Feb. 25, 2015 Inertial Balance Lab

Objective: Finding a set of equations describing the behavior of the pendulum

The Set-up:
We secure the inertial balance to the table using a C-clamp. We put a thin strip of tape to the end of the inertial balance. Then we set a photogate positioned in front of the inertial balance. The distance between the inertial balance and photogate is where the tape could be detected when the inertial balance begins to oscillate. Then we connect the photogate to the loggerpro on the laptop.

Data Collection:

We collect the data by measuring the period of different masses. We begin by measuring the period of the balance without any mass, then gradually increase the mass by 100g until we reach 800g. The data is show in the table below.

Mass in balance (g)	Period (sec.)	Mass in balance (g)	Period (sec.)
0	0.284	500	0.549
100	0.35	600	0.594
200	0.407	700	0.641
300	0.456	800	0.683
400	0.503

Next, we use the same method to collect the data from two random items. The items we choose are a wallet and a calculator.

Items	Period (sec.)
Wallet	0.454
Calculator	0.369

We purposely did not measure the masses of the two items because we will be using them to check our equations later on. 

Then we begin to formulate the equations. We first consider the equation:

T=A*(M) ^n

The M here not only refers to the mass we add but also the mass of the balance tray itself. Therefore, a more accurate equation would be:

T=A(M added+ M tray)^n

There are three unknowns here, they are A, M tray and n.

We guess the value of the M tray and input the data from the first data collection. The result is a curved graph. We need to continue to guess on the mass of the tray until we get a straight line, the straighter the line is, the more accurate the value of M tray becomes. 

We linearize the curve by taking the natural log of the equation, we get:

ln(T) = n* ln (M added + M tray) + ln A.

Not that this equation is very similar to slope intercept form: y=mx+b. We plot the equation by setting the x- axis to be ln(M added+ M tray) and the y-axis to be ln T. We linear fit the graph by maneuvering the parameter.   

Then we get this, it shows the A and n for our linearized equation.

This is the linearized graph for the high value of M tray

Now we have the value for both A and n. We still need a more accurate value for M tray since the M tray value we use is a guess and it has a correlation value of 0.9994. 

Next, we begin to find a range of value for the mass of the tray. Our goal here is to get the correlation value as close to 0.9999 as we can.

We find the range of the M tray to be 0.3 kg - 0.349 kg. We use this range and the data of the slope (m) and y-intercept (b) on the graph to find the two equations that uses the low and high value of the tray.

This is the format for the equations

We formulate the two equations, one for low and one for high, and we test the equations by calculating the mass of the two items that we choose earlier. 

The calculations show that whether using the high or low equation, the mass of same object remain close to one another. Since the results are not too far from one another, our equations are close to the actual behavior of the inertial balance.

Summary: 
The purpose of inertial balance lab is to find the relationship between the mass and period of the inertial balance. After setting up the balance, photogate, and loggerpro, we collect the data by measuring the period of the balance with various masses. Then we measure the period of two items with unknown mass. We begin the formulation of the equations with T= A(M added +M tray)^n. We take the natural log of both sides to linearize the equation, in order to find A and n. After we have the value for A and n, we input different value for the M tray until we get a correlation value at 0.9999. We then use the range of the M tray to write the two equations, one for the lowest value of M tray (0.3 kg) and one for the highest value of the M tray (0.349 kg). After finding the two equations for the period of the balance, we uses the periods of the two objects with unknown mass to test our equations. The result is that the mass for the same object is close to each other whether we use the high or low equation. This shows that we have successfully coming up with a set of equations describing the behavior of the pendulum!