Lab #3 Non-Constant Acceleration Problem

Purpose: To use excel to solve problems for us numerically because some physics problems are extremely difficult to solve analytically if it is even possible at all.

The problem we are to figure out is this:



To figure out how far the elephant goes before coming to rest we can use calculus to integrate the acceleration finction into a velocity function and further integrate that to come up with a position function.



Though the functions were able to be integrated, which is not always the case, we are now left with an incredibly hard functon to solve. Using calculus we can solve for the time when the velocity of the elephant equals zero. We then plug that time into our position function to come up with x=248.7m



Instead of going through all that trouble we can use excel to set up columns for time at and calculating the acceleration, average acceleration, change of velocity, velocity, average velocity, change in position and distance. We have to choose a very tiny time interval to make sure we do not miss the pont where velocity is closest to zero. We used a time interval of 0.1 seconds.



We then allowed excel to make as many rows as necessary to find the row where our Velocity is closest to zero.

 

We then followed the same process but simply changed the time interval we used to

0.05sec


0.2sec


1sec


Comparing our numerical results to our analytical results we can see we are right on the mark with our excel process.

01-Mar-2015: Free Fall Lab

Purpose: To experimentially calculate a value for the acceleration of a free-falling object due to gravity and use some statistics for analyzing our collective data.

This apparatus, allows a free-fall body to fall a distance of 1.5m while a spark generator zaps it every 1/60th of a second leaving a mark on the spark sensitive paper that lines the column.

Each lab group then recieved a seperate ribbon of spark sensetive paper that was used for a trial.

Since we know gravity accelerates the falling object, the direction where the consecutive marks are getting farther apart is our direction of fall. Each group then located their 1st mark(origin, x0) and measured the displacement from each of their following marks to their 1st mark. Using excel we created columns for time, total distance fallen, displacement from point to point, mid-interval time, and mid-interval speed. We then plotted a graph of Mid-interval speed vs. Mit interval time and set up a liner fit which gave us our slope(experimental acceleration due to gravity in cm/s^2).

We then compared our expiremental value of gravity with the other nine lab groups. We listed the ten values of g in one column and underneath had excel calculate the average. In the second column we found each values deviation from the mean and in the third column we squared each groups deviation. Finally we calculated the average of the squared deviations and square rooted that value to find our standard deviation for our values of gravity.

Each groups calculated a seperate value of g with no observable pattern possibly due to random errors. However, if we as a class omit my groups value of g the deviation would have been much smaller because ours was the most outlying value. I believe this might have been caused by the spark generator not being set to exactly 1/60 of a second but something slightly less making each one of our points farther apart then they should have been. Taking a look at the class' average value of g (942cm/s^2) compared to the accepted value of g (981cm/s^2) our value is still quite low. This may be due to systematic errors such ar the free-fall body draging against the wires, ribbon or column or of the apparatus itself.

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

Purpose: To derive a power-law equation that describes the relationship between mass and period of oscilation. This will be based on the data we collect and parameters we will create and adjust using Logger Pro. We will then use our equation and our calculated values of M-tray to find a tight range of mass for two unknown objects.

Procedure: We first set up an inertial balance and a photogate connected to a laptop to collect data. We used a c-clamps to hold the balance against the surface of the table so that all the motion is in horizontal oscilations. To collect data on each oscillation time we placed a piece of tape at the end of the balance and set up a stand to hold the photogate at the end. The piece of tape attached to the balance breaks the led light and records the time for each oscilation. We preformed several different trials using masses starting with zero mass on the tray and adding 100 grams each trial up to 800 grams. 

Below are the data table and logger pro graph from our trials.

The power-law type of equation we used, T=A(m+Mtray)^n, closely resembles the linear equation y=mx+b when we take the Natural Log of both sides.

 Using Logger Pro we entered the data and created additional columns for the Natural Logs of both the periods and masses and plotted out results.

 However, we did not weigh the actual tray, so we estimated and adjusted the Mtray parameter until it gave us the closest correlations possible. 

We chose to use 282 g as our lower and 311 g as our upper limits for our Mtray parameter. This later helped us select a very tight range when calculating the unknown masses of other objects by using the time each took for their oscillations.

 This lab seemed really tough because we started with so many unknown variables. Collecting data on different known masses with the Inertial Balance and photogate made things go smoothly when using Logger Pro. We also found the natural logs we needed, graphed the modified formula and found values for some of our variables. This also involved estimating, using and adjusting values for our Mtray parameter. Using very close values for our lower and upper limits of Mtray we were able to calculate a tight weight range for our unknown masses.