Eric Chong
Deriving a Power Law for an Intertial Pendulum
Lab Partners: Lynel Ornedo and Harvey Thai
Purpose: The goal of this lab is to see the relationship between the mass and period of an object by utilizing an inertial pendulum. An inertial pendulum is essentially a device that moves back and forth, creating an oscillatory motion, when pushed or pulled. Once an object is placed on top of the pendulum, the oscillatory motion becomes much slower, as the total inertia becomes much higher, making the device and the object harder to change its course of motion. This experiment is designed to derive power-law type of equations in order to determine the mass of an object from the period of the pendulum.
Theory/Introduction: For this lab, there are two fundamental concepts examined: inertia and the period of oscillation. Usually when objects move, some objects may have difficulty changing their course of motion more than others. This is due to the property of inertia of the objects. Objects with more inertia are harder to change their state of motion. Inertia is directly related to an object's mass. The more mass an object has, the higher the inertia. The next concept, the period of oscillation, is essentially the time it takes an oscillation to come back to its initial position. Combining these two concepts, it can be theoretically said that the period of an oscillation is dependent on the object's inertia, as objects with higher inertia placed in an oscillation will take more time returning to its initial position. Thus, the mass of the object is the leading factor that changes the period of oscillation.
Now, people might say, "why does this lab matter? Isn't a normal balance enough to determine the object's mass?" While that may be more practical, it is only true on Earth. A normal balance depends on the Earth's gravity in order to determine an object's gravitational mass. In an environment in which there is no gravity or the force of gravity is different, determining an object's mass is difficult. This is where this experiment comes into play. This lab essentially determines an object's inertial mass. Without gravity to aid us, we can determine an object's mass by oscillating it back and forth and measuring its period and using a relationship between the mass and the period shown in some power-law type of equation: T = A(m + Mtray)^n, where T is the period, A and n are constant values, m is the mass of the object, and Mtray is the mass of the tray. Here, we have three unknowns -- A, Mtray, and n. We can determine the value of all of these by manipulating the equation. By taking the natural log of both sides, we get lnT = n ln (m + Mtray) + lnA, which resembles a y = mx + b equation. Therefore, the slope of the line must be n, and the y-intercept is lnA. By graphing a plot of lnT vs. ln (m + Mtray), we can see that the line is a straight line with slope n and y-int lnA if the value of Mtray is correct. Thus, it is now possible to guess the value of Mtray, while also getting the value of A and n, since the end result should be a straight line. Any other values would produce a non-straight line.
Procedure and Data Analysis: First, we used a clamp to secure the inertial balance to the table and taped a thin piece of tape sticking out on the end of the inertial balance, so the motion sensor, that is secured onto a pole, can detect it oscillating. We set up the Labpro on the laptop using Logger Pro and connected it to the motion sensor, so we can see the graph and use it to analyze data.
Next, we did a trial without any mass on the inertial balance and oscillated and found its period. The result came out like so in this period vs. time plot:
After that, we added a 100 gram mass onto it and oscillated like the first trial. The rest of the trials were the same, but just added 100 more grams onto the inertial balance each time. The results are shown here:
Here is a summarized version of the results in the form of a data table:
We then unplugged the cable from the Labpro and set up a new Logger Pro document and set up a data table with X being Mass in g, and Y being Period in seconds. Afterwards, we set up a parameter called Mtray, and also set up 3 new columns -- m + Mtray, lnT, and ln (m + Mtray). Finally, we made a plot of lnT vs. ln (m + Mtray) and adjusted the parameter of Mtray until we ended up with a straight line with a correlation close to 1. Here is what the result is, with the value of Mtray that we guessed being 240 grams:
We also noticed that there is a range of values of Mtray that give the same correlation of 0.9994, our maximum correlation that we could get. So, we found the maximum and minimum values of Mtray, and determined that they were 210 grams and 250 grams. Here are the graphs respectively:
The reason why these graphs are important is that we need to find the values of A and N. Now, we know that A is the value of the y-intercept and n is the value of the slope. Therefore, we must look on those features of each graph. Here are the values in a data table:
Now that we know the values of A and n, we can set up the equation so we could solve for the mass of any object on the inertial balance. Here is how we did it for each value of Mtray in order from least to greatest:
And this is how we derived a power law for an inertial pendulum. To find the mass of an object, we simply find the object's period on the inertial balance and plug it into these 3 equations in order to find the range of possible masses it could have.
Extension: For this extension, the two items we chose were a tape dispenser and a stapler.
The period for the tape dispenser is 0.607 seconds, as measured on the inertial balance. To find the mass of the tape dispenser, we simply plugged it into those 3 equations for T. The same method is used on the stapler, with its period being 0.484 seconds. Here are the results in the form of a data table:
As seen, these values are the calculated masses of the objects, with the last row being the actual value on an electronic balance. Some values are accurate, but some are not, and this could be due to the presence of uncertainties and error in our method.
Conclusion: Through this method and data analysis, we can derive an equation for solving the mass of an object on an inertial balance. This is extremely useful for situations in which gravity is not a factor or an unreliable element in finding the mass. We found the values for A and n for each value of the mass of the tray, after guessing what it is, by looking at the Y-intercept and the slope. Then, we used those values and derived an equation and solved for the mass of the object.
Some problems that may have occurred in the data is that the period is not at all constant while measuring. This is probably due to the air resistance that slowed the period as the object was oscillating when we ran the trial. Hence, in the pictures of the period vs time plots, we could see that some of the values were slightly higher as the oscillation went on. To combat this, we simply took the middle values and used them in our data, though this in itself is an unreliable method. Because of this, our calculations are very uncertain and could have lead to the very high discrepancy between the maximum and minimum ranges of the masses in the end. Yet, despite this, there is some credibility in our data, as our intermediate values are still accurate enough to the real value of the mass of the objects, as seen in the last image posted.
Overall, this experiment showcased the value of an inertial balance and helped give us a better understanding of how we could find the mass without relying on gravity.
Now, people might say, "why does this lab matter? Isn't a normal balance enough to determine the object's mass?" While that may be more practical, it is only true on Earth. A normal balance depends on the Earth's gravity in order to determine an object's gravitational mass. In an environment in which there is no gravity or the force of gravity is different, determining an object's mass is difficult. This is where this experiment comes into play. This lab essentially determines an object's inertial mass. Without gravity to aid us, we can determine an object's mass by oscillating it back and forth and measuring its period and using a relationship between the mass and the period shown in some power-law type of equation: T = A(m + Mtray)^n, where T is the period, A and n are constant values, m is the mass of the object, and Mtray is the mass of the tray. Here, we have three unknowns -- A, Mtray, and n. We can determine the value of all of these by manipulating the equation. By taking the natural log of both sides, we get lnT = n ln (m + Mtray) + lnA, which resembles a y = mx + b equation. Therefore, the slope of the line must be n, and the y-intercept is lnA. By graphing a plot of lnT vs. ln (m + Mtray), we can see that the line is a straight line with slope n and y-int lnA if the value of Mtray is correct. Thus, it is now possible to guess the value of Mtray, while also getting the value of A and n, since the end result should be a straight line. Any other values would produce a non-straight line.
Procedure and Data Analysis: First, we used a clamp to secure the inertial balance to the table and taped a thin piece of tape sticking out on the end of the inertial balance, so the motion sensor, that is secured onto a pole, can detect it oscillating. We set up the Labpro on the laptop using Logger Pro and connected it to the motion sensor, so we can see the graph and use it to analyze data.
Next, we did a trial without any mass on the inertial balance and oscillated and found its period. The result came out like so in this period vs. time plot:
After that, we added a 100 gram mass onto it and oscillated like the first trial. The rest of the trials were the same, but just added 100 more grams onto the inertial balance each time. The results are shown here:
Here is a summarized version of the results in the form of a data table:
We then unplugged the cable from the Labpro and set up a new Logger Pro document and set up a data table with X being Mass in g, and Y being Period in seconds. Afterwards, we set up a parameter called Mtray, and also set up 3 new columns -- m + Mtray, lnT, and ln (m + Mtray). Finally, we made a plot of lnT vs. ln (m + Mtray) and adjusted the parameter of Mtray until we ended up with a straight line with a correlation close to 1. Here is what the result is, with the value of Mtray that we guessed being 240 grams:
We also noticed that there is a range of values of Mtray that give the same correlation of 0.9994, our maximum correlation that we could get. So, we found the maximum and minimum values of Mtray, and determined that they were 210 grams and 250 grams. Here are the graphs respectively:
The reason why these graphs are important is that we need to find the values of A and N. Now, we know that A is the value of the y-intercept and n is the value of the slope. Therefore, we must look on those features of each graph. Here are the values in a data table:
Now that we know the values of A and n, we can set up the equation so we could solve for the mass of any object on the inertial balance. Here is how we did it for each value of Mtray in order from least to greatest:
And this is how we derived a power law for an inertial pendulum. To find the mass of an object, we simply find the object's period on the inertial balance and plug it into these 3 equations in order to find the range of possible masses it could have.
Extension: For this extension, the two items we chose were a tape dispenser and a stapler.
The period for the tape dispenser is 0.607 seconds, as measured on the inertial balance. To find the mass of the tape dispenser, we simply plugged it into those 3 equations for T. The same method is used on the stapler, with its period being 0.484 seconds. Here are the results in the form of a data table:
As seen, these values are the calculated masses of the objects, with the last row being the actual value on an electronic balance. Some values are accurate, but some are not, and this could be due to the presence of uncertainties and error in our method.
Conclusion: Through this method and data analysis, we can derive an equation for solving the mass of an object on an inertial balance. This is extremely useful for situations in which gravity is not a factor or an unreliable element in finding the mass. We found the values for A and n for each value of the mass of the tray, after guessing what it is, by looking at the Y-intercept and the slope. Then, we used those values and derived an equation and solved for the mass of the object.
Some problems that may have occurred in the data is that the period is not at all constant while measuring. This is probably due to the air resistance that slowed the period as the object was oscillating when we ran the trial. Hence, in the pictures of the period vs time plots, we could see that some of the values were slightly higher as the oscillation went on. To combat this, we simply took the middle values and used them in our data, though this in itself is an unreliable method. Because of this, our calculations are very uncertain and could have lead to the very high discrepancy between the maximum and minimum ranges of the masses in the end. Yet, despite this, there is some credibility in our data, as our intermediate values are still accurate enough to the real value of the mass of the objects, as seen in the last image posted.
Overall, this experiment showcased the value of an inertial balance and helped give us a better understanding of how we could find the mass without relying on gravity.
This is super clear and it looks really good as far as the text goes.
ReplyDeleteI can't see any of the pictures you posted, just placeholders.
Try going directly to your blog http://phys4aehchong.blogspot.com/ without logging in to Blogger and see if you can see them.
Still nicely done.
ReplyDeleteA couple of things:
Once you show what the period timing looks like for the empty tray you can probably skip the rest of them because they all looks the same, as long as you have recorded the periods in the data tables (which you did).
Your rounding off of the y-intercepts is causing big problems with your answers. For the third whiteboard photo, for example, you have slope = -4.749.
e^(-4.749) = 0.00866 --> mass of 618.8 grams
e^(-4.4) = -.00823 --> mass of 697.8 grams
This is a pretty substantial difference! (>10%)
I'd encourage you to redo those calculations without rounding things. It doesn't quite make sense to do a really nice job writing up your lab and then blow it on sloppy calculations.
As far as sources of uncertainty go . . . When we set up our original equation all of the masses were cylinders centered in the tray.
Our unknown objects had different shapes and perhaps different placement in the tray. We didn't test separately to see if placement or shape made a difference. This is an assumption (that mass is the only variable) that maybe turns out not to be true.