Center of Mass
The average position of the mass in a body or system. A system will behave in response to external forces applied to any of its parts as if the entire mass of the body were concentrated there. The motion of the center of mass is unaffected by internal forces in the system (e.g. forces between the atoms, or collisions between different components of the system).
MotivationforConcept"> Motivation for Concept
MathematicalDefinition"> Mathematical Definition
BodyasSumofPointParticles"> Body as Sum of Point Particles
Under this condition, we can quickly derive the form and the utility of the moment of inertia by considering the body to be a collection of Np point particles. Each of the Np point particles (of mass mi where i runs from 1 to Np) will obey [Newton's 2nd Law|Newton's Second Law
CenterofMassasaSum"> Center of Mass as a Sum
We define the Center of Mass as:
\begin
[ \vec
= \frac{\sum_
^{N_{\rm p}} m_
\vec{r_
}}{\sum_
^{N_{\rm p}} m_{i}} ] \end
CalculatingCenterofMass"> Calculating Center of Mass
IntegralsinRectangularCoordinates"> Integrals in Rectangular Coordinates
For continuous objects, the summation in our definition of the center of mass must be converted to an integral. The Mass differential is given by:
\begin
[ dm = \rho(x,y,z) dx dy dz ] \end
To calculate the x - position of the center of mass we use:
\begin
[ CoM_
= \frac
]\end
We can obtain the y - and z - coordinates using similar expressions:
\begin
[ CoM_
= \frac
]\end
\begin
[ CoM_
= \frac
]\end