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Center of Mass

The point relative to a body which will behave in response to forces as if the entire mass of the body is concentrated there.


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:

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\begin

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[ \vec

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= \frac{\sum_

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^{N_{\rm p}} m_

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\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:

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\begin

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[ dm = \rho(x,y,z) dx dy dz ] \end

To calculate the x - position of the center of mass we use:

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\begin

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[ CoM_

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= \frac

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]\end

We can obtain the y - and z - coordinates using similar expressions:

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\begin

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[ CoM_

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= \frac

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]\end


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\begin

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[ CoM_

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= \frac

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]\end


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