Finding a relationship between mass and period for an inertial balance

First Lab

In the first lab of our physics 4A, we would determined the relationship between mass of object and period of the balance. Therefore, we already had several apparatus like a thin piece of masking tape, inertial balance, photogate, computer and weights to help us to finish our work. First, we attached the masking tape on the end of the inertial balance, and set on the beam of the photogate with iron pole so that the tape could completely passes through it when the balance is oscillating. After the photogate collected the data and sent it to computer for processing. And then, we could measure it as we added the weight in order to figure out the situation easily.

While we were taking the lab, we got the period when balance was oscillating of each different weights and wrote down to the data table. But if we wanted to figure out the relationship, we had to expect that the equation
 T=A(m+Mtray)^n
, and we would determine A, Mtray, and n. In Logger Pro, first step, we took the natural logarithm of each side we got: InT=nIn(m+Mtray)+InA, which looked like y=mx+b. Second step, making a new window, and we set the column to be Mass and Period,respectively. And then, we created  three new columns- m+Mtray, In T,and In(m+Mtray), adjusted the value of the Mtray until the correlation to be 0.99, and got the range the Mtray could be so the correlation is 0.99. 

1. Motion Sensor: 0.4020s, 191g

1. Wallet: 0.3513s, 101g

1. Eraser: 0.2979s, 18g

1. Motion Sensor:

1. Maximum: 0.4020 = 0.6452 * (m + 0.305)^0.6845

1. Minimum: 0.4020 = 0.6523 * (m + 0.285)^0.6555

1. Wallet:

1. Maximum: 0.3513 = 0.6452 * (m + 0.305)^0.6845

1. Minimum: 0.3513 = 0.6523 * (m + 0.285)^0.6555

1. Eraser:

1. Maximum: 0.2979 = 0.6452 * (m + 0.305)^0.6845

1. Minimum: 0.2979 = 0.6523 * (m + 0.285)^0.6555

1. Motion Sensor: 194.4g ± 1.6g (191g)

1. Wallet: 105.3g ± 1.2g (101g)

1. Eraser: 18.0g ± 0.4g (18g)

Third step, we could determine the value of the constant A and n, so we could select other stuff and measure their weight by this equation.
here is unknowed weight stuff:

And we used website-based calculator, WolframAlpha, by plugging in these values into the equation  T=A(m+Mtray)^n, and used the period measured from before, we are able to determine the range of the unknown masses finally.

As what we show above, we have already calculated and gotten the maximum and minimum of masses for each unknown weight object. We added them together and divided by two to get the mean values. The number in parentheses is the mass weighed from the balance scale. The difference between a maximum and mean value is the deviation.

Finally, in the lab, we got the equation about the period and mass in scientific way. Thank to the system, we could calculate and get the equation in short time.