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\begin{document}

\newcommand{\bold}[1]{\mbox{\boldmath $#1$}}

{\bf FORMLER TFY4125 2025} \\

{\bf Fete} symboler angir vektorer. Symboler med hatt over 
angir enhetsvektorer. Formlenes gyldighetsomr{\aa}de og de ulike symbolenes 
betydning antas for{\o}vrig {\aa} v{\ae}re kjent.
Symbolbruk og betegnelser som i forelesningene.  \\\\

{\bf MEKANIKK} 

\begin{itemize}

\item
$\begin{array}{l}
\displaystyle \mbox{Newtons andre lov: }
\displaystyle \bold{F} = d\bold{p}/dt \qquad 
\displaystyle \bold{p} = m\bold{v} = m\dot{\bold{r}}
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Konstant akselerasjon: }
\displaystyle v = v_0 + at \qquad 
\displaystyle x = x_0 + v_0 t + \frac{1}{2} a t^2
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Konstant vinkelakselerasjon: }
\displaystyle \omega = \omega_0 + \alpha t \qquad 
\displaystyle \theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Arbeid: } dW = \; \bold{F} \cdot d\bold{r} \qquad
\displaystyle \mbox{Kinetisk energi: } K = \frac{1}{2} m v^{2} \qquad
\displaystyle \mbox{Effekt: } P = \frac{dW}{dt} = \bold{F} \cdot \bold{v}
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Konservativ kraft og potensiell energi: }
U(\bold{r}) = -\int_{\bold{r}_0}^{\bold{r}} \bold{F} \cdot d\bold{r} \qquad
\displaystyle  \bold{F}= -\nabla U(\bold{r})
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Friksjon,} \qquad \mbox{ statisk: } \displaystyle f \le  \mu_s N \qquad 
\displaystyle \mbox{kinetisk: }\; f =  \mu_k N \quad
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Luftmotstand (liten $v$): } \bold{f}= - k \bold{v}  \qquad
\displaystyle \mbox{Luftmotstand (stor $v$): } \bold{f}= - D v^2 \hat{v}  
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Tyngdepunkt:}\quad  \bold{R}_{CM} = \frac{1}{M} 
\sum_i \bold{r_i} m_i  \rightarrow \frac{1}{M} \int \bold{r} \cdot dm
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Sirkelbevegelse: } v = r \omega \quad
\displaystyle \mbox{Sentripetalakselerasjon: } a  = v^2/r \quad
\displaystyle \mbox{Baneakselerasjon: } a = dv/dt = r\,d\omega/dt
\end{array}$

\item
$\begin{array}{l}
\mbox{Dreiemoment: } \displaystyle \bold{\tau} = \bold{r} \times \bold{F}
\qquad \displaystyle \mbox{Statisk likevekt: } \Sigma \bold{F}_i = 0 
\quad \Sigma \bold{\tau}_i = 0
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Dreieimpuls: }\quad  \bold{L}=\bold{r}\times\bold{p}  
\qquad \displaystyle \mbox{N2 rotasjon: } \bold{\tau} = d\bold{L}/dt 
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Stivt legeme, refleksjonssymmetri mhp rotasjonsaksen: } \;
\bold{L} = \bold{L}_b + \bold{L}_s =
\bold{R}_{CM} \times M\bold{V} + I_0 \bold{\omega}
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Kinetisk energi, stivt legeme: }  
K = \frac{1}{2} MV^2 + \frac{1}{2} I_0 \omega^2 
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Treghetsmoment: } 
I = \sum_i m_i r_i^2 \rightarrow \int r^2 dm
\end{array}$

$\begin{array}{l}
\displaystyle 
\mbox{Kompakt sylinder (skive): } I_0 = \frac{1}{2}MR^2 \qquad
\mbox{Kompakt kule: } I_0 = \frac{2}{5}MR^2 \qquad
\mbox{Kuleskall: } I_0 = \frac{2}{3}MR^2
\end{array}$

$\begin{array}{l}
\displaystyle 
\mbox{Tynn stang: } I_0 = \frac{1}{12}ML^2 \qquad
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Stivt legeme, rotasjon om fast akse: }  
K = \frac{1}{2} I \omega^2 
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{N2 rotasjon, akse med fast orientering: }  
\tau = I \frac{d\omega}{dt}
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Steiners sats (parallellakseteoremet): } \quad 
\displaystyle I = I_0 + M d^2
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Gravitasjon: }
\quad \displaystyle \bold{F} = -\frac{GMm}{r^2} \hat{r}
\quad \displaystyle U(r) = -\frac{GMm}{r}
\quad \displaystyle \bold{g} = \bold{F}/m
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Enkel harmonisk oscillator: }
\hspace{6mm} \displaystyle \ddot{x} + \omega_0^2 x =0 \quad
\hspace{6mm} \displaystyle T = 2\pi/\omega_0 \quad
\hspace{6mm} \displaystyle f = 1/T = \omega_0/2\pi
\end{array}$

$\begin{array}{l}
\displaystyle \mbox{Masse i fj{\ae}r: } \omega_0 = \sqrt{k/m}
\hspace{6mm} \displaystyle \mbox{Matematisk pendel: } \omega_0 = \sqrt{g/L}
\end{array}$  \\
$\begin{array}{l}
\displaystyle \mbox{Fysisk pendel: } \omega_0 = \sqrt{mgd/I}
\hspace{6mm} \displaystyle \mbox{Torsjonspendel: } \omega_0 = \sqrt{\kappa/I_0}
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Fri, dempet svingning, langsom bevegelse i fluid: }
\hspace{6mm} \displaystyle m \ddot{x} + b \dot{x} + k x =0 \quad
\end{array}$

$\begin{array}{l}
\hspace{6mm} \displaystyle \Rightarrow \ddot{x} + 2\gamma \dot{x} + \omega_0^2 x =0 \quad
\hspace{6mm} \displaystyle \omega_0^2 = k/m \quad
\hspace{6mm} \displaystyle \gamma = b/2m \quad
\end{array}$

$\begin{array}{l}
\displaystyle \mbox{Underkritisk demping}
\hspace{6mm} \displaystyle (\gamma &lt; \omega_0) \quad
\hspace{6mm} \displaystyle x(t) = A e^{-\gamma t}\sin(\omega t + \phi) \quad
\hspace{6mm} \displaystyle \omega = \sqrt{\omega_0^2 - \gamma^2} \quad
\end{array}$

$\begin{array}{l}
\displaystyle \mbox{Overkritisk demping}
\hspace{6mm} \displaystyle (\gamma &gt; \omega_0) \quad
\hspace{6mm} \displaystyle x(t) = A e^{-\alpha_1 t} + B e^{-\alpha_2 t} \quad
\hspace{6mm} \displaystyle \alpha_{1,2} = \gamma \pm \sqrt{\gamma^2 - \omega_0^2} \quad
\end{array}$

$\begin{array}{l}
\displaystyle \mbox{Kritisk demping}
\hspace{6mm} \displaystyle (\gamma = \omega_0) \quad
\hspace{6mm} \displaystyle x(t) = A e^{-\gamma t} + Bt e^{-\gamma t} \quad
\end{array}$

\item
$\begin{array}{l}
\displaystyle \mbox{Tvungen svingning, harmonisk ytre kraft: }
\hspace{6mm} \displaystyle m \ddot{x} + b \dot{x} + k x = F_0 \cos\omega t \quad
\end{array}$

$\begin{array}{l}
\hspace{10mm} \displaystyle \mbox{(partikul{\ae}r-)l{\o}sning: }
\hspace{6mm} \displaystyle x(t) = A(\omega) \sin(\omega t + \phi(\omega)) \quad
\end{array}$

$\begin{array}{l}
\hspace{10mm} \displaystyle \mbox{amplitude: }
\hspace{6mm} \displaystyle A(\omega) = \frac{F_0/m}{\sqrt{(\omega^2-\omega_0^2)^2+
             (2\gamma\omega)^2}} \quad
\end{array}$

$\begin{array}{l}
\hspace{10mm} \displaystyle \mbox{halvverdibredde: }
\hspace{6mm} \displaystyle \Delta \omega \simeq 2\gamma
\hspace{20mm} \displaystyle \mbox{Q-faktor: }
\hspace{6mm} \displaystyle Q = \omega_0/\Delta \omega
\end{array}$

\end{itemize}

\vspace{3mm}

{\bf ELEKTRISITET OG MAGNETISME}

\vspace{0.3cm}
\noindent
{\em Elektrostatikk}

\begin{itemize}
\item
Coulombs lov:
$$\bold{F}=\frac{qq_0}{4\pi\varepsilon_0 r^2}\hat{r}$$
\item
Elektrisk felt og potensial:
$$\bold{E}=\bold{F}/q_0 \;\;\;\;\;\;\;\;
\bold{E}=-\nabla V \;\;\;\;\;\;\;\;
\Delta V = \Delta U/q_0 \;\;\;\;\;\;\;\;
\Delta V = V_B - V_A = - \int_A^B \bold{E}\cdot d\bold{l}$$
\item
Elektrisk potensial fra punktladning:
$$ V=\frac{q}{4\pi\varepsilon_0 r} $$
\item
Elektrisk dipolmoment; for punktladninger $\pm q$ i innbyrdes avstand $\bold{d}$: \hspace{5mm}
$\bold{p}=q\bold{d}$
\item
Elektrisk dipol i ytre elektrisk felt: \hspace{5mm}
$\bold{\tau} = \bold{p} \times \bold{E}_0  \;\;\;\;\; , \;\;\;\;\;
  U = -\bold{p} \cdot \bold{E}_0$
\item
Line{\ae}r respons:
$$\bold{E} = \bold{E}_0/\varepsilon_r \;\;\;\;\;\;
\varepsilon = \varepsilon_r \varepsilon_0$$
\item
Kapasitans:
$$C=q/V$$
Seriekobling, parallellkobling:
$$C = (\sum_j C_j^{-1})^{-1} \;\;\;\;\;\;\;\;\;\;\;\; C = \sum_j C_j$$
\item
Parallellplatekondensator (ideell; feltstyrke $\sigma/2\varepsilon$ fra ett
stort og jevnt ladet plan):
$$E = \sigma/\varepsilon \;\;\; , \;\;\; C = \varepsilon A/d$$
\item
Energitetthet i elektrisk felt:
$$u_E = \frac{1}{2}\varepsilon_0 E^2$$
\end{itemize}

\vspace{3mm}

\noindent
{\em Elektrisk str{\o}m}

\begin{itemize}
\item
Str{\o}mstyrke, str{\o}mtetthet:
$$I = dQ/dt \;\;\; , \;\;\; j = I/A$$
\item
Ohms lov:
$$\bold{j} = \sigma \bold{E} \;\;\;,\;\;\; V = RI$$
\item
Drudemodellen:
$$\sigma = \frac{n e^2 \tau}{m_e}$$
\item
Resistans $R$ og konduktans $G$:
$$R = G^{-1} = l/\sigma A = \rho l/A, \;\; \sigma = {\rm konduktivitet}, \;\;
\rho = {\rm resistivitet}$$
$$R(T) = R_0 \left( 1 + \alpha (T-T_0) \right)$$
Seriekobling, parallellkobling:
$$R = \sum_j R_j \;\;\;\; , \;\;\;\; R = (\sum_j R_j^{-1})^{-1}$$
\item
Elektrisk effekt:
$$P = VI$$
\item
Midlere effekt med vekselspenning:
$$P = V_{\rm rms} I_{\rm rms} = \frac{1}{2} V_0 I_0$$
\end{itemize}

\vspace{3mm}

\noindent
{\em Magnetostatikk}

\begin{itemize}
\item
Magnetisk fluks:
$$\phi = \int\bold{B}\cdot d\bold{A}$$
\item
Magnetfelt fra str{\o}mf{\o}rende leder (Biot--Savarts lov):
$$\bold{B}=\frac{\mu_0}{4\pi}I\int\frac{d\bold{s}\times\hat{r}}{r^2} $$
\item
Lang rett leder:
$$B(x) = \frac{\mu_0 I}{2\pi x}$$
\item
P{\aa} aksen til sirkul{\ae}r str{\o}msl{\o}yfe:
$$B(x) = \frac{\mu_0 I R^2}{2(x^2 + R^2)^{3/2}}$$
\item
Magnetisk dipolmoment; for plan str{\o}msl{\o}yfe:
\hspace{5mm}
$\bold{m} = I \bold{A} = I A \hat{n}$
\item
Magnetisk dipol i ytre magnetfelt:
\hspace{5mm}
$\bold{\tau} = \bold{m} \times \bold{B}_0  \;\;\;\;\; , \;\;\;\;\;
  U = -\bold{m} \cdot \bold{B}_0$
\item
Line{\ae}r respons:
$$\bold{B} = \mu_r \bold{B}_0 \;\;\;\;\;\;\;
\mu = \mu_r \mu_0$$
\item
Lorentzkraften:
$$\bold{F} = q \bold{E} + q \bold{v} \times \bold{B}$$
\item
Magnetisk kraft p{\aa} str{\o}mf{\o}rende leder; generelt:
$$\bold{F} = \int_L\;d\bold{F} = I \int_L \;d\bold{s}\times\bold{B}$$
\item
Magnetisk kraft p{\aa} rett str{\o}mf{\o}rende leder:
$$\bold{F}=I\bold{L}\times\bold{B}$$
\item
Energitetthet i magnetfelt:
$$u_B=\frac{1}{2\mu_0} B^2$$
\end{itemize}

\vspace{3mm}

\noindent
{\em Elektrodynamikk og elektromagnetisk induksjon}

\begin{itemize}
\item
Faradays induksjonslov:
$$\Delta V = -\frac{d\phi}{dt}$$
\item
Selvinduktans: 
$$L=\frac{\phi}{I}$$
\item
Gjensidig induktans: 
$$M=\frac{\phi_2}{I_1} = \frac{\phi_1}{I_2}$$
\item
Transformator:
$$V_2/V_1 = N_2/N_1$$
\item
Spole (ideell):
$$B = \mu (N/l) I \;\;\; , \;\;\; L = \mu N^2 A/l$$
\item
Energitetthet i elektromagnetisk felt:
$$u=\frac{1}{2} \varepsilon_0 E^2 + \frac{1}{2\mu_0} B^2$$
\end{itemize}

\vspace{3mm}

\noindent
{\em Kretser}

\begin{itemize}
\item
Spenning over motstand, kapasitans, induktans:
$$RI \;\;\;\;\;\;\; Q/C \;\;\;\;\;\;\; L\,dI/dt$$
\item
Tidskonstanter, $RC$-krets og $RL$-krets:
$$\tau = RC \;\;\;\;\;\;\; \tau = L/R$$
\end{itemize}
\begin{itemize}
\item
Opplading av kondensator i $RC$-krets:
$$Q(t) = Q_0 \left(1 - e^{-t/\tau}\right)$$
\item
Oppbygging av str{\o}m i $RL$-krets:
$$I(t) = I_0 \left(1 - e^{-t/\tau}\right)$$
\end{itemize}

\vspace{3mm}

{\bf KONSTANTER, OMREGNINGSFAKTORER OG DEKADISKE PREFIKSER} 

\begin{itemize}
\item
Fundamentale konstanter:
\begin{eqnarray*}
G &amp;=&amp; 6.67 \cdot 10^{-11} \; {\rm Nm}^2/{\rm kg}^2 \;\;\; (g = 9.81 \; {\rm m/s}^2) \\
m_e &amp;=&amp; 9.11 \cdot 10^{-31} \; {\rm kg} \\
m_p = m_n &amp;=&amp; 1.67 \cdot 10^{-27} \; {\rm kg} \\
u &amp;=&amp; 1.66 \cdot 10^{-27} \; {\rm kg} \\
e &amp;=&amp; 1.60 \cdot 10^{-19} \; {\rm C} \\
\varepsilon_0 &amp;=&amp; 8.85 \cdot 10^{-12} \; {\rm F/m} \\
\frac{1}{4\pi\varepsilon_0} &amp;=&amp; 9 \cdot 10^9 \; {\rm Nm}^2/{\rm C}^2 \\
\mu_0 &amp;=&amp; 4\pi \cdot 10^{-7} \; {\rm H/m} \\
k_B &amp;=&amp; 1.38 \cdot 10^{-23} \; {\rm J/K} \\
N_A &amp;=&amp; 6.02 \cdot 10^{23} \; {\rm mol}^{-1} \\
h &amp;=&amp; 6.63 \cdot 10^{-34} \; {\rm Js} \\
c &amp;=&amp; 3.00 \cdot 10^8 \; {\rm m/s} \\
\end{eqnarray*}

\item
Omregningsfaktorer:
\begin{eqnarray*}
1 \; {\rm eV} &amp;=&amp; 1.60 \cdot 10^{-19} \; {\rm J} \\
1 \; {\rm {\AA}} &amp;=&amp; 10^{-10} \; {\rm m} \\
\end{eqnarray*}

\item
Dekadiske prefikser:
p = piko = $10^{-12}$, n = nano = $10^{-9}$, $\mu$ = mikro = $10^{-6}$, m = milli = $10^{-3}$, 

c = centi = $10^{-2}$, k = kilo = $10^3$, M = mega = $10^6$, G = giga = $10^9$, T = tera = $10^{12}$

\item
Geometri: \\
Areal, sirkul{\ae}r skive: $\pi r^2$. Kuleflateareal: $4\pi r^2$. Kulevolum: $4\pi r^3/3$. \\

\end{itemize}

\vspace{3mm}

{\bf MATEMATIKK} 

\begin{itemize}
\item
Krumningsradius:
$$\rho = \frac{[1 + (dy/dx)^2]^{3/2}}{|d^2 y/dx^2|}$$
\item
$$\frac{d}{dx}\, e^{\alpha x} = \alpha e^{\alpha x}$$
\item
$$\int e^{\alpha x}\,dx = \frac{1}{\alpha} e^{\alpha x}$$
\item
$$\int \frac{dx}{x} = \ln x$$
\item
Gradient:
$$\nabla f(x) = \hat{x}\,df/dx \;\;\;\; , \;\;\;\;
  \nabla f(r) = \hat{r}\,df/dr$$
\end{itemize}

\vspace{3mm}

{\bf MIDDELVERDI OG FEIL I M{\AA}LINGER}

\begin{itemize}
%\item
%Gauss' feilforplantningslov:
%$\left(\Delta q \right)^2 = \sum_{i=1}^n \left( \frac{\partial q}{\partial a_i}\,
%\Delta a_i \right)^2$
%\item
%Hvis st{\o}rrelsen $q$ er et produkt av potenser av $a_i$ 
%($q = a_1^{N_1}\cdot a_2^{N_2} \cdot \dots$):
%$$\frac{\Delta q}{q} = \sqrt{\left(\frac{N_1 \Delta a_1}{a_1}\right)^2 + 
%                             \left(\frac{N_2 \Delta a_2}{a_2}\right)^2 + \dots}$$
\item
Middelverdi (gjennomsnittsverdi):
$\overline{x} = \frac{1}{N} \sum_{i=1}^N \, x_i$
\item
Standardavvik (feil i enkeltm{\aa}ling):
$\delta_x = \sqrt{\left(\frac{1}{N-1} \sum_{i=1}^N \, \left(x_i -
\overline{x} \right)^2 \right)}$
\item
Standardfeil (feil i middelverdi):
$\delta_{\overline{x}} = \delta_x/\sqrt{N}$
\end{itemize}

\end{document}
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