December- 4 2014 Semicircle Physical Pendulum

Semicircle Physical Pendulum 

Objective:
  Find the period of a half circle mass acting as a physical pendulum at two different orientations.

Set up:
  As seen in the photo to the right, we (each group) had to cut semicircles our of a light material and use calipers to measure its dimensions and weight its mass with scales. We then attached light rings to the edge of orientations and record its motion at a small angle.

Data Collection:
  Firstly we had to find the moment of inertia of the center of mass of the object, but in order to do that we had to find the moment of inertia about the flat side of the semicircle and the location of the Center of Mass to apply the parallel axis theorem:
Iflat= ICOM + m(d^2)
ICOM= Iflat  - m(d^2)

As seen below, we found the center of mass first which is at the middle of the x axis parallel to the flat side and (4R/3pi) to the y direction toward the curved side. Then using what we found our moment of inertia to be about the flat side and subtracting according to parallel axis theorem we found the moment of inertia about its center of mass. We used this to then find our moment of inertia about the middle of the curved edge of the semicircle. We found that about the flat side has less inertia than id the semicircle were rotating about the curved side.

From there we set up our scenario as a dynamics problem of unequal non- constant torque to meet the form of simple harmonic motion assuming a small theta or angle. We found that our theoretical calculation for period if it were turning on its flat side is less than 1% off as is expected for using a taylor series on the sine value to assume it to be just theta. However in the actual for the curved side rotating, the period is exactly the same as if it were to be about its flat side which the theoretical calculations did not suggest however we were only 1.9% off.

Conclusion:
It seems friction or damping of the system may have altered the results. Another possibility is that we may have had a too large of a theta during our test for the curved side of the circle trial otherwise the sensor may not have read such a high average for its period.

Mass Spring Oscillations Lab Nov.- 20- 2014

Mass Spring Oscillations Lab

Objective: Find the relationship between period and spring constant as well as with the amount of mass that is oscillating.

Set up: There were a total of four groups each with a different spring, the appropriate mass so that way each group would have the same amount of oscillating mass determined by the equation (mosc= mhang+ mspring/3). We used a motion sensor and lab Pro to graph our results and place an appropriate fit.

Data Collection: The first step for our group to do is to find the spring mass of our given spring. To do so we weighed the spring for its mass, We then hung the spring to measure its length at a relaxed state, then once more with a .1 kg mass hanging at the end without oscillation:

Fnet= 0N= mg - kxstretch               xstretch= x- xo = .067 m

                                                   m= 26 g

mg= k(x -xo) => k= mg              k= 14.64 N/m

                              (x- xo)

We then went on to derive our equation for omega and, eventually, for our period using Newton's second Law as seen in (fig. 1).

Figure 1. All my derivations and notes from the set up as well as comparisons to our actual
unfortunately there is a mistake in the percentage comparison on theoretical period to actual.

Figure 2. Results of our 4 trials to find the average period for the set up.

We now had our theoretical equation for period of our set up and we tested with 4 trials for consistency. We counted the periods with in a 5 second trial and found that our theoretical calculation was almost 100% accurate to our actual. When we graphed our results compared to others and graphed them with an appropriate curve fit we find something interesting. First, how we analytically found the curve fit to be a power fit was this observation:

Our period came to a form of:

T= 2pi*(m/k)^.5

It seems to follow the power fit seeing as there is a constant followed by a value with a variable (either k or m) multiplied to a certain power. Units -wise is simple, since the constant is unitless or in radians so all that remains is to show that what's in the parenthesis needs to be s^2:

T= C*[(kg )]^.5 => [(kg*m) ]^.5        =>  [1 * (s^2)]^.5 = s

          [N/m ]            [kg*m/(s^2)]             [1       1   ]

Seeing this sort of relationship with mass and spring constant we can predict as k=> infinity with a constant mosc T would approach a period of zero at an curve similar to y= (1/x)^.5 seeing as k being in the denominator. However, in that respect, if mosc => infinity with a constant k then T will also approach infinity at an inverse rate compared to the predicted behavior of T vs. k, y= (x)^.5 , seeing as m being in the numerator. According to the graphs below with appropriate curve fit it seems as though our predictions were correct.

Figure 2A. Our graph of T vs. oscillating mass with appropriate curve fit.

Figure 2B. Our T vs. k graph with appropriate powerfit.

Lastly, there was the regard to see how much a 5% spring constant error would affect a calculated period in which for our set up we found the error to be around only 3% difference compared to both actual and correctly calculated spring constant as seen in (fig 1.).

Conclusion:
We found the our source of any uncertainty would lie in the calculations of other groups as well as any inconsistencies in the spring seeing as it swung side to side shortly after the 5 seconds of our trials.