29-Aug-2016: Finding a relationship between mass and period for an inertial balance

1. Title: Finding a relationship between mass and period for an inertial balance
    Name: Qiwen Ye (Sherry)
    Lab partners names: Eugene, Chandler
    Date: 29-Aug-2016

2. Purpose
To use an inertial balance to measure mass, and find relationship between T and m for inertial pendulum equation that predicts well. We calibrate the balance using known masses, and use the balance to find the mass of unknown objects.

3. Theory/Introduction
The concept of inertia originated from Newton's First Law of Motion. Mass is measured by comparing the fore of attraction due to gravitation between earth and the object and that between earth and comparing masses.
Here is, Power-law type of equation:

T=A(m+Mtray)^n

If we take the natural logarithm of each side we can get,

ln T=n ln (m+Mtray) + ln A, which looks like y=mx+b

therefore,

y: ln T       slope: n        x: ln (m+Mtray)         y-intercept: ln A 

4. Apparatus/Experimental procedure
Use a C-clamp to set up the inertial balance on the tabletop. Put a thin piece of masking tape on the end of the inertial balance. Then, set up photogate and LabPro like those picture one.

In the picture two, we tape on the end of the tray in order to pass through photogate to measure period. Add 0 to 800 gram to the tray, 100 gram at a time, to measure period each time.

5. Data

Here is a date table for recording each period by 0 to 800 gram. According to power law type of equation, we have three unknown values - A, Mtray, and n. We use above data in order to make a plot of ln T vs. ln (m+Mtray).

6. Results/Graphs
This is a graph about ln T and ln (m+Mtray). We use "Linear Fit" to make our graph become a beautiful straight line.

Here is a process that we adjust our data:

225g  correlation   0.9994

240g  correlation   0.9995

250g  correlation   0.9995

270g  correlation   0.9995

295g  correlation   0.9994

240g

250g

270g

Also, through the above equation, we can find the mass of two other unknown objects by measuring their periods of oscillation.

the period of staple = 0.340s

the period of pencil case = 0.422s

Therefore, the mass of staple is 88.72g, and the mass of pencil case is 88.29g.

 7. Analysis
In that graph, we can get constants A and n:

ln A= -4.767 , A= e^-4.767       and        n= 0.6316

So, the equation is:

T = e^-4.767 (m+Mtray)^0.6316

We can see that ln T increasing when ln (m+Mtray) increasing. It means that with the mass of objects increasing, the period of the object will be increased.
By finding two unknown objects of masses, we can know that if we have object's periods, we will get a mass of objects.

8. Conclusion
In this lab, the relationship between mass and period was found using an inertial balance and a photo gate. We calculated the period(T) and compared it the measure period(T). The values also allowed for the mass of 3 "unknown" objects to be calculated and compared to the measured masses. When comparing the period and masses with the calculated and compared, the differences between them were less 10%. Before we even performed any calculations, we made an assumption that when mass increasing, the period will increasing. We had measured the period of different mass and notices that assumption is correct. Also, by the power law equation, we can get the mass of unknown object through its period.