02-Mar-2015: Non-Constant acceleration problem solved on excel

Purpose: We will understand the convenience of how excel can calculate solutions from difficult and tedious problems.

We were given a scenario. A 5000-kg elephant on frictionless roller skates is going 25 m/s when it gets to the bottom of a hill and arrives on level ground. At that point a rocket mounted on the elephant's back generates a constant 8000 N thrust opposite the elephant's direction of motion.
The mass of the rocket changes with time (due to burning the fuel at a rate of 20 kg/s) so that the
m(t) = 1500 kg - 20 kg/s*t

Find how far the elephant goes before coming to rest.

By hand

Above image: Drawing a picture enabled us to better view the problem. Gathering the given information, we rearranged our force equals mass multiplied by acceleration. Solving for acceleration we came up with our acceleration function of time "a(t)". With a(t) we integrated the equation so that we can get our velocity function of time "v(t)". 

 Above image: Integrating again with the v(t) became even longer because given the way the function was we had to integrate by parts. Doing so eventually lead to our position function of time "x(t)".

With excel

Above Image: We made eight columns; time, acceleration, acceleration average, change in velocity, velocity, average velocity, change in position, and position. Utilizing the functions of excel we where able to essential set up a calculator. B1 was our time interval we set to have excel calculate our values we are searching for. Row 3 is the initial of time, acceleration, velocity and position. B3 we imputed our a(t) we figured out.


The idea of excel is to get the boxes small enough and find a better value.

Left Image: If we image a graph of A vs. T and change in time is small enough. We can approximate the curve as a line. This will give us a trapezoid for a nice formula in finding our velocity. The area underneath the curve will give us the change in velocity. The formula for change in velocity is a result of average acceleration multiplied by change in time. Imagining a graph for V vs. T will similarly follow our last imagined graph. Underneath the V vs. T curve gives us the change in position between the interval.

Right Image: Without integrating, (highlighted in red) excel gave us our answer for the elephant's position before coming to rest. 248.69m

Conclusion: Comparing the way to solve the elephant scenario by hand and using excel is vastly different. Allowing excel to calculate the solution ultimately will be faster and just as accurate if not more. Choosing a small time interval will yield a closer result to the solution by the hundredth decimal place. As you can see from the image above rows 197 to 200 do not change at the hundredths decimal place. Thus finding the velocity as it reaches zero or close to (0.118884) tells us when the elephant is coming to rest. If you come across numbers that do not make sense it could be an error in applying the correct value in excel. A way to tell is if you are given negative signs for a scenario that does not apply.
The logic behind using excel for calculations is to simplify the work load. Typing initial values into the spread sheet is broadly simpler than going through the integration by hand. Ultimately the accomplishments of allowing a computer program to do your calculations will save more time and importantly give you more time to do more.

02-Mar-2015: Free Fall Lab (Learning Excel)

Purpose -  In this lab we will determine the value of gravity while using excel to analyze some data.

The sturdy column provides a 1.5 m falling distance. This device will plot points on a film of paper as a metal piece free falls perpendicular to the ground. We will use the data to make a distance versus time graph and a velocity versus time graph.

The film of paper had dots generated by sparks that we measured with a ruler. At the beginning the dots we close to one another. As we measured the dots they became more spread apart. Each dot was 1/60th of a second apart from one another.

We logged our time, distance, delta x, mid-interval time and mid-interval speed onto excel. With excel we are able to use the program to calculate and input the data in a timely manner. In the time column A3 we entered =A2+1/60 and then dragged the corresponding cells to have excel input the numbers for us. For distance the data was from the paper film we measured. Delta x the formula was =(B3-B2). Mid-interval time was =A2+1/120. Mid-interval speed was =C2/(1/60).

Velocity vs Time
Graphing our speed versus time, we highlighted our data from columns D and E. Selecting the XY scatter graph we were able to obtain what is showing on the left. Using excel to graph our data also allowed us to gain our slope by adding a trend-line in the options.

Mid-interval is the point between each of the 1/60th of a second.

Position vs Time
We obtained the position over time graph the same way as above. The difference is we highlighted our data from column A and B.











            You can see that constant acceleration is shown on the velocity over time graph by looking at the trend-line. The line moving in a straight line upward indicates that the speed is constant. When the speed is constant we know that acceleration is zero. The times at which we have acceleration is the moment the metal piece dropped and when it stops. In this scenario it seems we are trying to determine that gravity will ultimately be our acceleration value.
            To gain the value for acceleration is velocity over time. 

            To gain the value for acceleration from our position over time. 

Column A we took the g values from every group and in the red obtain the average of them all. (=average(A2:A11)
Column B was the deviation from the mean.
(=A2-$A$12 the dollar sign locks in the value we want.)
Column C is the deviation squared. (the symbol to square is "^")











       The pattern of all the values were all lower than the actual value of gravity. Same with the average value of gravity compared to the actual. They are lower due to the air resistance from the metal piece free falling or maybe the sparking from each dot gathering may have caused a slight friction force.
      Systematic errors in experimental observations usually come from the measuring instruments. The calibration of the sparks to annotate the dots could be a factor for this experiment. The ploting of our graphs in excel could be involved with these types of errors.
      Random errors in experimental measurements are caused by unknown and unpredictable changes in the experiment. These changes may occur in the measuring instruments or in the environmental conditions. Air drag due to a gust of wind from the air conditioning system could play a role in determining points. A surge of electricity from the equipment being quite old.

       The value of this lab is to proficiently use excel. If some of the student did not quite understand the commands excel provided, this lab fixed that. Allowing us to find gravity and use excel to assist is another highlight to how this program is efficient. By plotting graphs to obtain an equation that gave us slope. The calculation that excel can do for the students is convenient. On top of all the nice features excel can do for us. It is imperative to understand each of the calculations and the meanings of graphs and the errors that may impact data.

23-Feb-2015 Deriving a power law for an inertial pendulum

Purpose - The purpose of today's lab is to find a relationship between mass and period from an inertial balance. We gather the period during the oscillation of a horizontal tool by adding mass. We are to record 9 trials when adding an addition 100g mass for each trial beginning with 0 mass. During this lab we obtain a power law equation for an inertial pendulum so that we can determine the 2 unknown masses with a low and high guess of our mass of the tray.
T=A(m+Mtray)^n

We taped the end of the apparatus so the photogate sensor will read each oscillation. At the end of the device we add 100g weight for each trial. At each trial we slightly pulled the device so that it may sway back and forth until we felt comfortable to begin collecting data.

Here is one of our nine data collections we obtained during the experiment. At this moment we just started and had one 100g weight added to the device.

This graph (T vs M) shows the raw data being applied to a curve fit diagram. You can see our first record period with no mass was 0.286s. Each record we added 100g mass.

This graph (lnT vs ln(m+Mtray)) had the mass of the tray added into the expression. We also add several columns; total mass, ln T, and ln (m+Mtray). Then we received the curve fit equation. For our correlation to become 0.9999 we gave a mass of 250g to our tray. Which also ended up being our lowest guess for mass of tray. For our highest mass of tray we assumed 310g. In finding these masses we maintained our 0.9999 correlation.

From the slope we obtained our "n" value. With the y-intercept we were able to translate our "A" into e to the power of our y-intercept. Plugging in the period from our unknown mass we were able to calculate a range that the actual mass may be.





















Given this experiment our group was able to plot points that gave us key values into using period as a means to obtaining information that was unknown to us. This helped understand the concept of deriving a power law for an inertial pendulum and the relationships involved. Though some errors may have interfered with a true value. For example, the placement of the masses during our nine trials, the clocking of the oscillations, how far the apparatus was pulled each time may have been different, old and used sensors, and other outside sources that may impact the experiment. Overall we manged to obtain the unknown mass using the period from our apparatus.