Vectors:

Can be defined in 3D as: $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$ $=\begin{pmatrix}x\\y\\z\end{pmatrix}$ $=(x,y,z)$

Derivative: $\dfrac{d\vec{r}}{dt}=\dfrac{dx}{dt}\hat{i}+\dfrac{dy}{dt}\hat{j}+\dfrac{dz}{dt}\hat{k}$

Magnitude: $|\vec{v}|=\sqrt{v_x^2+v_y^2+v_z^2}$

Dot product: $\vec{a}\cdot\vec{b}=a_xb_x+a_yb_y+a_zb_z=|\vec{a}|*|\vec{b}|*cos\lgroup\widehat{\vec{a}\vec{b}}\rgroup$

Motion:

                                                         $\\begin{aligned} \\vec{a}&=a_x \\hat{i}
    +a_y \\hat{j}
    +a_z \\hat{k}\\\\
    &= \\frac{dv_x}{dt} \\hat{i}
    +\\frac{dv_y}{dt} \\hat{j}
    +\\frac{dv_z}{dt} \\hat{k}\\\\
    &= \\frac{d^2x}{dt^2}\\hat{i}
    +\\frac{d^2y}{dt^2} \\hat{j}
    +\\frac{d^2z}{dt^2} \\hat{k} \\end{aligned}$

$a_{||}=$ same direction as instantinous velocity

tangent to the actual path

$a_{\perp}=$ perpendicular to instantinous velocity

perpendicular to path

➖ In the case of linear motion:

• direction is constant

⭕ In the case of circular motion:

• speed is constant ****

📃 Example:

PHYS10101 Dynamics

Circular motion:

$$ \frac{|\Delta\vec{v}|}{\vec{v_1}}=\frac{\Delta\vec{s}}{R}=\Delta\varphi\hspace{1cm}(\text{if }\Delta\varphi\to0) $$

Using this we can conclude that for uniform acceleration: $a=\frac{v^2}{R}$

$$ \begin{aligned} \vec{a}&=\vec{a_{rad}}+\vec{a_{tan}}\\ &= -\frac{v^2}{R} \hat{r} +\frac{d|v|}{dt} \hat{\theta} \end{aligned} $$

Where $\hat{r}\text{ and }\hat{\theta}$ change with the rotation:

• $\hat{r}$ is from the origin to the point
• $\hat{\theta}$ is tangent to the circle in the direction of the motion

Polar coordinates:

$$ \begin{aligned} r&=\sqrt{x^2+y^2}\\ \theta&=arctan(\frac{y}{x}) \end{aligned} $$

$$ \begin{aligned} x&=rcos(\theta)\\ y&=rsin(\theta) \end{aligned} $$

The unit vectors in the polar coordinates are:

• $\hat{r}$ which is from the center outwards
• $\hat\theta$ pointing clockwise & tangent

❗They are dependent on each other❗

$$ \begin{aligned} \hat{r}&=cos(\theta)\hat{i}+sin(\theta)\hat{j}\\ \hat\theta&=-sin(\theta)\hat{i}+cos(\theta)\hat{j} \end{aligned} $$

Vectors:

Can be defined in 2D as: $\vec{a}=a_r\hat{r}+a_{\theta}\hat\theta$

Dot product: $\vec{a}\cdot\vec{b}=a_rb_r+a_{\theta}b_{\theta}$

derivative: $\frac{d\vec{r}}{dt}=\frac{dr}{dt}\hat{r}+r\frac{d\theta}{dt}\hat{\theta}$