Tuesday, March 31, 2015

03-Mar-2015: Trajectories

Purpose: The purpose for this lab was for us to use our understanding of projectile motion to predict the impact point of a ball on an inclined board.

Procedure: Materials required to perform this lab included an aluminum "v-channel", steel ball, board, ring stand, clamp, paper, and carbon paper. The set-up required that we make an initial incline ramp connected to a horizontal ramp on a table which shoots the steel ball off of the table onto the floor. After witnessing where the ball seems to hit the floor, we taped a sheet of paper to the floor and layered it with the carbon paper, which is used to make a mark wherever an impact occurs.
Ramp set up on table with carbon paper on floor
You can see the lab apparatus in the picture above and the close up of just the ramp below. We taped the ramps together with regular masking tape. Then we measured the angle of the incline and found it to be 31.5 degrees. We also measured the height from the end of the ramp and the floor and after 5 separate runs with the steel ball from the same starting point, we measured the average horizontal distance the ball traveled from the end of the ramp. These variables are important pieces for our calculations we do solve for the velocity of the steel ball as it exits the ramp.

closer view of ramp 31.5 degree angle 
After 5 runs you can see the 5 separate marks made onto the carbon paper by the steel ball's trajectory.
carbon paper with 5 separate ball marks

removed carbon paper
Results/Analytics: After removing the carbon paper we can find the dots which were all relatively within the same spot. We took the average distance as our value for x in our calculations for velocity. Using our known values, we were then able to create a system of equations that allowed us to solve for an initial velocity of 1.55 m/s.

Calculation for finding velocity(initial) off table
After finding the initial velocity of the ball as it exits the ramp, we had to put a board inclined at the edge of the table, such that now the ball would hit some point on the board instead of the floor. That distance from the end of the ramp to the impact on the board is our distance d. We were tasked with deriving an expression that would allow you to determine the value of d given that we knew v initial and the angle of the inclined board and the floor.


Deriving distance on board with propagated error
calculations for individual derivatives

our analytical results compared to actual measured

Our predicted value of 0.8143 plus/minus 0.0155 cm did not catch the 0.857 measured distance of the mark on the board from the end of the ramp by a few hundreths of a cm.

Conclusion/Uncertainty: Through this lab, we were able to derive the initial velocity of a steel ball as it left the end of a ramp off a table by measuring the horizontal and vertical distance it traveled and the angle of the incline that provided the steel ball with a force. After that we could calculate the exactly where the ball would land if a board was in the way of the floor as long as we were given an angle of incline on the board. Although we did not get the distance to match our range of distances, the results were not too far off and reaffirm the precision of the analytical data and its relation to the physical world.

As far as uncertainty goes, there are several factors that we do neglect like the release point of the ball not always precisely the same every time. Also keeping the carbon paper on the board was a little harder to keep stable. I dont think air resistance makes a difference in our results, it was more about understanding projectile motion in two dimensions. Of course lab equipment has certain limits which is why we rely on propagated uncertainty in order to maintain integrity in our end results.

Thursday, March 26, 2015

16-Mar-2015: Modeling Friction Forces

Purpose: For this lab we had 5 different experiments that all involved friction. We wanted to physically see the properties of static and kinetic friction in different systems.

Part 1: Static Friction

Procedure: Static friction is the friction force acting between two bodies when they are not moving relative to one another. For this part of the lab we are going to measure the maximum static friction of a block on a table top being pulled by cup of water on a pulley. We kept adding water to the cup until the weight surpassed the maximum static friction from the block. The threshold of motion in the experiment can be characterized by u(static) = f(static, maximum)/Normal. We weighed a block with felt on one side, tied a string on it and and put the string over a pulley to a paperclip attached to a cup. We patiently added water to the cup (very little at a time) until the block started to slide. Then recorded the mass in of the cup required to move the block. We weighed another block and added to the first block and repeated the steps all the up to four blocks.

lab apparatus with four blocks


Results/Analysis:  
Our results were put into a data chart and converted into a friction/Normal graph. When we applied the linear fit you can see that the slope of the linear fit is very consistent with the points on the graph. The slope of the graph is our coefficient of static friction. Below you can see that our u(static) was A = .2912, which is a reasonable number for a coefficient of static friction.

Graph of friction (static)/ Normal force. also data table and slope
Part 2: For part two we wanted to measure the kinetic friction of an object by building a model that is in motion. The kinetic friction of an object can be measured with the equation f(kinetic) = u(kinetic)/Normal. In our model, the kinetic friction force is a fixed value for a given N, regardless of the speed of the motion. Like static friction, the coefficient of kinetic friction is not dependent on weight or area of contact, but the on the surface material.

For this lab we needed a force sensor tied to a block, and required us to apply a pull force onto the block with a constant speed along the surface of the table. First we calibrated the force sensor by hanging a 500 N mass on the end. Then we measured the mass of the wooden block. with a horizontal pull applied, we hit collect and and stored the run under from the experiment menu. Using the analyze menu and choosing statistics, we recorded the mean value of the pulling force for the block while it moved horizontally on the surface. We then repeated the steps with a second, third, and fourth block.


4 Blocks tied to a force sensor

After saving our four runs, we were able to find the mean value of the pulling force so that we can plot the friction force over the normal force in another graph.
Graphs of force/time for the four runs
Below you can see we were able to find the coefficient of kinetic friction by using the linear fit and looking at the slope of the graph.


graph of kinetic friction/ Normal force
Part 3: The next part of the lab required us to put the block on a dynamic surface where we could have it horizontal initially and slowly raise one side of the surface until the block breaks static friction and slides down the surface. We carefully measured the angle at which the block surpassed the threshold for maximum static friction with our phone app. We were then able to solve for the coefficient of static friction of the block using the mass of the block and the angle theta of the slope. Notice how our result is similar to that of the static friction for the first block in the first experiment.

Diagram of experiment of static to static friction


Part 4: Here we wanted to measure kinetic friction from sliding a block down an incline. Using a motion detector fixed at the top of the ramp, we recorded the position of the block as it slid down the ramp over a period of time. we measured the weight of the block, the angle of the incline and also found the acceleration of the block as it slid down the ramp. We found that the acceleration of the block was .4164 m/s^2 at a 19.3 degree incline. 


block sliding down incline with motion sensor
Below shows our calculations that we did analytically using our values of mass and the angle of the incline. We wanted to find the answer for the coefficient of static friction analytically and compare it to the results on our graph .

calculating coefficient of kinetic friction

Part 5: The final part of our lab required us to predict the acceleration of a two mass system. Using our coefficient for kinetic friction from part 4, we were tasked to to derive an expression for what the acceleration of the block would be if we used the hanging mass sufficiently heavy enough to accelerate the system down.

lab setup for two mass system
We drew the system up in order to understand the forces being applied to the block so we can solve for the acceleration. we created an equation and plugged in our values and figured out that theoretically our acceleration of the mass should be around 0.459 m/s^2. We want to compare this answer to what we get from our graphs formed by the motion sensor and logger pro.

calculating acceleration of the block
graph of position/time and velocity/time. shows slope which is the acceleration
Above are the results of both our calculated results and the results from the motion sensor. Our calculated answer was only 1/100ths of a decimal off, which showed that we were very accurate with our prediction.

Conclusion/Uncertainty: Through this experiment, we were able to confirm friction in action and relate our calculated results with results from our lab equipment. Our results showed that the coefficient of static friction is always greater than that of the kinetic friction. This is inline with how it should work theoretically and in the real world because it should be harder to move something that is still than to move something that is already in motion. Also the coefficients of friction are always under 1 because it the friction was greater than the normal force being applied, the system would not move. Our results also proved that the coefficient of friction is only affected by the surface of the objects and not the speed of the object or the weight. 

Uncertainties in our lab experiments include things such as our measuring tools, which are not precise to the third or fourth decimal place. also the table at times could very in smoothness and texture. Things like the air resistance are ignored when pulling our objects down inclines so our answers are not exact.

09-Mar-2015: Propagated uncertainty in measurements

Purpose: The purpose of this lab was understand the meaning of propagated uncertainty and applying the idea by comparing measurements and results from the experiment with accepted values from the scientific community.

Procedure: This was a two part lab, the first part being calculating the propagated uncertainty in our measurements of density of metal cylinders. The second one being the determination of two unknown masses. But before delving into the two experiments, we must first understand the term propagated uncertainty. Propagated uncertainty is how uncertainty in measurements leads to uncertainty in final result. Basically, the more unsure or unspecific you are about your measurements, the even more uncertain you will be about your result. It is sort of a compounded effect of error leading to a sort of range for the possibility of an outcome. Its pretty similar to the idea of significant figures because the name of the game is accuracy.

Part 1: For the first part, we were given three metal cylinders and were asked to use a calipers to measure the cylinders' height and diameter, and a scale to measure density. A caliper is a similar to a ruler but more precise. it can clamp the ends of the object you are measuring or go inside to measure things such as the inside of a tube. The vernier caliper measures down to 1/100th of a cm so the uncertainty in our measurements is just that. The electronic scale we used measured down to the tenth of a gram so that is the level of uncertainty for our mass measurements.

Lab equipement(scale, masses, caliper)
We created a data chart showing the height, diameter, and mass of the three metals iron, zinc, and copper.
Data table of measurements of metal cylinders
We used the formula density = (4*m)/(pi*diameter^2*height) to find the densities of our metals. We then needed to calculate the range of result by finding our propagated uncertainty. In order to find the propagated uncertainty of our density, we had to find the derivative for each variable within the equation of density first. Then the propagated uncertainty for our density dp = | dp/dm |*dm + | dp/dh |*dh + | dp/dd |*dd. where dm, dh, and dd are the propagated uncertainty of the mass, height, and diameters. 

propagated uncertainty of Fe
Propagated uncertainty of Zn
Propagated uncertainty of Cu

Result/Analysis: The actual accepted densities are here. Fe is 7.874 c/cm^3, Zn is 7.14 g/cm^3, and Cu is 8.92 g/cm^3. So we were not within the range for any of the three metals, although the ranges were not that wide. This shows that even with very precise tools, it was difficult for us to get precise measurements.

Conclusion/Uncertainty: The first part of the experiment showed that we as humans are not perfect in measuring objects and that it is important to set a range for uncertainty in order to maintain integrity in all of our experiments. Without having that integrity to say when there is a certain uncertainty in our calculations, our evidence is not strong. In order for our experiment and results to be part of what the scientific community can call evidence we must be accurate in our measuring and experiments.

Part 2: For the second part of the lab, we were asked to determine the mass of two unknown objects.
We had an unknown mass hang on a string that was tied to two separate spring scales. these spring scales, 10N, had to be adjusted before putting the masses on so it would read 0 with no mass hanging. The mass hangs down and we made sure that the springs were held at asymmetrical angles as shown in the lab.

Lab Apparatus with scales and hanging mass
tool that measures angles
We measured the angles of the string and recorded the readings on the white board. We then estimated the uncertainty in our angle readings and scale readings. We used the measured values to determine the mass of the unknown mass.
measurements of readings
Using the formula for forces we calculated the mass with our measurements and also the derivatives in respect to each variable.

calculation for derivatives of forces and angles
Plugging in the values, we were able to solve for what the mass should be with the range of propagated uncertainty.

propagated uncertainty for first unknown mass

Results/Analysis: Just like the first part of the lab, the second showed that it hard to hit within the range of the propagated error, even with careful measuring techniques. Our calculations of the mass using our values proved that we were off a little from the actual measured mass using a scale. It is also understood that the equipment we used is also not 100% reliable.

Conclusion/Uncertainty: The final conclusion through this two part lab showed just how important it is to solve with a propagated uncertainty to insure quality of our data. It was very hard with our experience and our tools to perfectly match the accepted values but the gap was also not too far off. Just enough that it was outside the range of the propagated uncertainty. Again, the importance of a lab like this is to show just how skewed a final result can be from a few hundredths of a cm or N in prior measurements. Also how our experiments mean nothing to other scholars if we don't follow strict guidelines to insure to quality of our data. The integrity of our experiments determines the quality of our results and how accurate they really are.

Wednesday, March 25, 2015

04-Mar-2015: Non-constant acceleration problem/ Activity solution

Purpose: The purpose for the Non-constant acceleration lab was first, to solve the given non-constant acceleration problem analytically. And second, to show how instead of struggling with complex equations, that by using a spreadsheet, we can easily organize and calculate the answer numerically.

Procedure: We were given a problem involving the non-constant acceleration of a 5000 kg elephant on frictionless skates going 25 m/s down a hill and then on ground level. Then a rocket mounted on the elephant's back generates a constant 8000 N thrust opposite the elephants direction of motion. Where the mass of the rocket changes with time so m(t) = 1500kg - 20kg/s*t. We wanted to know how far the elephant travels before coming to rest. Using Newtons second law and integration skills we aimed to derive the answer analytically.

Below is the analytical process required to get from the given variables to answering the question of how far the elephant traveled. Newton's second law gave us the acceleration of the elephant + the rocket as a function of time. Integrating the function from o to t to find change in velocity and then derive the equation for v(t).  Integrating that equation form 0 to t to find the change in x then deriving an equation for x(t) using that equation. After simplifying the equation using integration by parts we get a cleaner equation for x(t). In order to find the distance of the elephant we have to find t when the velocity reaches 0. Plugging in that time to the x(t) equation gave us x =248
.7 m. 
sideways image of problem setup and integration
This long, strung out process can be very time consuming and luckily for us we are able to do this problem numerically with the help of Microsoft Excel.

First we opened up a new Excel spreadsheet and made a series of columns such as change in t, a, a avg, change in v, v, v avg, change in x, and x. We began setting up the time increments by 0.1 for at least 220 rows. Using a change in t small enough was crucial in finding area under the curve by approximating a straight line. We assumed the function of the acceleration as 2t^2. Then we input a formula into cell B2 so that we can calculate the acceleration at any given time. Fill the formula down. Next in cell C3 we calculate the average acceleration for that first 0.1s interval. In cell D3 we calculated the change in velocity for that first time interval. (Using A3-A2 for the time rather than 0.1 s, so we could calculate this stuff using different time intervals if we chose.) In Cell E3 we calculated the speed at the end of that time interval. In cell F3 we calculated the position of the elephant (x at +v interval *change in t). we then filled down the data of row 3 to the rest of the spreadsheet in order to find when and where the elephant came to rest.


Results/Analysis: The spreadsheet gave us an answer that was extremely close to what we came up with analytically, proving that we can use a spreadsheet as a shortcut to solve messy problems. Two separate processes were used to solve the same problem and arrived at virtually the same answer.




Data table 

Conclusion/Uncertainty: This lab really showed us the logic of using a spread sheet to solve problems numerically rather than analytically in order to simplify a complicated process of integration and save time doing so. Plugging in simple equations into Excel spit out answers for an entire column of data. We were able to accomplish our goal of reaching the same conclusion using two separate processes and seeing how easy it is to use Excel to solve complex problems such as a non-constant acceleration.

A problem like this does not have much uncertainty because we are not physically measuring anything. The problem was to solve a complex problem both analytically and numerically using a spread sheet and comparing the results. Some error can occur in both processes but the coming up with similar results from both processes proved that an error, if any, was negligible.









02-Mar-2015: Free Fall Lab - determination of g (and learning a bit about excel) and some statistics for analyzing data

Purpose: The objective of the free fall lab was to examine the validity of the statement "In the absence of all other external forces except gravity, a falling body will accelerate at 9.8 m/s^2". We also wanted to mess with Excel in order to get more comfortable with the program.

Procedure: In order to demonstrate the motion of the falling object, determine g, and study the basic laws of motion, we used an apparatus that provided an object 1.5 meters falling distance in order to get an accurate reading. When the object is released from the electromagnet, its fall is recorded by a spark generator that puts marks on a strip of spark sensitive tape. The marks, which are made at consistent time intervals, were used by us students to record distance/time and velocity/time graphs. Once recorded, we can begin to understand the system and how gravity acts on an object.

So using a two-meter stick we measured the distance between the dots and the 0-cm mark. We then opened up Microsoft Excel and began plugging in the data starting with the time intervals which were 1/60 of a sec apart. Using the names of the cells (for example cell A1, A2, A3), we were able to short cut using the Edit/Fill/Down menu and let excel fill in the cells below. Next we inputed distance in cell B1 and the data for distance under in the B column. For cell C We entered change in x. ( getting the triangle symbol by holding option and pressing j). By putting =(B3-B2) in cell C2 and using the edit/fill/down menu you can quickly input all the data for that column. In cell D1 we entered Mid-interval time and fill downed =A2+1/120 to give the time for the middle of each 1/60th's interval). In cell E1 we input Mid-interval speed and fill down from E2 =C2/(1/60).

Our data table showing time, distance, change in distance, mid interval time, and velocity.
Next we wanted to graph our data in columns D and E. We clicked on the Chart tab and chose scatter then Marked Scatter to five an XY (scatter) graph with point not connected. Using Chart Layout tab, we can give the appropriate titles with units for the graph and graph axes. Then under Trendline we chose Trendline Options to linear fit the graph. We then took the same steps for the A and B columns except we used a polynomial fit of order 2.
Velocity and Mid-interval time graph

Distance over time graph

Results/Analysis: The experiment proved that gravity was accelerating the object at a constant 9.8 m/s^2 with pretty good accuracy and agreed with the theoretical prediction because we got 9.27 m/s^2. We can get the acceleration due to gravity from our velocity/time graph by finding derivative of the equation of the graph or using the slope of the graph at any given time. Getting acceleration from the position graph takes two steps of derivatives.

Conclusion/Uncertainty: Our values were pretty close but not exactly the numbers we suspected. this is due to the fact that there were certain assumptions we made in our lab. Things such as our measurements and ignoring factors such as air resistance and friction play a role in skewing our results. Although we assumed these things to be too minuscule to take into account, they did play a small factor in deriving our results. Our result of 9.27 m/s^2 which is our experimental value, divided by the expected value (9.8 m/s^2) multiplied by 100% give us the relative difference. There was a little uncertainty in measuring our distances because of human error and our measuring apparatus to give us measurement= best estimate + or - some uncertainty. Systematic error and random error have an effect on the experiment if you don't account for all variables in the system. This is why it is important to take note of all the uncertainties and errors because it is crucial to maintain integrity in our experiments. Without proper documentation, the experiment has no real value for the scientific community.

11-Mar-2015: Modeling the fall of an object falling with air resistance

Purpose:

Monday, March 2, 2015

23-Feb-2015: Finding a relationship between mass and period for an inertia balance


     For our first lab we were given the task to understand the relationship between mass and period for an inertia balance. This procedure required us to use an Inertia Balance, which measures inertia mass by an object's resistance to change in its motion. The objective was to pull and release the inertia balance so the computer could record a reasonable period for the oscillation of the device. Data was recorded with no mass on the tray and then repeated 8 more times, adding 100 grams to the tray each time.

image of inertia balance with added mass on tray
The image above shows the set up of the apparatus clamped to the table and a photogate clamped on the opposite side that recorded the oscillation of the device as it passed through a laser sensor.

period of oscillation being recorded into logger pro
The data was compiled into a data chart where we could see the relationship between the addition of mass and the length of the period for the oscillation. The data was then graphed. We were also given the task to measure the period of two unknown masses in order to see how precise our data was.

graph of mass and period 
Given the equation T = A(m+Mtray)^n, we took the natural logarithm of both sides in order to get an equation that looks similar to y=mx+b. we then plotted the ln T and ln of (m +Mtray) in order to determine the values of the slopes and y-intercepts.

data chart of mass, period, and total mass. also graph of ln of period and ln of total mass
Finding the values of the unknowns in the equation gives us the mathematical model for the behavior of our inertia pendulum. Using this equation, we were able to estimate and verify the masses of the two unknown objects by plugging in their respective periods of oscillation.


high and low calculations of light unknown object and actual mass

high low calculations of heavy unknown and actual mass
The results showed that the estimated range of masses of the unknown objects were nearly consistent with the actual masses of the objects. The lab procedure proved that an increase in mass on the inertia balance equaled greater periods of oscillations.

As a side note, there were certain uncertainties that came along with the lab procedure. For one it was difficult to get consistent pull and releases on the inertia balance due to slight human errors. Even distribution of the mass and consistency of stability of the mass were also slight issues. Also, we had to find a range of masses for the unknown objects due to uncertainty in the equations we came up with  for the relationship  between mass and period.