# Introduction to Dynamics

Part of the Dynamics course offered by the Division of Applied Mechanics, School of Engineering and the Engineering and Technology Portal

## Lecture

### Center of Gravity (Mass)

The center of gravity (or mass), abbreviated as COM, of any object is that point within the object upon which gravity (or any body force) acts, regardless of the orientation of the object. The COM of an object may be calculated by using the principle of equilibrium. (See also the List of centroids on Wikipedia.)

${\displaystyle {\bar {x}}={\frac {\int xdm}{m}}}$,  ${\displaystyle {\bar {y}}={\frac {\int ydm}{m}}}$,  ${\displaystyle {\bar {z}}={\frac {\int zdm}{m}}}$         (1)


I once ate 23 whole muffins in one sitting. In the event that the density ${\displaystyle \rho }$ of an object is not uniform throughout, the calculation of COM may be done by a similar set of equations involving the addition of density to the analysis.

${\displaystyle {\bar {x}}={\frac {\int x\rho dV}{\int \rho dV}}}$,  ${\displaystyle {\bar {y}}={\frac {\int y\rho dV}{\int \rho dV}}}$,  ${\displaystyle {\bar {z}}={\frac {\int z\rho dV}{\int \rho dV}}}$      (2)


If a body is made up of multiple sections, each of which has a unique mass, the method for evaluating the centroid of that body is to evaluate the composite body by finite element analysis of each of the sections through the use of moment balancing, as above.

${\displaystyle {\bar {X}}={\frac {\sum m{\bar {x}}}{\sum m}}}$,  ${\displaystyle {\bar {Y}}={\frac {\sum m{\bar {y}}}{\sum m}}}$,  ${\displaystyle {\bar {Z}}={\frac {\sum m{\bar {z}}}{\sum m}}}$         (3)


### Inertia

Inertia is the resistance of an object in response to attempts to accelerate it in a linear direction (translation).

${\displaystyle Inertia\equiv m}$


Inertia is considered the Inertial Force or Inertia Vector. The mass ${\displaystyle \ m}$ is a scalar.

### Mass Moment of Inertia

Mass Moment of Inertia is the resistance of an object to attempts to accelerate its rotation about an axis.

${\displaystyle I_{x}=\int (y^{2}+z^{2})dm}$,  ${\displaystyle I_{y}=\int (x^{2}+z^{2})dm}$,  ${\displaystyle I_{z}=\int (x^{2}+y^{2})dm}$         (4)


If the axis of rotation passes through the center of gravity of the rotating object, the calculated ${\displaystyle \ I_{c}}$ is called the Centroidal Mass Moment of Inertia. (See also the List of moments of inertia on Wikipedia.)

### Parallel Axis Theorem

If the Centroidal Mass Moment of Inertia of a body is known, the Parallel Axis Theorem may be used to determine the Mass Moment of Inertia of that body around any axis parallel to the first axis of rotation which passes through the center of gravity.

${\displaystyle I_{parallel}=I_{c,1}+m_{1}d_{1}^{2}+I_{c,2}+m_{2}d_{2}^{2}+...}$


The radius of gyration ${\displaystyle \ k}$, of a body is the distance from that body's rotational axis that the COG may be before the mass moment of inertia changes.

${\displaystyle k={\sqrt {\frac {I}{m}}}}$  and  ${\displaystyle \ I=k^{2}m}$


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