16 March 2015: Coffee filters

Lab 5: Coffee Filters

Purpose: Today we used coffee filters to determine the relationship between air resistance force, as something is falling, and the objects speed as it free falls. We are using the equation Fresistance = kv^n which should support the relationship we are looking to find.

Procedure: Beginning the experiment, we dropped coffee filters from the second floor of the balcony in the information technology building at Mt.SAC. We used video capture in order to project the fall of the coffee filters by marking every position every 1/10 of a second. We did this five times and each time we increased the number of coffee filters until we did 5 coffee filters. The fall was then analyzed and recorded by logger-pro were we found our data that we were going to test out using excel by analyzing our answers.

This pictures above show our data recording on logger-pro(left-side) and the data taken with the different number of filters(right-side).


Logger-pro is able to make a graph by marking the distances after each time frame that the filter fell. Which is how we came up with the position vs. time graph below. We also used a curve fit to find the slope which will help us to eventually find the acceleration since we now know the velocity.

We then used the force and velocity information to make a graph that would help us toy solve for our k and n values. We would have then got several values and compared them with our values of the function we created by solving for our own k and n values in terms of coffee filters. When using excel the values should be similar if not then you most likely had an air in your experiment with measuring the the position of the coffee filters as they fell to the ground. 

Final Thoughts: All in all, the lab had some errors in calculations due to the lack of equipment , if the equipment was better our data could have been more accurate and since the filters were white they were very hard to see as they fell closer to the ground. We still were able to find a relationship between the force resistance of air and kv^n but even though our calculated value wasn't exact to the expected value they were very similar. 

11 March 2015: Propagated Uncertainty

Lab 4: Propagated Uncertainty

Purpose: The purpose of this lab is to understand how to mathematically solve for the propagated uncertainty of several masses. Then use this knowledge in order to calculate three masses of different densities and find the uncertainties of this density.

Part 1: We took three metal cylinders that are made of copper, aluminum, and steel.We then used a scale and found the mass of each object and used calipers which helped us to find the dimensions of each object which we would then use to solve for the density of each of the cylinders.

The picture above shows the materials we used, the three masses are steel,copper, and aluminum which are on the bottom left. The scale we used to find the masses is also at the bottom left. And the caliper is the measuring instrument also on the bottom left in the picture. All of our data was recorded and the next steps taken was determining the densities which we found by using the volume of a cylinder (pi*d^2h/4). 

This is how we solved for the densities of each of the cylinders. We can then begin to find the propagated uncertainty by then differentiating the diameter, height, and mass, we were able to add up those uncertainties to figure out the exact propagated uncertainty.

These calculated values will now replace the original uncertainties of .1g,.01cm, and .01cm.

aluminum: 2.61g/cm^3 +/- .03

copper: 8.73g/cm^3 +/- .12

steel: 7.66g/cm^3 +/- .11

Part 2: The second part of the lab was found by choosing a tension diagram made by the professor in class. Where we first found the able of measurement of tensions and then had to solve for the mass of the equation. 

This is the work above on how to solve for the hanging mass by setting up vectors directions for the tensions. I solved for the unknown ganging mass by solving for the mass in both the y and x directions of vectors. 

Final Thoughts: The main purpose of this lab was for us to practice our free body diagrams for part two and also being able to solve for an unknown mass given two tensions and a hanging mass. We learned in part one that most things have propagated uncertainty which we now know how to solve using calculus.

02 March 2015: Non-Constant Acceleration

Lab 3: Non- Constant Acceleration (Rocket/Elephant)

Purpose: The purpose of the lab is to solve an equation with a non-constant acceleration mathematically by using an excel spreadsheet.

Problem: There is a 5000 kg elephant on skates who is on a hill which is frictionless, as he comes to the bottom of the hill. The rocket is then thrusted on and lets out a force of 8000N(newtons) constantly, but it is pointed in the opposite direction of where the elephant is moving. The mas of the rocket is its mass before the rocket is on, 1500kg, minus the fuel which leaves the rocket causing it to lose fuel at 20kgs a second. Meaning mathematically the rockets mass is 1500-20kg/s*t=M(rocket).
The question is asking us to find the distance the elephant has traveled before running out of fuel or coming to rest.

Procedure: In order to solve for the equation we worked it out mathematically and we also used excel to start a spreadsheet in order to input different data and test our results to find a time to then find a distance that the elephant has traveled. Mathematically speaking we labeled all of our known values.
M(rocket)=1500kg-20t
elphant mass=5000kg
v(intitial)= 25m/s
masses together= 5000+1500-20t=6500-20t
We can use the equation of the masses to solve for a force value which gives us our a(t), acceleration function.

The picture above shows the integral of a(t)= Fnet/M(together) which equals F(t)= -8000/6500-20t(all functions integrated from 0 to t) a(t)= -400/325-t. Leading to the equation v(t)=25-400ln(325)+400ln(325-t). After taking the integral of this function we find the position equation x(t)= (25-400ln(325))t+400(t-325)*ln(325-t)-t+325ln(325), which we need to find the time he traveled in order to find the distance that the rocket and elephant traveled. We found that time was equal to 19.69 seconds by solving for velocity at V(0). We use this t value to then solve for the distance the elephant had traveled which approximated to about 248.7. To prove our mathematical work we used excel spreadsheet to calculate the a(t),v(t), and x(t) by setting up several different columns where we input equations with respect to what they represent. 

Each column was labeled,( t) represents time, (a) represents acceleration, (a-avg) represents the average acceleration on a given time interval, (deltaV) change in velocity on a given time interval,(v) represents velocity, (Va) represents the average velocity on a time interval, (deltaX) represents the distance traveled in a time interval, (X) represents the total distance traveled completely. We used a small time interval to guarantee accuracy. A was represented by the acceleration equation we integrated. A-Average was found by a(final)-a(initial)/2. deltaVelocity = acc*time.Velocity = V(initial)+ deltaV(change in v).Velocity (average)= Vf-Vi/2. DeltaX= A-avg*t. X is the position which is solved by plugging in each time for x(t) equation above.

Class data: In class we were asked to have a very large time scale in order to check our work so we had the columns go down an addition two hundred rows. By simply highlighting the row and the equation used to solve for the respective value we got around 230 new columns with new data. We continued to do so with every row until our entire data had each column down to about 230 rows each. 

The second way of solving is by going to the velocity column and checking where time is equal to zero because we want to know the distance that the rocket and elephant traveled after coming to rest when the rocket runs out of fuel. We find that our data matches!!!! The excel sheet calculated a 19.6-19.8 time window since we made our time interval so small were were able to see just what a tenth of a second would change. Our time we found was 19.69 seconds approximately which is between the time window we found on excel. 

Final Thoughts: Sometimes when solving numerically and analytically the resultant is equal making your working easy knowing that you did the correct steps mathematically to solve for the problem in this case a non-constant acceleration problem involving an elephant on skates thrusted by a rocket strapped to his back. The elephant traveled a distance of 248.7 m before coming to rest which we proved in both methods, numerically and analytically. The thing that allowed us to get such accurate data was the fact that our time interval was shortened to a tenth of a second allowing more data and more accurate numbers to be calculated. This lab helped to retouch on skills on excel to analytically solve something by first proving it mathematically. 

02 March 2015: Free Fall Lab and determining gravity(g)

Lab 2- determining free fall (g= 9.81m/s^2)

Purpose: The purpose of the lab is to prove the theory and validate the statement that, "In the absence of all other external forces except gravity, a falling body will accelerate at 9.8m/s^2.

Materials/ Procedure: The apparatus consisted of many parts including a long metal tripod stand which is used as a track for the falling electromagnet. Before beginning the process the wooden cylinder with a ring attached to it must be attached to the electromagnet, acting as a conductor for the electric spark to hit the paper strip. This electromagnet is powered by a spark generator which allows it to mark its distance as it free falls from the top of the tripod to the bottom. There is a long strip of spark paper which is zapped with around 60Hz of electricity leaving dark enough dots to see where it was zapped by the electromagnet. After the apparatus is done recording the dots zapped on the paper the first part is done now the mathematical solving and analyzing comes into play.

part 2 procedure:  Since we were limited to only one apparatus we were given strips that were made from previous classes in order to give us more time to work on the mathematical part of the lab by using excel. Every 1/60th of a second was the time frame used for the electromagnet to shock the paper strip. We determined a starting point of the paper and then measured the distances apart from each of the dots generated by the spark. The direction is shown but can also be found by analyzing the distances from left to right as they get larger. We were then asked to make at least 20 or more data points that would be plugged into an excel sheet and used to solve mathematically.














 









The table to the right is a list of the data we took. The 1st column is representational of time which in our case we took every sixtieth of a second which explains why we got such small decimals for time in column 1. The 2nd column is the distances we recorded between each point of the falling electromagnet the first point we recorded being the origin. The change in distance is represented by the 3rd column which means how much more it traveled from the previous distance. The 4th column is when when we doubled our time frame for the shock to hit the strip which is every hundred and twentieth of a second. Which would be the midpoint between two of the sixtieth of a second intervals. The 5th column was the average velocity of the falling object on a hundred and twentieth of a second time interval. Using this data we set up a time vs distance graph.

Distance vs. Time

We found out by using curve fit that the line is representational of a parabola. This equation is almost identical to the position equation used in physics 2A (x(t)= Xo+Vot+1/2at^2 ). By using this graph we can now have an equation where we can derive the v(t)(velocity vs. time) equation and also derive the a(t)(acceleration vs. time) equation which will give us an acceleration of 9.56m/s^2. This differs from the already given value of g= 9.8m/s^2. This tells us that somewhere along the way we had error or the computer may have miscalculated. 

Velocity vs. Time

We derived the acceleration from here by taking the derivative of the equation given. 

V(t)= 954x+42.529

Giving us a g value of 9.54m/s^2 which isn't the g value of 9.8m/s^2.

Class data: We did an entire graph together as a class and found that the average class values were also off from the actual value of g (9.8m/s^2). The picture on the bottom shows the first column as the solved g values for each group.  The second is the difference between the class average at the bottom of column one and the actual deviations of groups. The third column shows the average of the second column squared and square rooted to find the mean of the deviation from the class average to their solved g value. The class average was about 20.12. 

We were able to find the range of the g value as a class which was between 916 to 996 giving a possible error of about +/- 2. 

Given by the lab packet, we found the percentage of it being between our data was around 95 percent. An error could have been made with measuring accurately causing our numbers to be off by a few tenths. 

Final Thoughts: Although we did not solve an exact value for g we did get fairly close in our data and identified a relationship using excel spreadsheet to solve for the g of our strip paper. Our class average of about 950 and our groups value of 956 differs in the excepted value of 980. Which gives us reason to believe there was a small error in recording the data causing our numbers to easily differ from group to group. We learned that by doing this lab we will run into some trial and error after analyzing our data. We also retouched on our skills with excel spreadsheet and how to input equations to get the results, although we didn't solve for g exactly we did learn about excel and how solving mathematically may not always get you the exact value, but doesn't mean your wrong. 

23 February 2015- Inertia Balance Lab

Lab 1: Inertia Balance

Purpose- In class we used a known mass and c- clamp to measure the period of an inertia balance, as we added weight(kg) to see how it affected the period(t) of the inertia balance. We found that they mathematically share a relationship within the graphs and will use the data from logger pro to prove that the data matches a power law equation which best fit the curve of the graph as we analyzed on logger pro.

Materials: Inertia balance, masses 0-800g, eraser, calculator, c-clamp,LabPro, and Photogate.




Procedure: The first part of the procedure was to set up the c-clamp stand and the inertia balance so we could use a photo-gate as a motion detector. This helps keeping track of the period by picking up light rays as the pendulum goes through its complete oscillation the flashing red light on the photo-gate acts as the sensor  showing the time it takes for it to go back forth( one cycle). We did this eight times adding 100 g masses for every new trail we did. After recording the data from logger pro we decided to also find a curve fit to see how we would go about solving the equation of the inertia balance with increasing masses. We found that a power law equation fit best. We recorded the data of the inertia balances period(seconds) with respect to the mass(0-800g).

There are also two random objects we found and measured there masses. These two masses we ended up solving for there periods as if they were in our graphs by using the power law equation which fit our graph almost perfectly. By finding there periods we can analyze our data and see if they match up.

 This photo above shows all the work we did in order to find the time of calculator and the eraser.
Equation: lnt= nln(mass+Mtray) + lna
T for the period was equal too T= A(low/high)(m+Mtray(low/high))^n(low/high)
The low is representational to the lowest period with the lowest mass and high is the highest period with the heaviest mass. We solved for the slope of the equations first so we could find there periods. After doing so we used the period to estimate the lowest and highest possible masses they could be. by doing this we were able to later analyze our data and find that we had an error.

Data: All in all our data table worked out nice but there was an error in calculations due to the fact the we were probably not being consistent with the amount of strength we used on the inertia balance or a miscalculation done by the computer. By working out the answersby hand of two random masses we were able to compare it with the data from picture 2 (chart) which showed that our masses did not fit into the correct period.

Final Thoughts: We found the relationship between masses and its period depend highly on the amount of mass you use. The more you use the longer the period of the object will take in order to complete a full oscillation. We mathematically solved an equation that roughly calculated the mass of an object with respect to its period.