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Study Blog Part 3: Center of mass

Blogs > Mr.X546
Post a Reply
Zhazulo
Profile Blog Joined February 2011
Sweden226 Posts
February 03 2013 20:27 GMT
#1
Hey all!

This week will be a rather short blog. I've had plenty to do (trying out new study habits, playing SC2, and else) but I wrote together a bit of the week!

Code S has really occupied me daily, hehe. With it being on at 10:10 AM more or less every day and our school having WiFi in every building, it's a bit too easy to tune in ^^ oh well!

I'll be playing at UniSL FInals next week. You know, the guys who made the "Land of the Zerg" video. I believe I'll be playing first on Saturday at 1 pm. Be sure to tune in then!

1) What have you done this week?

+ Show Spoiler +
i) Topics covered

Center of mass

ii) Time spent/what sessions?

Theory reading (1 x 2h)

4 calculation sessions (4 x 2h)

Repetition (~2 h)

Total=12 h

iii) Topics explained

I was supposed to go through free body diagrams in 2D as well, but there was a lot to do with just center of mass, so I ended up doing only that. Luckily, next week is only about FBDs and I managed to read the theory about FBDs in 2D, so I think it will be ok in the end.

Center of mass is about calculating the center of mass of an object or several objects. This is relevant because mechanics is about simplifying complex situations, and in many cases, objects can be approximated by being a single point object, which would be located at it's center of mass.

Calculations include two areas; finding the center of mass of a system of particles or the center of mass for a body. In this case, we approximate the body to be a rigid body; it will NOT change form (more about that later on!)

Finding the center of mass of a particle system can be done with a straightforward formula; finding the sum of the particles' position vectors, multiplied by their corresponding mass, and then dividing the sum with the total amount of mass of the system.

A more common situation is to find the center of mass of a body. It get's a little trickier, but there are several techniques. Basically, the same principle applies; approximate the body to a particle. If the body is complex, the idea is to divide it into sub-bodies, calculate their center and then add them all up as a particle system.

To find the center of mass of a body (or a sub-body!) we can use Pappus Rule. Pappus rule relates the area or volume of rotation around an axis to the center of mass and the line or area which is to be rotated. The equations are as following:

Volume of rotation=2(pi)*Yg*Area
Area of rotation=2(pi)*Yg*Length

where:
Yg - is the y-coordinate of the center of mass (if the figure is to be rotated around the x-axis)

Most of this theory will become understandable when it comes to solving problems, that's when the learning really happens!

iv) What challenges did you face?
Most of the exercises were tedious and quite daunting to start at, but it got easier and easier the more comfortable I got! The main focal point was just to do calculations


2) Problems/Exercises

+ Show Spoiler +
No exercises this week either maybe I'll have to rethink how I should approach this section, as I have yet to upload a single exercise ^^


3) Reflection Questions
+ Show Spoiler +
How would Pappus Rule be applied if the rotation would only be half-way?

How can Pappus Rule be used to find the area and/or the volume of a figure?


4) For Next week
+ Show Spoiler +
i) What topics? What do you know about this?
Next week will be about free body diagrams, but I discussed this last week, so nothing really new to add here!


Have a week!

ThePhan2m
Profile Blog Joined September 2004
Norway2750 Posts
Last Edited: 2013-02-03 21:58:04
February 03 2013 21:57 GMT
#2
Good job, expecting some exercise pictures next week!!
RoyGBiv_13
Profile Blog Joined August 2010
United States1275 Posts
February 04 2013 18:32 GMT
#3
Sort of related to your blog:

A homework short story:
+ Show Spoiler +
As a freshman at engineering college, I had fast-tracked my way into taking multi-dimensional calculus through AP classes in high school. The calculus professor, Dr. Alden, used the Mathematica-like program, Maple, nearly every day to show us examples of how to do incredibly complicated equations using the school-issued (and tuition bloating) laptops. The brightest students quickly caught on that you could use this program to do your homework. You could just print out the results instead of breaking your wrist writing page after page of equations that you inevitably just plugged into the computer anyway.

There was an extensive library as part of Maple. It could do all sorts of animations, differential equations, symbolic logic and really anything a mathematician or engineer could need. There was even keyboard shortcuts to quickly format the worksheet to make it organized and able to be submitted for homework. Part of the skill in using the program was to learn how to ask questions of it. The help file was accessed by a quick "?<command>".

There was one notorious week, two weeks before the final, where Dr Alden had to teach several topics per day to accommodate the schedule. He would teach one subject during the lecture, then assign another for homework. Naturally, calculating the center of mass was a homework assignment. One not even covered in the textbook. The entire class was confused over the instructions and examples (the topic itself isn't too difficult, but there are a number of tricky things in multivariable land that quickly catch up with you). Only three of us turned in the homework on time.

It was my shortest homework assignment since elementary school. It took me under 3 minutes to complete.

The trick? "?Centerofmass".... Turned out that Maple had a CenterOfMass(f(x,y,z),x=A..B,y=C..D,z=E..F) command. I transcribed the functions into the program, ran the worksheet, and printed it. The average length of that homework (the prof recorded these stats, being a Math prof and all) was 11 pages. He had exclude my 100% homework of 2 pages as an outlier.

Any sufficiently advanced technology is indistinguishable from magic
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