# Kinematics of particles

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

## Lecture

### Equations of Motion

Rectilinear Motion
The motion of any particle is most easily described by using the the equations of Rectilinear Motion. Where ${\displaystyle {\boldsymbol {s}}}$ represents distance or displacement, ${\displaystyle {\vec {\boldsymbol {v}}}}$ represents velocity and ${\displaystyle {\vec {\boldsymbol {a}}}}$ represents acceleration, it may be remembered from Physics that:

${\displaystyle {\vec {v}}={\frac {ds}{dt}}=s'}$   and   ${\displaystyle {\vec {a}}={\frac {d{\vec {v}}}{dt}}={\vec {v}}'=s''}$


Curvilinear Motion
The motion of any particle along a curved path is most easily described by using the the equations of Curvilinear Motion. Where ${\displaystyle {\boldsymbol {\vec {r}}}}$ represents the position of a particle in cartesian or polar coordinates and ${\displaystyle {\boldsymbol {\Delta {\vec {r}}}}}$ is the displacement of said particle, the scalar quantity ${\displaystyle {\boldsymbol {s=|{\vec {r}}|}}}$ represents the distance of the displacement and ${\displaystyle {\boldsymbol {\vec {v}}}}$ is the instantaneous velocity of the particle:

${\displaystyle {\vec {v}}={\frac {d{\vec {r}}}{dt}}=r'}$


### 3D Motion

Motion in three dimensions may be described by the following equations:
Rectangular Coordinates (Cartesian) - (x,y,z)

${\displaystyle {\vec {R}}=x{\hat {i}}+y{\hat {j}}+z{\hat {k}}}$
${\displaystyle {\vec {v}}={\vec {R}}'=x'{\hat {i}}+y'{\hat {j}}+z'{\hat {k}}}$
${\displaystyle {\vec {a}}={\vec {v}}'={\vec {R}}''=x''{\hat {i}}+y''{\hat {j}}+z''{\hat {k}}}$


Cylindrical Coordinates - (r, ${\displaystyle \theta }$, z)
Also see Polar Coordinates (r, ${\displaystyle \theta }$)

${\displaystyle {\vec {R}}=r{\hat {e}}_{r}+{\hat {e}}_{\theta }+z{\hat {k}}}$
${\displaystyle {\vec {v}}={\vec {R}}'=r'{\hat {e}}_{r}+r\theta '{\hat {e}}_{\theta }+z'{\hat {k}}}$
${\displaystyle {\vec {a}}={\vec {v}}'={\vec {R}}''=(r''-r\theta '^{2}){\hat {e}}_{r}+(r\theta ''+2r'\theta '){\hat {e}}_{\theta }+z''{\hat {k}}}$


Spherical Coordinates - (R, ${\displaystyle \theta }$, ${\displaystyle \phi }$)

${\displaystyle {\vec {R}}=R{\hat {e}}_{R}+{\hat {e}}_{\theta }+{\hat {e}}_{\phi }}$
${\displaystyle {\vec {v}}=R'{\hat {e}}_{R}+(R\theta '\cos \phi ){\hat {e}}_{\theta }+R\phi {\hat {e}}_{\phi }'}$
${\displaystyle {\vec {a}}=(R''-R\phi '^{2}-R\theta '^{2}\cos ^{2}\phi ){\hat {e}}_{R}+\left({\frac {\cos \phi }{R}}{\frac {d}{dt}}(R^{2}\theta ')-2R\theta '\phi '\sin \phi \right){\hat {e}}_{\theta }+\left({\frac {1}{R}}{\frac {d}{dt}}(R^{2}\phi ')+R\theta '^{2}\sin \phi \cos \phi \right){\hat {e}}_{\phi }}$


### 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)

### Additive Motion & Relative Motion

For a system of two vectors oriented in different directions, the relationsip between the two may be establshed through vector addition. For vectors ${\displaystyle {\vec {\boldsymbol {v_{A}}}}}$ and ${\displaystyle {\vec {\boldsymbol {v_{B}}}}}$, the relative motion of ${\displaystyle {\vec {\boldsymbol {v_{A}}}}}$ from the frame of reference of ${\displaystyle {\vec {\boldsymbol {v_{B}}}}}$ is:

${\displaystyle {\vec {\boldsymbol {v_{A}}}}={\vec {\boldsymbol {v_{B}}}}+{\vec {\boldsymbol {v_{A/B}}}}}$


The calculation of relative motion is completed similarly for acceleration.

Activities: