16-Mar-2015: Modeling friction forces

     The purpose of this experiment was to model friction forces. It was split into five different parts. The first part asked to find the coefficient of static friction between a block and the lab table. The second part asked to find the coefficient of kinetic friction. The third part asked to find the coefficient of static friction from a sloped surface. The fourth part asked to find the coefficient of kinetic friction from a sloped surface. Finally, the fifth part asked to predict the acceleration of a two-mass system.

Part 1

     The first part of the experiment was to find a coefficient of static friction between a block and a lab table. The setup had a block resting on a table, with a piece of string attached to it. The piece of string went over a pulley at the end of the table and was attached to a cup hanging off the table.

Setup for experiment 1.

     The mass of the block was weighed. Then, water was poured into the hanging cup until the block moved the slightest bit. The mass of the cup plus water was weighed.This was done three more times for a total of four blocks on top of each other. The resulting data was plugged into a table in Logger Pro.

Since it is known that static friction force is proportional to the normal force of an object by equation,

then the coefficient of static friction must be the slope of a friction vs normal force graph given by,

     So, the friction force and normal force needed to be found and plotted. In this case, the normal force of the blocks is equal to the weight of the blocks (mg) and the friction force is equal to the weight of the cup plus water (mg). Doing the calculations gives a new table of:

Taking the values of friction and normal force and plotting them gives a graph of:

And a linear fit with an equation of y=Ax, similar to the friction force equation, where A is the slope.

                                             

This means the coefficient of static friction between the block and the table was found to be:

coefficient of static friction=0.3093.

Part 2

     The second part of the experiment asked to find the coefficient of kinetic friction between the same block and table. This was achieved by using the same block and string but this time the end of the string was attached to a force sensor. The force sensor was connected to the Logger Pro system. The force sensor was calibrated by hanging a known mass from it and telling the computer what the mass was. The Force sensor was then held horizontally and zeroed out. The block was then pulled horizontally at a constant speed by the person holding the force sensor. Logger Pro recorded the data and produced a graph. This was repeated for up to 5 blocks sitting on top of each other.

The graphs of all five trials were stored.

     Each graph was then analyzed using the statistics tool in Logger Pro. The mean value of the pulling force of each individual graph was recorded. Since it was assumed there was no acceleration in the system, the pulling force was equal to the friction force. This means if a graph of friction vs normal force is plotted then the slope of the line is the coefficient of kinetic friction. Much like the static friction in the first part of the experiment. 

The friction force seen in the table is the mean of each force graph (analyzed above). The normal force is equal to the weight of each mass of blocks. With this data a graph of friction vs normal force was produced:

with a best fit line that produces an equation y=Ax, with A being the slope which is the coefficient of kinetic friction.

This means the coefficient of kinetic friction was between the block and table was found to be:

coefficient of kinetic friction=0.2531.

This value is lower than the coefficient of static friction, which agrees with the model of friction.

Part 3

     The third part of the experiment asked to find the coefficient of static friction from a sloped surface. The block was placed on one end of a horizontal track. The end of the track where the block was placed was lifted up, slowly, until the block starts to slip. The angle the track made with the horizontal was then measured and used to calculate the coefficient of static friction.

The angle was measured to be 24 degrees. The mass of the block was measured to be .111kg.

 A free-body diagram and the sum of forces were needed to find the coefficient of static friction.

free body diagram of the block on a sloped surface.

Below are the sum of forces and the steps taken to solve for the coefficient of static friction.

Plugging in the angle of 24 degrees into the above equation yields a coefficient of static friction of: 0.445

Part 4

     Part four of the experiment asked to find the coefficient of kinetic friction from sliding a block down an incline. The setup was much the same as part 3, except this time there was a motion sensor attached to the ramp and the ramp was already inclined enough so that the block would automatically slide down. 

     To find the coefficient of kinetic friction, the acceleration of the block and the angle of the slope was needed. The motion sensor was used to find the acceleration. The motion sensor was setup with Logger Pro. When it was activated, the block was released. The following position vs time and velocity vs time graphs were produced:

The velocity vs time graph was analyzed with a linear fit to produce a slope, which is the acceleration of the block.

An acceleration of 0.3377 m/s^2 was recorded. The slope was measured with a smart phone to be 16 degrees.

     After the acceleration and slope was found, the sum of forces was found to solve for the coefficient of kinetic friction.The free-body diagram was the same as the one in part 3, but the sum of forces in the x-direction were slightly different. Here are the steps that were taken to solve for the coefficient of kinetic friction:

     Since the block was moving this time, the sum of the forces in the x-direction were equal to (ma). After solving for the coefficient of kinetic friction and plugging in the measured angle and the acceleration, a value of 0.251 was found.

Part 5

     Part five of the experiment asked to predict the acceleration of the block when attached to some amount of mass over a pulley. After the prediction, it asked to test the experiment with a motion sensor to find the acceleration. Since the same mass was being used on the same surface as part 4, the mass measurement and coefficient of kinetic friction values would be used. The hanging mass was simply weighed on a scale to find its value. In this trial, a hanging mass of 0.08kg, a block mass of .111kg, and a coefficient of kinetic friction value of .251 was used.

     The first thing that needed to be done was draw the free body diagrams of the block, string, and hanging mass. Then, the sum of forces needed to be found. Solving for acceleration and simply plugging in numbers was the last step. Here are those steps:

Free-body diagrams for each mass.

     The sum of forces were found and solved for acceleration. The acceleration of the system was then predicted by plugging in the masses into the derived acceleration equation. Then the trial run was done. A graph on logger pro popped up and a linear fit was done for the velocity vs time graph. The slope of the line gave the acceleration of the system. The predicted acceleration was not close to the actual acceleration. Each trial gave a sloppy velocity vs time graph so a best fit line had a poor correlation. This might have been caused by the surface of the ramp not being smooth enough.

09-Mar-2015: Modeling air resistance using coffee filters and logger pro video analysis.

     The purpose of this experiment was to find a relationship between the force of air-resistance and speed. We know that Newtons 2nd Law states that for every reaction there is an equal and opposite reaction. This applies to an object at free-fall under the influence of the force known as weight, calculated by mass times gravity (mg). The force opposing weight is defined as drag, or air resistance. A model of Fairresistance=kv^n was used to determine the relationship between air resistance and speed, where k takes into account the shape and area of the object and n takes into account the material the object is moving through. 

   

     Coffee filters were used as the objects in free-fall. The idea was to drop coffee filters off of a balcony and record the fall. 

Balcony used to drop coffee filters off of. 

     The recorded fall was analyzed by Logger Pro software with the help of human input. Logger Pro allowed people to plot a dot at the location of the coffee filter at any point in time. A position vs time graph comes up. Setting the origin and direction of both axis' gives a graph to each person's liking. Shown here is the analyzed video with the positive y-axis pointing down (the yellow lines). The green line was the reference length measured at 1.46m input into Logger Pro to get accurate data. 

Analyzed recording of the free-fall of 2 coffee filters.

Best fit line showing terminal velocity of 2 coffee filters to be 1.323 m/s.

     Analyzing a best fit line of the resultant position vs time graph (shown above) gave back a terminal velocity of the coffee filters. The terminal velocity being the largest magnitude of velocity when the force of air resistance equals the force of weight. This entire process was repeated 5 times, for 1, 2, 3, 4, and 5 coffee filters falling from the balcony. After those trials, there was enough data to create a force vs velocity graph. Below is the data set input to Logger Pro. In the left column are the terminal velocities of the coffee filters. In the right column is the force of weight acting on the coffee filters, calculated by simply multiplying the mass of 1, 2, 3, 4, or 5 coffee filters by gravity.

     The resultant graph was analyzed with a best fit line. A "power fit" was done since the best fit line needed to correspond with the mathematical model introduced earlier and the points appear to curve up. 

Power fit analyzation provided values for constants k and n.

This power fit produced numbers for k and n that allowed for calculations of terminal velocity. 

k=0.01105

n=1.725

Now that there are numbers for the mass of a coffee filter, k, and n, an excel spreadsheet was opened to help model and predict future experiments. 

There were 6 columns set up in the spreadsheet. They were composed of variables such as time and velocity and contained initial parameters in the free-fall. It looked like:

     Above the variables were reference cells for time, mass, k, and n. This allowed for the change of those variables without having to redo any formulas. This looked like:

     A formula for acceleration was needed to plot into the acceleration column. Doing a simple free-body diagram for the force of air-resistance opposing weight gives an equation for net force of:

Rewriting force as mass times acceleration and solving for acceleration gives:

The spreadsheet was then modeled to make velocity and position as functions of time. Numbers were plugged in and the resulting data was produced. 

Data for mass of 2 coffee filters.

     At first glance, the data might not say much, but after scrolling down the data shows a couple of interesting and useful things.

Data for mass of 2 coffee filters continued..

    First, notice that as time goes on, acceleration approaches zero. Second, notice that as acceleration approaches zero , velocity rises and levels off. 

     Acceleration approaching zero means that the force of air-resistance is becoming equal to the force of weight. And if those two forces are becoming equal to each other, then the object must be approaching terminal velocity. So the excel model works.

     

     How did it compare to the experiment data? 

The excel model compares well with the experiment data. The largest difference in terminal velocities was for one coffee filter.

Experiment terminal velocity=0.9578 m/s

Excel model terminal velocity=0.8745 m/s

Difference in velocities=0.0833 m/s

The smallest difference in terminal velocities was for 2 coffee filters, a difference of 0.016 m/s.

So, the excel model has a close enough approximation to be able to predict how experiments will turn out.

04-Mar-2015: Determining the density of unknown metals/Determination of an unknown mass and finding the propagated uncertainty in both parts.

     The purpose of this experiment was to understand how uncertainties in measurements affect the final answer of calculations. The experiment was divided into two parts. In the first part, three unknown cylindrical metals were given and the problem was to find their densities and the uncertainty in the measurements. Finding the densities would allow the elements to be identified. In the second part, there were masses being hung in equilibrium. The problem was to find their masses and the uncertainty in those measurements.
     Take a look at the first part of the experiment.
Vernier calipers were used to make measurements of the unknown metals. The calipers have centimeter marks on them (shown below).

     Measurements with the calipers are made by placing the object between the teeth (shown below).

     Then, you take the measurement based on the vernier scale. On the calipers is another set of marks in addition to the centimeter marks. These marks are called the vernier scale.

The first mark indicates where on the centimeter scale the object is (to the tenth place). In the picture shown to the left, it would measure at 8.5 centimeters. Then, you find where the next centimeter and vernier mark line up. The vernier mark indicates the centimeter to the hundredths place. In the picture the object would then measure 8.58 since the 8th vernier mark lines up with a centimeter mark.






     For this experiment, in order to find the density of the three metals, the mass and volume needed to be found. The mass was weighed using a scale and the volume was found by measuring the height and diameter using the vernier calipers. Once these measurements were made they were plugged into the formula for density, which is density=mass/volume.
     Since the measurements made by the scale and calipers had a certain degree of uncertainty to them, the uncertainty needed to be taken into account. This is because uncertainties follow the calculations throughout the problem and eventually affect the final answer. It usually results in a final answer with a range of values (example, density=6.00 (+-) 0.03 gives a range of densities of 5.97 to 6.03).
     To find the uncertainty, a function for density that contains all the variables measured needed to be found. Simply plugging the formula for the volume of a cylinder into the formula for density does the trick. The result looks like:

From now on since "d" represents diameter, density will be represented by the greek letter "rho".

To find the uncertainty in the density, the summation of the partial derivative of density with respect to mass, diameter, and height must be taken and multiplied by the uncertainty in each respective measurement. This looks like:

     Once the partial derivatives were found, all appropriate numbers were plugged into the formula and out came the uncertainty in the density.

Here are the measurements and results for the three metals:

The uncertainty in mass was 0.1g

The uncertainty in diameter was 0.01cm

The uncertainty height was 0.01 cm

Table 1. Final calculated values vs accepted values

     The calculations for the aluminum metal and the copper metal were both outside of the accepted value. The calculations for the iron metal, however were in the range of the accepted value. The discrepancies for the first two metals might have been caused by a few different types of errors. The vernier calipers might have been read wrong or they might have not been used properly. The tools used to measure the metals might not have been precise enough. One way to find out is to use better tools and redo all measurements, but those tools would cost a lot more money. Overall, the results are not bad considering the equipment provided.

Now take a look at the second part of the experiment.

     The second part of the experiment asked to observe two different masses, each in its own system in equilibrium. The process is very similar to the first part of the experiment, in that measurements were taken and uncertainties found.

Here are the two systems in question.

Figure 1. System 1 in equilibrium

Figure 2. System 2 in equilibrium

     The masses in both systems are held by strings connected to poles. Each string has a certain amount of tension in them measured in Newtons. Below is the tool connected to the string that measures the force of tension. As you can see, there will be uncertainties accompanied by this tool. 

     The angle of each string also needed to be measured. Here is the tool used to measure those angles. This measures angles in degrees which means its uncertainty is in degrees. When it comes time to find the uncertainty, the degrees will need to be converted into radians in order to do calculus.

     Since the mass was being looked for, a function for mass was needed. This was achieved by creating free-body diagrams for each system and then finding the sum of the forces. Since both systems look the same, one free-body diagram can be used to find the sum of forces. Here is the diagram:

Since the sum of forces in the x-axis does not contain mass, the sum of forces in the y-axis was used. This looks like:

Solving for mass gives:

After mass was solved for, the measurements were plugged in and mass was calculated. Next the uncertainty in mass was calculated. The formula looks like:

Once the partial derivatives were found, all numbers were plugged in and the uncertainty in mass was found.

Here are the measurements and the final calculation for the mass of each system:

Table 2. Measurements and calculated mass

Table 3. Uncertainties in measurements

     After plugging all numbers in, the calculated mass for system 1 was 843.98 grams (plus or minus 77.55 grams). The calculated mass for system 2 was 518.97 (plus or minus 98.70 grams).

As you can see the propagated uncertainty was quite large for both systems. Like the first part of the experiment, this can be due to incorrect readings of measuring instruments and imprecise instruments.  

02-Mar-2015: Using Microsoft Excel to numerically compute a non-constant acceleration problem.

     The purpose of this lab was to develop skills using Microsoft Excel to help solve difficult problems numerically instead of analytically. The problem given is a non-constant acceleration problem. A 5000-kg elephant is riding on frictionless roller skates with an initial velocity of 25 m/s. The elephant reaches the bottom of a hill and arrives on level ground. A rocket, which is strapped to its back, then produces a constant 8000 N thrust in the opposite direction. The mass of the rocket changes with time due to burning fuel, which is given by

     The questions asks to find the distance the elephant travels before it comes to rest. 

   
     Since the rocket is strapped to the elephant, they can be treated as one system. This gives a new function of

 

     The famous formula of force=mass*acceleration is given from Newton's second law of motion. In this problem, since acceleration is a function of time, it is rearranged so that

     Since the elephant is on level ground riding on frictionless skates, the net force would be -8000 N caused by the thrust of the rocket. Now that Fnet and m(t) are known they can be plugged in, which gives:

which simplifies to

     A normal analytical approach to this problem would be to integrate this function twice to derive an equation first for v(t) and then for x(t). Then solve for t using the velocity function by plugging in zero for final velocity. Doing that gives a time of 19.69075 seconds. Plug that time into the position function and solve. That gives a position of 248.7 meters. Although this problem can be solved analytically, it gets quite messy and requires more time than a person would want to spend on a problem. Below are the messy integrations we hope to avoid by using a numerical approach. (click the pictures to expand them for a closer look)

     

     Now take a look at the numerical approach. 

Microsoft Excel was used for this portion of the lab. After a new worksheet was opened, the first and second rows were filled in with the six variables and initial parameters from the problem, as shown below.

• Next, a time interval of 0.1 seconds was chosen and filled down to over 220 rows. 
• Then, the acceleration function from earlier was plugged into cell B2 and filled down to B3. This enables the user to find the acceleration at any point in time. 
• In cell C3 the average acceleration was found for that time interval. 
• In cell D3 the change in velocity was found for the first time interval. 
• In cell E3 the final velocity was calculated at the end of the time interval. 
• Finally, in cell F3 the position of the elephant was calculated by adding the average velocity times the change in time to the position at the start of the time interval. 

     After all formulas were plugged in, a simple fill down to the desired amount of rows was done (all formulas are shown below).

After "filling down", the point where velocity v is closest to zero was found. 

     As you can see from the above data, the numerical approach gives a very close approximation to the analytical approach. Changing the time interval to a smaller one to get a better approximation can be done, but there is a certain point where not much will change except in the thousandths place. For instance, below is the data for a time interval of 0.05 seconds.

     Although it does give a better approximation, unless you are overly concerned about the number of significant figures then a 0.1 second time interval is enough. Using a time interval of 1 second, however, is too big. The data given by that time interval is too far off from the results given by the analytical calculation (it gave a distance of 248.3 meters). By using Microsoft Excel to do this problem numerically, we were able to let the computer do the integration for us and give an extremely close approximation. This saves time and brain power!