8/31/2014 deriving formula for Inertial balance's SHM

Inertial balance lab

Introduction: Mass is the inertia of an object, and is an independent value that will not be affected by the strength of gravity. Inertia is the ability of an object to resist the change in an object's state of motion, so we can expect the motion of the inertial balance to be slower when the mass in the inertial balance is large. In this lab, we are going to find out how mass in the inertial balance affect the period of the simple harmonic motion of the inertial balance, and generate an function to find out the relationship between mass and period of an inertial balance.
Apparatus:

Inertial balance: which will do simple harmonic motion once we give it a start. We will put different weight on it to make the inertia of the entire system different, and observe the relationship between its total mass and period of its motion.

Photogate: This device will help us record the period of each one of complete harmonic motion of inertial balance. We will place inertial balance in between its C-clamp.

Thesis: Because mass is actually inertia of an object, and inertia is the quantitative measure of an object's ability to resist change in motion, we know there must be some kind of relationship between the mass of the object on the inertial balance and the period the inertial balance finish each one of complete simple harmonic motion.
               We assume the relationship between the mass and the period can be showed in power-law type equation:                                                 T= A(m+Mtray)^n
               In this experiment, we will have three unknown number: A, Mtray, and n. To make it easier to get those three value, we will take the natural logarithm of each side :
                                                              lnT=n*ln(m+Mtray)+lnA
               We will then be able to find out n and lnA through graphing lnT vs. ln(m+Mtray) using Logger Pro, and to get an accurate value of Mtray, we will need to plug in different parameter numbers until we get correlation coefficient that is very close to 1. Since the we can not find the exact number of Mtray, all of our unknown number can only be defined to a very small interval instead of exact number.
Data and calculation:
Here is the data we collected in class with a photogate:

If we let lnT=y; ln(m+Mtray)=x; n=a, and lnA=b,
we can look at the equation in the form of y=ax+b, which means:
the slope of the graph m=a=n;
y intercept of the graph b=lnA,
and now we can try to different number to be parameter Mtray until we find the numbers that will make correlation coefficient to be as close to 1 as possible. Once we make the correlation coefficient to be close enough to 0, we can narrow lnA and n to a very tiny interval.

In these two graphs, the correlation coefficients are both 0.9947, which is the best correlation coefficient my team and I could found.

lnT vs. ln(m+Mtray), Mtray=0.397

Parameter Mtray=0.397, and we found that b=lnA=-0.5109, and m=n=0.7138
which indicate that A=e^(lnA)= 0.6000

lnT vs. ln(m+Mtray), Mtray=0.397

Parameter Mtray=0.399, and we found that b=lnA=-0.5123, and m=n=0.7157
which indicate that A=e^(lnA)= 0.5991

Conclusion:

Conclude the number we got from both graphs, we get ab estimate equation that tells us the relationship between the mass and the period of the inertial balance: 

T=(0.5995±0.005)(m+Mtray±0.001)^(0.7148± 0.010)

With the apparatus that is available to us for this lab, this is the best equation that tells the relationship of mass and period of the inertial balance our team get. It obviously shows that when the total mass is greater, the period is also larger.