to other classmates data and see if we are in each others range of uncertainty.

3/2 Non-constant Acceleration Problem/Activity Solution

Purpose: To prove that non-constant acceleration problems can be done numerically besides just analytically. 

Procedure:In the first part of the lab we were given a problem (FIGURE (1)) that turned out to be very difficult to integrate.

FIGURE (1)

Luckily, we are able to use Excel to solve the problem numerically. We created a spreadsheet (FIGURE (2)) and set up a time interval, instantaneous acceleration, average acceleration, ∆velocity, instantaneous velocity, and distance. 

FIGURE (2)

To set up the spreadsheet, we started with acceleration because virutually all the other equations we need will need acceleration. In order to calculate the acceleration at each time interval we used the equation: 

a= 2t²

For average acceleration we used the equation:

 

For ∆velocity we used the equation: 

∆v= 

For instantaneous velocity:

v=v₀+∆v

And for distance: 

x= 

After entering all these equations in excel, we can evaluate any of the variables we have at any time interval, acceleration, velocity, etc.. 

Conclusion:

In the problem, we are trying to find distance before coming to rest. So our velocity should be zero. In the analytically found result, this gives us a time of 19.69 seconds, and if we plug that into our expression for x, then we will get 248.7 meters.
As shown in FIGURE (2), at 19.6 seconds, our distance is 248.7 meters. We can also tell that 19.6 seconds is the correct time to choose because our velocity is very near zero (the closest on our spreadsheet). These results prove that our spreadsheet works!

2/25 Determination of g and some statistics for analyzing data

Purpose/Goal: To prove that, without any other forces, the value of a free falling object is 9.8 m/s², as well as to analyze the class' data to find the average standard deviation. 

FIGURE (1)
http://hyperphysics.phy-astr.gsu.edu
/hbase/mechanics/ffallex.html

Procedure: We started by using an apparatus that marks the position of a falling object on a paper strip every 1/60^th of a second while the object falls freely. The object falls a distance of 1.5m while its fall is recorded by a spark generator that makes marks on spark sensitive tape (FIGURE (1)). Once the fall is recorded onto the strip of paper, we can tear off the strip of paper to measure the distance between marks.

 After we measured the distance between each mark we recorded our data into excel:

FIGURE (2)

In our data table (FIGURE (2)), time is in increments of 1/60^th of a second and distance is in cm.

After we entered all of our data, we graphed it. 

First, we graphed Mid-interval speed v. Mid-interval time: 

FIGURE (3)

Then we graphed Position v. Time:

FIGURE (4)

For constant acceleration, the velocity in the middle of a time interval is practically the same as the average velocity for the time interval. To prove this, I derived an average velocity equation from the calculated position equation in FIGURE (4):

v(t)=2(470.13986)t+100.20629 

I entered this equation into excel, and got a velocity very similar to the velocity in the middle of a time interval (as shown in FIGURE (5)).

FIGURE (5)

To obtain the value for acceleration due to gravity, I derived the velocity equation from the midinterval speed graph (FIGURE (3)) to get acceleration in terms of time. This left me with the value of 9.33 m/s², which is pretty close to the accepted value of gravity: 9.8.

Another way to obtain the value for acceleration due to gravity is to derive the position equation from our position v. time graph (FIGURE (4)) twice. When I did this I got a value of 9.4 m/s², which is very close to the 9.33 m/s² I got from deriving the velocity equation. 

Conclusion: 

Overall, this was a successful lab. The value that we got for acceleration due to gravity was fairly close to the accepted value. The relative difference was -4.08%, which isn't too high. 

After the class had all obtained their experimental values of gravity, we were able to analyze the class' data. In excel, we found the standard deviation from the mean to be ±32. (FIGURE (6) & FIGURE (7))

FIGURE (6)

FIGURE (7)

Our class' average value of gravity is closer to the accepted value of gravity than my group's experimental value. The accepted value of gravity is within one standard deviation of our mean, so we weren't too far off. Some of the possible errors in this lab include systematic errors such as the equipment we used, measurement error, or flaws in assumption (such as the assumption of no air resistance) or simply random errors. 

Analyzing a group's data is an important part of experiments because it gives a range of values instead of just one. It also allows us to find the standard deviation of the mean of the group. This gives us more confidence in our experimental value of g and apparatus. The class mean was closer to the accepted value of gravity than my lab group's experimental value was. Analyzing group data also helps eliminate random errors in our data. 

2/23 Finding a Relationship Between Mass and Period for an Inertial Balance

Purpose/Goal: To determine the relationship between mass and period by measuring the period of different masses using an inertial balance and comparing the results with a period vs. mass equation [T=A(m+M_tray)ⁿ]

Procedure:

The first part of the procedure involved setting up the inertial balance and the Lab Pro to record the data. Below is the setup.

Inertial Balance and Lab Pro setup

We had different masses that we would put into the mass tray. They ranged from 100g-800g. We stuck a piece of tape on the end of the inertial balance so the motion sensor could measure the period. 

After the inertial balance was set up, we put masses in the mass tray and measured the period using the information the Lab Pro sent to Logger Pro on the computer and recorded it on a data table. We repeated this step with different masses until we reached the mass of 800 g.

Data Table

We also measured the mass and period of two "unknown" objects. For us, those objects were a water bottle and a roll of tape.

"unknown" object values

Once we had all of our data we entered it in Logger Pro which created a graph.

Data in Logger Pro

After we finished entering data we took a look at our equation (T=A(m+M_tray)ⁿ) and came up with a way to solve for the three unknowns- A, M_tray, and n. In order to solve for those variables, we took the natural log of the equation:

lnT=n ln (m+M_tray)ⁿ)+ln A

Once we had this equation, we plotted it in Logger Pro in order to solve for M_tray.

ln T vs. ln (m+M_tray)

In order to get a "beautiful straight line", we had to adjust the value of the parameter M_tray until the correlation coefficient was as close to 1 as possible. After many tested numbers, and a few mistakes, we found the lowest M_tray value that would give us the closest correlation coefficient and the highest M_tray value.

Lowest M_tray value

Highest M_tray value

After we had these values we could plug them into the ln T vs. ln(m+M_tray) equation with the period value for the water bottle. 

Lowest M_tray value:

Highest M_tray value:

Our actual mass value was .496 kg. We were close but not in the range that the equation gave us. There could be multiple reasons for this error. Some of those reasons could include not measuring the period correctly (letting go of the pendulum at a different distance each time or when we let go of the pendulum), not rounding to enough significant figures, or errors in measurements.

Summary:

In this lab we tested the relationship between period and mass as well as the equation to get us from one to the other. We confirmed that period and mass have a direct relationship. We were also able to get an answer with our equation that was reasonably close to the measured values. This proves that the equation relating period and mass is a suitable representation.