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 $𝒔$ represents distance or displacement, $\overrightarrow{𝒗}$ represents velocity and $\overrightarrow{𝒂}$ represents acceleration, it may be remembered from Physics that:

$\overrightarrow{v}=\frac{ds}{dt}=s^{\prime }$   and   $\overrightarrow{a}=\frac{d\overrightarrow{v}}{dt}={\overrightarrow{v}}^{\prime }=s^{″}$

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

$\overrightarrow{v}=\frac{d\overrightarrow{r}}{dt}=r^{\prime }$

3D Motion

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

$\overrightarrow{R}=x\hat{i}+y\hat{j}+z\hat{k}$ 
$\overrightarrow{v}={\overrightarrow{R}}^{\prime }=x^{\prime }\hat{i}+y^{\prime }\hat{j}+z^{\prime }\hat{k}$ 
$\overrightarrow{a}={\overrightarrow{v}}^{\prime }={\overrightarrow{R}}^{″}=x^{″}\hat{i}+y^{″}\hat{j}+z^{″}\hat{k}$ 

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

$\overrightarrow{R}=r{\hat{e}}_r+{\hat{e}}_{\theta }+z\hat{k}$ 
$\overrightarrow{v}={\overrightarrow{R}}^{\prime }=r^{\prime }{\hat{e}}_r+r{\theta }^{\prime }{\hat{e}}_{\theta }+z^{\prime }\hat{k}$ 
$\overrightarrow{a}={\overrightarrow{v}}^{\prime }={\overrightarrow{R}}^{″}=(r^{″}-r\theta {\text{'}}^2){\hat{e}}_r+(r{\theta }^{″}+2r^{\prime }{\theta }^{\prime }){\hat{e}}_{\theta }+z^{″}\hat{k}$

Spherical Coordinates - (R, $\theta$, $\varphi$)

$\overrightarrow{R}=R{\hat{e}}_R+{\hat{e}}_{\theta }+{\hat{e}}_{\varphi }$ 
$\overrightarrow{v}=R^{\prime }{\hat{e}}_R+(R{\theta }^{\prime }\cos \varphi ){\hat{e}}_{\theta }+R\varphi {{\hat{e}}_{\varphi }}^{\prime }$
$\overrightarrow{a}=(R^{″}-R\varphi {\text{'}}^2-R\theta {\text{'}}^2{\cos }^2\varphi ){\hat{e}}_R+(\frac{\cos \varphi }{R}\frac{d}{dt}(R^2{\theta }^{\prime })-2R{\theta }^{\prime }{\varphi }^{\prime }\sin \varphi ){\hat{e}}_{\theta }+(\frac{1}{R}\frac{d}{dt}(R^2{\varphi }^{\prime })+R\theta {\text{'}}^2\sin \varphi \cos \varphi ){\hat{e}}_{\varphi }$

Mass Moment of Inertia

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

$I_x=\int (y^2+z^2)dm$,  $I_y=\int (x^2+z^2)dm$,  $I_z=\int (x^2+y^2)dm$         (4)

If the axis of rotation passes through the center of gravity of the rotating object, the calculated $I_c$ 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 $\overrightarrow{{𝒗}_{𝑨}}$ and $\overrightarrow{{𝒗}_{𝑩}}$, the relative motion of $\overrightarrow{{𝒗}_{𝑨}}$ from the frame of reference of $\overrightarrow{{𝒗}_{𝑩}}$ is:

$\overrightarrow{{𝒗}_{𝑨}}=\overrightarrow{{𝒗}_{𝑩}}+\overrightarrow{{𝒗}_{𝑨/𝑩}}$ 

The calculation of relative motion is completed similarly for acceleration.

Assignments

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