26 August 2014: Finding a relationship between mass and period for an inertial balance Lab 1

Imagine if the world suddenly lost all it's gravity. The idea has been entertained in countless storybooks, television shows, and even a nightmare or two. In an instant, everyone and everything would be tumbling into space with absolutely no hope of return or survival. On the bright side, dieting would become obsolete as weight loss and weight in general would no longer be an issue. When stepping on a scale, we're taking advantage of gravity to measure our mass. Mass, however, is a measure of an object's inertia and is not dependent on gravity. In other words, your mass is the same here safe at home on Earth as it is on the moon; it is constant regardless of the presence or lack of gravitational pull. To prove this, in this lab we used an inertial balance which measures inertial mass instead of gravitational mass.

Materials:

This is the setup of the inertial balance. It contains a c-clamp to secure the inertial balance, masking tape that was placed at the edge of the inertial balance, and a photogate along with the program LoggerPro to record the oscillations and the mass's resistance to change.

Procedure:

To complete this experiment, first we recorded the period of the oscillations of the inertial balance by itself. Then, we added masses to the tray of the balance, securing it with tape, and recorded their periods. We started with 100 g then incremented each mass by 100 g till we had recorded the period of an 800 g mass. 

100 g mass

800 g mass

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To later test the validity of our data, we recorded the period of two unknown objects. We also recorded their weights using a normal scale. In our lab our unknowns were the cell phone of a very trusting classmate and a water bottle. 

Cell phone

water bottle

The resulting data table looked like this:

Data Analysis:

To evaluate our data, we opened up Logger Pro and entered in the mass on the balance (in kg) in column X and their respective periods (in sec) in column Y. Using the period, we created a column of data called "ln T" that utilized the equation "ln "period"". We also created another column of data called "ln (M+mtray)" utilizing the equation "ln ("Mass" + mtray)". Using these two columns, we plotted "ln T" vs. "ln (M+mtray)". The goal was to create as straight a line as possible. This is accomplished when the linear fit correlation coefficient is very close to 1. We discovered the best coefficient was given when the value for "mtray" was between 0.240 kg and 0.300 kg. The resulting graphs looked like so:

mtray= .240 kg

mtray=.300 kg

 From these graphs we can gather that when mtray=0.300 kg, slope (n)= 0.661 and the y-intercept (ln A)= -0.4294. Also when mtray=0.240 kg, slope (n)=0.5895 and the y-intercept (ln A)=-0.4003. Using these numbers we can calculate the mass of the two unknowns and compare them to the values gotten from directly measuring them using the gravitational balances. The equations used to calculate these values was:

The results were as follows:

Unknown 1 (phone)=0.130 kg when Mtray=0.240 kg

Unknown 1 (phone)=0.134 kg when Mtray=0.300 kg

Unknown 2 (water bottle)=0.390 kg when Mtray= 0.240 kg

Unknown 2 (water bottle)= 0.392 kg when Mtray= 0.300 kg

Conclusions:

From our calculations, we found that the mass of unknown 1 (phone) was between 0.130-0.134 kg. When compared to the value obtained from the gravitational balance of 0.133 kg, we found that we were closer to the true value when "Mtray"= 0.300 g.

Also, from our calculations, we found the mas of unknown 2 (water bottle) was between 0.390-0.392 kg. When compared to the value obtained from the gravitational balance of 0.365 kg, we found that we were closer to the true value when "Mtray"=0.240 kg

Overall this lab was both fun and educational. It was interesting to be able to come up with an equation and use it to determine how objects behave and work in the real world.