10. Formúlublað

10.1. Vigrar

\[\begin{split}\begin{aligned} \bar{a} &= \begin{pmatrix} a_x \\ a_y \\ a_z \end{pmatrix} = a_x \hat{\imath} + a_y \hat{\jmath} + a_z \hat{k} \qquad & \text{Mismunandi táknmál}\\ |\bar{a}| &= a = \sqrt{a_x^2 + a_y^2 + a_z^2} \qquad & \text{Lengd vigurs}\\ \bar{a} + \bar{b} &= \begin{pmatrix}a_x+b_x \\ a_y+b_y \\ a_z+b_z \end{pmatrix} \qquad & \text{Samlagning vigra}\\ c \cdot \bar{v} &= \begin{pmatrix}c \cdot v_x\\ c \cdot v_y \\ c \cdot v_z \end{pmatrix} \qquad & \text{Margföldun með fasta}\\ \bar{a} \cdot \bar{b} &= a_x b_x + a_y b_y \\ &= a b \cos{\phi} \qquad & \text{Innfeldi vigra}\\ \bar{a} \times \bar{b} &= (a_x \hat{\imath} + a_y \hat{\jmath} + a_z \hat{k}) \times (b_x \hat{\imath} + b_y \hat{\jmath} + b_z \hat{k}) \\ &= (a_y b_z - a_z b_y)\hat{\imath} + (a_z b_x - a_x b_z)\hat{\jmath} + (a_x b_y - a_y b_x)\hat{k} \qquad & \text{Krossfeldi}\\ |\bar{a} \times \bar{b}| &= ab\sin \phi \qquad & \text{Lengd krossfeldis}\\ \bar{a} \times \bar{b} &= -\bar{b} \times \bar{a} \qquad & \text{Regla um krossfeldi} \end{aligned}\end{split}\]

10.2. Hraði

\[\begin{split}\begin{aligned} v &= \frac{x_2-x_1}{t_2-t_1} = \frac{\Delta x}{\Delta t} \qquad & \text{Meðalhraði}\\ v & = \lim_{h \to 0} \frac{x_{t+h}-x_{t}}{(t+h) - t} = \lim_{h\to 0}\frac{x_{t+h}-x_{t}}{h} = \frac{dx}{dt} \qquad & \text{Augnablikshraði} \\ \end{aligned}\end{split}\]

10.3. Hröðun

\[\begin{split}\begin{aligned} a &= \frac{v_2 - v_1}{t_2 - t_1} \qquad & \text{Meðalhröðun}\\ a &= \lim_{h \to 0} \frac{v_{t+h}-v_{t}}{(t+h) - t} = \lim_{h\to 0}\frac{v_{t+h}-v_{t}}{h} = \frac{dv}{dt} \qquad &\text{Augnablikshröðun}\\ \end{aligned}\end{split}\]

10.4. Hreyfijöfnur

\[\begin{split}\begin{aligned} v&=v_0+a\cdot t \qquad &\text{Hraði sem fall af tíma} \\ x&=x_0 + v_0 \cdot t + \frac{1}{2} a\cdot t^2 \qquad & \text{Staðsetning sem fall af tíma} \\ v^2 &=v_0^2+2a\cdot (x-x_0) \qquad & \text{Tímaóháða jafnan} \\ x-x_0 &=\frac{1}{2}(v_0 + v)\cdot t \qquad & \text{Færsla} \end{aligned}\end{split}\]

10.5. Tvívíð hreyfing

\[\begin{split}\begin{aligned} y&=\tan(\alpha_0) x - \frac{g}{2v_0\cos^2(\alpha_0)}x^2 \qquad & \text{Ferill kasthreyfingar} \\ a &= \frac{v^2}{R} = \frac{4 \pi^2 R}{T^2} \qquad & \text{Jöfn hringhreyfing} \end{aligned}\end{split}\]

10.6. Kraftar

\[\begin{split}\begin{aligned} \Sigma \bar{F} = 0 &\Rightarrow v = \text{fasti} \qquad & \text{1. lögmál Newtons}\\ \Sigma \bar{F} &= m\bar{a} \qquad & \text{2. lögmál Newtons} \\ \bar{F}_{AB} &= -\bar{F}_{BA}\qquad & \text{3. lögmál Newtons} \\ \end{aligned}\end{split}\]

10.7. Orka

\[\begin{split}\begin{aligned} W &=\bar{F}\cdot \bar{s} \qquad & \text{Vinna}\\ P &= \frac{dW}{dt} \qquad & \text{Afl}\\ K &=\frac{1}{2}mv^2 \qquad & \text{Hreyfiorka}\\ U &= mgy \qquad & \text{Stöðuorka í þyngdarsviði}\\ p &= mv \qquad & \text{Skriðþungi}\\ \end{aligned}\end{split}\]