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

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### Equations of Motion

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

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

### 3D Motion

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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,  θ {\displaystyle \theta } , z)
Also see Polar Coordinates (r,  θ {\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,  θ {\displaystyle \theta } ,  ϕ {\displaystyle \phi } )

{\displaystyle {\vec {R}}=R{\hat {e}}_{R}+{\hat {e}}_{\theta }+{\hat {e}}_{\phi }}

cos

{\displaystyle {\vec {v}}=R'{\hat {e}}_{R}+(R\theta '\cos \phi ){\hat {e}}_{\theta }+R\phi {\hat {e}}_{\phi }'}

cos

cos

sin

sin

cos

{\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

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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  I c {\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

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

## Assignments

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Activities:

* Create an activity

Readings:

* Peruse the appropriate sections of b:Mechanics

Study guide:

1. Wikipedia article:Cartesian coordinates
2. Wikipedia article:Polar coordinates
3. Wikipedia article:Cylindrical coordinates
4. Wikipedia article:Spherical coordinates
5. Wikipedia article:Displacement
6. Wikipedia article:Distance
7. Wikipedia article:Velocity
8. Wikipedia article:Speed
9. Wikipedia article:Acceleration