Center of Mass

Overview

So what exactly is center of mass? According to the Barron’s AP Physics C book, it’s the weighted average of the location of mass in a system. The book also provides a complicated mathematical formula to solve for center of mass.

So now, my definition of center of mass is simply that, it is the physical point that represents the center of some object (mass) taking into account all abnormalities of the shape and relative weights of all parts of the object.

RomeoJuliet.jpg

Equations

$\vec{r} = \frac{\sum m_i\vec{r}_i}{\sum m_i} = \frac{\sum m_i\vec{r}_i}{total mass = M} \iff \left\{ \begin{array}{rl} x_{CM} & = \frac{\sum m_ix_i}{M}\\\ y_{CM} & = \frac{\sum m_iy_i}{M}\\\ z_{CM} & = \frac{\sum m_iz_i}{M}\end{array} \right.$

I'm here to make this formula make more sense and teach you all there is to know about CENTER OF MASS! Raise your hand if you're excited. =) I know I am.

The formula above helps to find the center of mass for an object that has a continuous mass distribution. This means that dividing the object into infinitesimally small pieces would give all pieces the same mass.

A simpler way to think about the general formula is to draw the image of the object on a coordinate plane with one corner at the origin. Then, for the horizontal the y-axis will be used at the base and all pieces of the object not entirely on the y-axis are used in the calculation. The mass of each piece is then multiplied by the distance from the y-axis and all the pieces are added together. The same is done for the vertical; except the x-axis is the base and all pieces not entirely on the x-axis are used in the calculation.

Examples

A rectangular wire frame has a width of 2 and a length of 3. Each unit length has a unit mass except for the bottom width of length 2 which has a mass that is twice that of the other unit lengths. Find the center of mass.

A wire mass is shaped into a cross with unit lengths at every bend. There is one side missing. Find the center of mass.

Special Cases

What happens when the small pieces of an object have different masses, just as in an object like a chair? The formula above must then be altered to find the masses of each small piece. Thus, it becomes:

$\vec{r}_{CM} = \frac{\int \vec{r} dm}{M} \iff \left\{ \begin{array}{rl} x_{CM} & = \frac{\int x dm}{M}\\\ y_{CM} & = \frac{\int y dm}{M}\\\ z_{CM} & = \frac{\int z dm}{M}\end{array} \right.$

This formula is needed to find the center of mass of irregular objects, but it is rarely, if ever, seen on the AP Physics C exam and so it does not necessarily need to be learned. It is simple to use though.

Application

Center of mass is useful in many large scale things, such as orbiting planets and moons. When a planet and its moon are considered, there is a center of mass between the two that helps determine the center point of the two orbits. The moon does not actually orbit around the planet. Both masses orbit around the center of mass point, but because of their relative sizes, it appears to be that the moon is orbiting the planet.

Joke

QUESTION:

What do physicists enjoy doing the most at baseball games?

ANSWER:

The 'wave'.

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