Mr. P: Good morning. Bobby, please read the problem and Bo, please translate.
Flipping Physics
Bobby: Three point objects are located at various locations on a Cartesian coordinate system.
Bo: Stop. What's a Cartesian coordinate system?
Billy: The Cartesian coordinate system is the coordinate system invented by René Descartes in the 17th century.
It describes the locations of objects on a two-dimensional plane with the location zero, zero,
or the origin, usually in the middle. It also has a horizontal x-axis and a vertical y-axis.
Actually, it can also describe locations of objects in three dimensions with the...
Bo: That’s right. I remember. Thanks Billy.
Billy: You are welcome.
Bo: Bobby, please keep going. Bobby: Actually, what is a point object?
Mr.P: Sorry, a point object, or point particle, is an object that is small enough,
relative to the overall geometry of the situation,
such that the size and shape of the object are irrelevant.
In other words, a point particle can be treated as if all of its mass is located at a single point in space.
Bo: Yeah, Bobby?
Bobby: Mass one with a mass of 1.1 kilograms is located at one point (1.0, 1.5) meters.
Bo: Stop.
Mass one equals 1.1 kilograms,
the position of mass one is at a point 1.0 meters to the right of the origin, and
1.5 meters above the origin.
Bobby: Mass two with a mass of 3.4 kilograms Is located at (3.0, 1.0) meters.
Bo: Mass two equals 3.4 kilograms,
the position of mass two is 3.0 meters to the right and 1.0 meters up.
Bobby: Mass three with a mass of one point three kilograms is located at (1.5, 2.5) meters.
Bo: Mass three equals 1.3 kilograms,
position of mass three is one point five meters to the right and two point five meters up.
Bobby: Where is the center of mass of the three object system?
Bo: Position of the system center of mass equals question mark.
Mr. P: Believe it or not, the letter r is generally used to describe the overall
location of an object in two or three dimensional space
and it is always good to draw a picture of problems like these.
We have yet to discuss the general equation for the center of mass of a system of particles. So here it is:
The position of the center of mass of the system equals the mass of object one times the position of object one
plus the mass of object two times the position of object two, plus as many of those mass times position
expressions as exists in the system,
all divided by the quantity mass one plus mass 2 plus as many masses as exist in the system.
For example, in this 3 point object system the equation has three expressions, one for each object.
Notice, I have specifically identified this as the x-position center of mass.
We will also determine the y-position center of mass in a bit
and we could also determine the z-position center of mass if
we were doing a problem in a three dimensional space. However, I don't think that is necessary right now.
Also realize the position of each object is relative to a zero reference point, which needs to be clearly identified.
In this problem the (0, 0) position, or origin, was already clearly identified.
However, it might not always be and you may have to identify the zero reference point.
Bobby: Mr. P? Mr. P: Yes, Bobby
Bobby: So basically, the center of mass position equation locates the average position of every object.
Bo: And objects with a larger mass have a larger influence over where the center of mass is.
Billy: And the denominator of the equation is just the total mass of the system.
Mr. P: I think it is important to reiterate that the center of mass is not just the
average position of every object, but rather the mass weighted average position of every object.
In other words, the mass of the object is taken into account when determining
how much effect the object has over where the center of mass is.
Billy please begin solving the problem. Billy: Okay. Sure
So, well, you already wrote down the equation for the exposition of the center of mass of the system.
We just needs to substitute in all the known values. So it equals mass 1 which is 1.1,
times the x-position of mass 1, which is 1 plus mass 2,
which is 3.4 times the x-position of mass 2 which is 3
mass 3 which is 1.3 times the x-position of mass 3 which is 1.5,
all divided by the total mass, which is 1.1 plus 3.4 plus 1.3.
That works out to be
2.28448 or 2.3 meters with 2 significant digits.
Mr. P: Correct Billy. Bo, please compute the y-position center of mass.
Bo: Ok. Well we use the same equation we just used only replace all of the x's with y's.
So it equals 1.1 times 1.5 plus 3.4 times 1 plus 1.3 times 2.5. All divided by the quantity 1.1 plus 3.4 plus 1.3.
That equals 1.43103 or 1.4 meters with two significant digits
Mr. P: Therefore, the center of mass of this three point particle system of particles is at the location
2.3 meters to the right of and 1.4 meters above the origin.
Notice how the center of mass is different than the geometric center of the three particle system.
Which would be located right here at an x-position of 1.8 and a y-position of 1.7 meters.
This geometric center is called the centroid.
The geometric center, or centroid, does not account for the different masses of the particles
and is where the center of mass would be if they all had the same mass.
Visually that may be where you expect to the center of mass to be,
however, because our three objects have different masses,
the center of mass and geometric center, or centroid, are not in the same location.
Thank you very much for thumbing with me today. I enjoyed learning with you