Lab 8 - March 28, 2015 - Centripetal Acceleration vs. Angular Frequency

Lab 8:
March 28, 2015
Brandon Elder

Centripetal Acceleration vs. Angular Frequency

Fig. 1 Apparatus used to measure centripetal
 acceleration and time of rotations.

Purpose: Using an accelerometer to measure the amount of time and acceleration of a rotating disk in order to determine the relationship between centripetal acceleration and angular speed.

Set-Up: The apparatus was set up in the front of the classroom and the experiment was performed by the instructor. We recorded all the data as a class so we all have the same numbers. This lab was very short and not much to it other than a few calculations. See Fig. 1 below for a picture of the set-up and apparatus.

The wheel was spun around at various speeds and the amount of time that it took to complete 10 different rotations was recorded. The accelerometer reading corresponding to each rotational speed was also recorded. Lastly, the distance of the accelerometer from the center of the rotating disk was calculated.

Calculations: Six different trials were performed at various speeds set by voltages from the source causing the wheel to spin the disk. See Fig. 2 for all of the data saved and used in the calculations.

Fig. 2 All calculations and data taken for this lab. Each highlighted voltage represents a
different trial of data recorded. The t(0) represents the time the photogate on the first rotation
was passed and the t(10) represents the time the tenth pass took place. The difference of the two
represents the amount of time it took for 10 rotations around the axis and is recorded
next to each trial. The "a = acceleration" which is the centripetal acceleration recorded. The 

The formula used in calculating the centripetal acceleration is written at the top of Fig. 2. It is "a=r*omega^2.  Omega is equal to "2*pi radians / time for 1 rotation". The omegas are all calculated and displayed as well above for each speed.  During the lab, acceleration was calculated for us as well as the time for one rotation. We rearranged the formula to solve for the radius. Each calculation was performed and the radius was recorded above as well. The average was taken and determined to be .13685 meters or 13.685 centimeters. We were told by the pro-FESS-OR that the acceptable range was between 13.8 and 14 centimeters.

Lastly, we inputted all our data into a data table in LoggerPro to create a graph of data with acceleration on one axis and the omega value on the other side (see Fig. 3). The data table underneath it (Fig. 4) are the same numbers that were calculated on paper above. The only slight variance is that the best fit line of the graph, which displays the average r value is off by a few thousandths from the calculated r average value.

Fig. 3 Graph of acceleration vs. Omega squared. The average value for the radius is displayed here on the best-fit line.
The value is 0.1372 or 13.72 centimeters. This varies slightly from the value of 13.685 calculated in the notes above.

Fig. 4 Data table with calculated values inputted. The R values are in purple to the right.

Conclusion:

There are a few reasons why the values of R do not add up to what we were told is the acceptable range of R values. Perhaps the tape for the photogate apparatus caused a fluctuation in the acceleration values or the time values. There could have been friction inputted into the system on the wheel causing a force to slow down the wheel more than was expected.

Lab 5 - March 27, 2015 - Trajectories

Lab 5
Brandon Elder
3/27/2015

Trajectories

Purpose: To determine the impact location of a ball launched from a ramp onto a slanted platform that will be placed at the base of the metal ramp. The location will be determined by observing the ball's distance first without the slanted platform and then using kinematic equations to determine the impact location with the platform. 

Fig. 1 The apparatus was set up
as seen above. The top
metal ramp was taped
to the bottom ramp. The angle
does not matter for the two
metal pieces so long as it
remains the same.

Fig. 2 Carbon paper on the
ground with the paper to
mark the location of the
ball drops.

Procedure: Set up of the apparatus should be similar to Fig. 1. Place a piece of paper on the ground and use tape to keep it in place. The paper should be in the location that you expect the ball to land at. Let the ball drop from the apparatus once to estimate the correct location for the paper.  Next, tape a piece of carbon paper to the paper so that the location of the ball falling on the paper will be marked from the carbon (see Fig. 2). Once the correct location has been determined and the paper has been taped, choose a starting location for the ball on the top metal ramp to drop the ball from. It is important that the starting location for the ball is the same each time data is collected. This will allow for relatively the same location being marked on the paper as the ball will launch off the ramp with roughly the same velocity. Place the ball and let it roll down and launch it off five different times. The ball will hit the carbon paper and mark a dot on the paper. After five runs there will be five dots on the paper all in relatively close proximity (see Fig. 3). Watch the video below to observe exactly how the apparatus works and notice the ball bounce right on the paper with the carbon taped to it.

Fig. 3 Five dots on the paper relatively
close with one dot a little off.
 This one dot is off because we
ran a 6th trial with the ball at a different
starting location. This proved to
 us that it was critical to have the
starting location be the same for all the runs. 

Data Analysis:

Fig. 4 Diagram of apparatus with the
measured height and horizontal distances.

The next step in the process will be to analyze the test results. Measure the horizontal distance on the ground from the base of the ramp edge to the center of the five dots. Record this measurement. Measure the vertical distance from the edge of the ramp to the floor. Record this measurement. Then draw a diagram with the measured distances inputted (see Fig. 4). 

Next, using kinematic equations, calculate the time that it took the ball to travel and hit the ground. Once you have the time, you can calculate the speed at which the ball is launching from the ramp. The equation used and the calculations completed can be seen in Fig. 5.

Fig. 5 The equation used to determine
 the time and then the equation used
 to determine the velocity.

Fig. 6 Set up of board to the
apparatus. It is important to note
that the board must rest right up
against the edge of the ramp
from the first part of the experiment.

Now that you have calculated the time and the velocity at which the ball left the ramp, you can perform the next part of the lab. The next part of the lab will be to attach a slanted board to the end of the apparatus and to determine where along the board the ball will land. Measure the angle at which the board is laying against the apparatus. See Fig. 6 for the set up of the new board to the apparatus. The object here is to derive an expression that would allow you to determine the value of "d" (see Fig. 7) given that you know the initial velocity and the angle at which the board is slanted. The formula will look like Fig. 8. This gives you an expression for "d" in terms of the velocity.

Fig. 7 "d" is the distance down the ramp that we are
estimating with our formulas from below.



Fig. 8 Calculations used to determine the formula for "d"     

However, velocity is an expression that came from the x distance measurement as well as the y distance which was used to calculate time. The formula must be re-written in terms of all calculated expressions. The calculated measurements were the x-distance, y-distance, and the angle of the ramp. Re-writing velocity results in the expression below, (see Fig. 9).

Fig. 9 The formula for "d" in terms of theta, x, and y.

All that is left at this point is to plug in numbers to decide what the distance of "d" will be, and THEN calculate the uncertainty of course. We just need to take partial derivatives of the equation that we have for d with respect to the three measurements (theta, x, and y). Each partial derivative result will be multiplied by the known uncertainty in each measurement. The sum of all three products will result in the total uncertainty in our calculations. Finally, perform the experiment once more and measure the distance down the board to determine the actual distance. Compare that value to the value that you calculated. Is the data within the amount of "+/-" uncertainty? If so then you have a pretty good proof of just how accurate this experiment was.

Conclusion:

Our data gave us a calculated result of 1.12 meters for how far the point of impact down the board "d" should be. Our uncertainty calculated out to be +/- .0068 meters. When we actually ran the experiment and measured the data we recorded the ball landing at 1.06 meters down the board. This is not within the amount that we had calculated. :(

After recovering from a few days of severe depression following the realization that our experiment had failed, I started to reflect on possible issues for this non-concurrence. One main issue was that the board had been bumped and re-adjusted in-between the measurement of the angle and the performing of the experiment. This would have totally caused the type of error that we witnessed in our final results. The calculations for "d" were determined after we measured our angle theta but before we ran the actual experiment. 

Lab 4 - March 22, 2015 - Modeling the Fall of an Object with Air Resistance

Lab 4:
3/22/2015
Brandon Elder

Modeling the Fall of an Object with Air Resistance

Purpose: This lab is two-part, first we will determine the relationship between air resistance and speed, second we will model the fall of an object including air resistance.

Part 1: First of all, we expect that the force of air resistance can be modeled by some exponential equation, such as below (See Fig. 1).

Fig. 1 The force of air resistance is equal to some constant
times the velocity to some degree.

Fig. 3 The filter reaches
terminal velocity and
no longer requires
acceleration to be factored
into the equation.

In order to measure these unknown terms and to determine the air resistance, we will drop coffee filters (see Fig. 2) from a set height and record the speed at which they float down. At some point along this drop down, they will reach their terminal velocity, where there will no longer be any acceleration (see Fig. 3) acting on the filters, so the sum of forces will be the air resistance equal to some equation we will determine.

Fig. 2 We used a total of 5 coffee filters.

We headed over to the technology building, building 13 because there is a good place to drop the filters from inside where we can record them falling down uninterrupted (see Fig. 4). We set up LoggerPro to capture video for each drop. We started with one filter. Then we added another filter to the first one. We were careful to add each additional filter inside the first one, so we could limit the change in the surface area and shape of the filter being recorded as it dropped down. We repeated this process until we had a video for one to five filters, for a total of five drops.

Fig. 4 The white ledge at the bottom of our feet was measured so that the known distance could be inputted into LoggerPro later as a reference mark for distance. The filters were dropped and allowed to float all the way down to the ground in a straight line.

A video was taken of this entire process and each drop was stored in it's own file. Back in the lab, we evaluated each of these videos. Through LoggerPro, we placed a dot on the falling coffee filter every three frames. The software assigns a time value to the location we click on. The computer stores these points in a location and time table. This is done for each video until the filter either reaches the ground or is very near to the ground (see Fig. 5).

Fig. 5 A dot was placed every few frames
capturing the location of the coffee filter vs time.

Fig. 6 Slope of graphs has the
terminal velocity.

The picture below shows the videos we captured during our data collection and the graph shows our data points acquired through the video capture. The slope near the end of the curve signifies the terminal velocity reached by each of the coffee filters (see Fig. 6). We need the terminal velocity in order to graph the Position vs Time. This process was done 5 times for each of the coffee filters used.

Below is the graph of Speed vs. Force that is used to determine k and n in the equation Air Resistance Force = kv^n (see Fig. 7).  The third data point was not included because we thought we would get a better correlation if we excluded it. We got 0.007 for k and 1.793 for n.

Fig. 7 Graph of Speed vs Force used to determine
k and n in the equation F=kv^n

Fig. 8 Data plugged into an excel spreadhseet
 set to calculate the terminal velocity.

Calculations: In order to verify that we were able to get the correct values, we put the values of k and n into excel (see Fig. 8) in order to find the final velocity. The results proved to be the same, so it was accurate. Below are our calculations that we did for the experiments. (See Fig. 9). Fig 10 has the equation that we used to determine the weight of one coffee filter.

Fig. 9 All data calculated
from the experiment.

Fig. 10 Calculation used to determine
the weight of one coffee filter.

Summary: First, we dropped the the coffee filter from the top of the balcony so that we could gather the data in order to find the terminal velocity. This was done by capturing videos of the filters falling. Then we found the final velocity by graphing the Position vs. Time for each of the coffee filters. Then we used the terminal velocities to graph the Speed vs. Force which gave us the k and n for the equation Air Resistance Force = kv^n. We then used Excel in order to verify our k and n values which proved to be correct. Overall, this lab proved to be successful since we came to the same results and there could be some uncertainty because interpreting the data can not be 100% accurate. All in all, the air resistance force is directly related to the speed as proved by the lab results.

Lab 7 - March 18, 2015 - Modeling Friction Forces

Lab 7:

March 18, 2015

Brandon Elder

Modeling Friction Forces

Purpose: This lab is split into five parts. The purpose of the first two parts is to use a system of masses, string, wooden blocks, force sensor, and a pulley to determine the coefficients of static and kinetic friction. The purpose of the third and forth part of the lab is to measure the angle at which a block slides down a ramp and measure a mass of a block and use this information to determine the coefficients of friction, some static and some kinematic. The last part's purpose is to use one of the coefficients of kinetic friction we determined to then predict the acceleration of a two-mass system.

Part 1: Measuring Static Friction

Set up the apparatus to look like Fig. 1. Measure the block before you begin. We measured a mass of 131 grams. 

Fig. 1 The apparatus includes a wooden block, with felt, connected by a string to a cup with
water. The cup was filled with water slowly until the block began to move. The amount of water
that was added to the cup is the amount of force that was required to make the system move.

The following steps were taken for the rest of the process.

• Gradually fill the foam cup, the one suspended in the system, with water until the block(s) resting on the table, begin moving. This point represents the value for the max. static friction. Stop adding water the very instant that the block begins to move. 
• Weigh the mass of the cup, with the water still in it. 
• Stack another block on top of the block which is tied to End A of the string. 
• Repeat all previous steps. 

We repeated the experiment, adding one block to the system with each repeat until we had four blocks stacked. The data that we recorded is below, in Fig. 2. The free body diagram of the system is shown below in Fig. 2 as well.

Fig. 2 The recorded values of our block weights and the cup weight when filled with enough water to make the system move. This experiment was performed four times.

Even though we used LoggerPro to determine the coefficient of static friction, we also performed all calculations in order to double check (See Fig 2a and 2b).

Fig. 2a Calculations for the coefficient of static friction. The free body diagram and sum
of forces equations are all drawn on this paper.

Fig 2b The remaining calculations for the 3rd and 4th block trials. The average coefficient
was determined here as well.

Fig. 3 Data inputted into LoggerPro.

In order to graph this data to determine the coefficient of the maximum static friction, we needed to input this data into LoggerPro (See Fig. 3). The slope of the graph of the friction force and the normal force data gives up our coefficient of static friction. In this case, our coefficient of static friction was .305 (See Fig. 4).

Fig. 4 Slope of the line on the graph determines our coefficient of friction static in this experiment.

Part 2: Kinetic Friction

Fig. 5 Force sensor attached to wooden block and pulling it
 across the table at a constant velocity.

Next, we determined the coefficient of kinetic friction. We accomplished this by attaching a force sensor to a string to a block. We then pulled the block across a table at a constant velocity (no acceleration). (See Fig. 5). LoggerPro measure the force that was required to pull the block across the table. Fig. 6 shows the data inputted into LoggerPro and the graphs of the data. From this graph we can determine the average force required to pull each block across the table at a constant speed. We then took that data and made a new data table and graphed the data. From that table, (See Fig. 6a) we were able to look at the slope of the best fit line, which is the coefficient of the kinetic energy.

Fig. 6 These four graphs represent the four trials performed with the four blocks with the force sensor. 

Fig. 6 These four graphs represent the four trials performed with the four blocks with the force sensor. 

Fig. 6a All four pulls are graphed here and the slope of the line is the coefficient of friction - kinetic.

Part 3: Static Friction from a Sloped Surface.

The purpose of this part is to measure the coefficient of static friction, once again. Except this time, we will be using an inclined ramp to track the block. The process is simple. We placed a block with a felt bottom on a board and slowly raised the board until the block started to slide down. Once the block slid down we measured the angle at which the board was at (See Fig. 7 and 8).

Fig. 7 The angle is being measured of the ramp. The block slid down the ramp at this angle.

Fig. 8 The block sliding down the ramp.

The angle at which the block begins to slide is the angle we will use for theta in our sum of forces calculations. Start by drawing out a free body diagram of the set up. The sum of forces in the x-direction (parallel to the ramp) are going to be mg(sin(theta)) = the static friction, because we are looking for the amount of f(static) there is no acceleration to take into account for.  The angle we measured was 19.8 degrees and the mass of the block was 122 grams. With these two pieces of information we were able to perform the calculations (See Fig. 9) and determine the coefficient of static friction to be 0.36. Funny thing is that the coefficient ended up being just the tangent of theta... LOL

Fig. 9 The sum of forces calculations and free body diagram used to determine the coefficient of static friction.

Part 4: Kinetic Friction from Sliding a Block Down an Incline

The purpose of this portion of the lab is to determine the coefficient of kinetic friction of the block as it slides. Because we know that it takes more force to get something to move than to keep it in motion, we are expecting that this coefficient of kinetic friction will be less that what we discovered in part 3 above. The way we will determine the friction is by using a motion detector to track the acceleration of the block as it slides down. We will also record the angle at which the ramp is placed while the block slides down. Once we have those two pieces of information we can set up our free body diagrams and our sum of forces equations (See Fig. 10). The coefficient of kinetic friction for this experiment was 0.337, which is in line with our assumption from above. The kinetic friction force is less than the static friction force.

Fig. 10 The free body diagrams and the sum of forces equations and calculations.

Part 5: Predicting the Acceleration of a Two-Mass System

In this final part of the lab we will take the coefficient of kinetic friction from above, .337, and use it to determine an equation for the acceleration of a block being pulled across a table in a two-mass system exactly like the set-up of part 1 (See Fig. 1). The mass hanging from the string must be big enough to move the system (because we are using the kinetic friction). We will then measure the acceleration in LoggerPro and compare the measured acceleration to our calculated acceleration. The diagrams, formulas, and calculations are below (See Fig. 11). We determined the acceleration on the system to be -.087 m/s2. When we plugged in the motion detector to LoggerPro to measure the acceleration of the block as it slid across the table, the value for acceleration that we received was 0.9458, which is very close.

Fig. 11 The formulas used and diagrams used to estimate what the acceleration on the system would be. 

Conclusion:

We started by using the block-water pulley system to find the coefficient of static friction between the block and the table and graphing the force of static friction with the normal force of the block to find a relationship. The relationship was the coefficient of static friction.

Then we used the force sensor to calculate the kinetic friction of the experiment using the force graph of the recorded data. After that we found the coefficient of static friction of the wooden block on top of an inclined ramp. Next, we used the acceleration found from the block sliding down a steep ramp with the motion sensor to calculate the coefficient of static friction for the ramp. Finally we predicted the acceleration of the system by using the previous coefficient of static friction.

Lab 6 - March 11, 2015 - Propagated Uncertainty in Measurements

Lab 6:
Brandon Elder
3/11/2015

Propagated Uncertainty in Measurements

Purpose: To learn how to calculate the propagated error in each of our density measurements of three pieces of metal. An additional purpose is to calculate the mass of two unknown objects hanging from spring scales and to determine the propagated uncertainty in the calculated value of the mass.

Fig. 1 Calipers used to measure the length and diameter
of all three masses.

Gathering Data: Obtain a pair of calipers (see Fig. 1) a box of masses. Choose three of the masses and measure the height and diameter of all three. Put together the weights with the measurements in a table (see Fig. 2). After all the measurements are taken, weigh the objects on a scale to get the mass.

Fig. 2 Graph of three different masses. Aluminum, Copper,
and Lead.

Fig. 3 Formula for the
volume of a cylinder.

Fig. 4 Formula used to calculate
density. (D=M/V)

Calculations: 

Volume: First, calculate the volume for all three objects. Volume of a cylinder is: (pi * radius squared * height) (see Fig. 3). All calculations were recorded on our white board (see Fig. 2).

Density: The formula for calculating density is (mass / volume) (see Fig. 4).

Fig. 5 Known uncertainties in measurements. the uncertainty
in these calipers is .1 mm because the measurement will
always be between .1 mm increments. 

 Ok, that was the easy part. Now we have a table of data that has some numbers on it. But we must ask ourselves, how certain are we that these numbers are accurate? Anytime that a measurement is taken there will be some degree of uncertainty. For example, when we take a measurement, the object being measured will most likely fall between two divisions on our scale. The reading will come down to a judgement call on the user's end about which division the object is closer to and this is the uncertainty that is always present in any measurement (see Fig. 5). These uncertainties are called known uncertainties. It is a combination of known uncertainties in measurements that leads to unknown uncertainty in the final result.

Fig. 6 In the top of the picture we see the formula used to calculate density.
Underneath it we see the formula that will be used to calculate "dp" the
uncertainty in density.

Propagated Uncertainty: Propagate means to spread out. When we spread out known uncertainties we create unknown uncertainty. Uncertainty in density calculations comes from three different calculations, the mass, diameter, and the height calculations. We must create an equation for density that we can use to calculate the amount of uncertainty in our density measurement. We know that we have three uncertainties, m, l, and h. We also know that density = m/v. We need an equation that has all of our uncertainties in it. If we substitute the formula used to calculate the volume, we will have density = m/(pi*(d/2)^2*h. This simplifies to (4/pi)*(m/(d^2*h) (See Fig. 6).

Take the derivative of the function for each different variable, m, d, and h, respectively. Then, plug the numbers into the new equations and this will give you the "dp/dm", "dp/dd", and "dp/dh" portions of the equation (middle of Fig. 6). The formulas used to calculate these are in Fig. 7.

Fig. 7 Formulas used from the partial derivatives.

The known uncertainty in mass, diameter, and height are taken form the devices used to measure the numbers. The mass is .1 grams, the diameter is .1 cm, and the height is .01 cm (See Fig. 7a). See Fig. 8 and Fig. 9 for the calculations of all three object's unknown, propagated uncertainties. As you can see from our measurements, the calculated densities plus the propagated uncertainties are all within the ranges of the known values of their densities.

Fig. 7a Entire calculation with all formulas for propagated uncertainties.

Fig. 8 Aluminum and Copper Density with uncertainties.

Fig. 9 Lead's density with propagated uncertainty calculated.

Determination of an Unknown Mass:

Next, we are going to apply our knowledge of uncertainty to an unknown mass hanging from a pair of spring scales (see Fig. 10). We will need to measure the angles and record the spring scale readings (see Fig. 11). Then, we used the measured values to determine the mass of the unknowns. The formula used to determine the mass is written down in Fig 11a. The partial derivatives were taken (See Fig. 12) to calculate the propagated error for each variable: F1, F2, theta1 and theta2. Fig. 13 and 14 has the calculations completed for both unknown masses and the amount of uncertain error.

Fig. 11a Formula used to calculate mass is located to the
right of the diagram.

Fig. 10 Unknown mass hanging from
two spring scales.

Fig. 11 Measured angles and
Spring Scale readings.

Fig. 12 Partial derivatives.

Fig. 13 Unknown mass #2 with
propagated uncertainty formula.

Fig. 14 Unknown mass #1 with
propagated uncertainty.