Meccanica:
Moti
Moto Rettilineo Uniforme
$$s = vt\\
\Delta v = 0\\
v = \frac{ s }{ t }\\
t = \frac{ s }{ v }$$
Moto Uniformemente Accelerato
$$s = s_0 + v_0t + \frac{1}{2}at^2\\
v = v_0 + at\\
\Delta a = 0\\
a = \frac{v^2 - v_0^2}{2\Delta s}$$
Moto Parabolico
$$\mathrm{s:}\begin{eqnarray}
\left\{
\begin{array}{l}
x = x_0 + v_{0x}t \\
y = y_0 (h) + v_{0y}t - \frac{1}{2}gt^2
\end{array}
\right.
\end{eqnarray}\\
v_x = v_{0x} \Rightarrow \Delta v_x = 0\\
v_y = v_{0y} - gt\\
v_{0y} \neq 0 \Rightarrow R = 2\frac{v_{0x}v_{0y}}{g}\\
v_{0y} = 0 \Rightarrow R = v_{0x}t_f\\
h = y_0 + \frac{v_0^2\sin(2\alpha)}{g}\\
\mathrm{Traiettoria:}
y = \frac{v_{0y}}{v_{0x}}x - \frac{g}{2v_{0x}^2}x^2$$
Moto Circolare Uniforme
$$s = \alpha r\\
\omega = \frac{\alpha}{\Delta t} = \frac{2\pi}{T} = 2\pi f\\
f = T^{-1}\\
v = \omega^2 r\\
a_c = \frac{v^2}{r} = \omega^2 r$$
Moto Armonico Semplice
$$s = A\cos (\omega t)\\
v = -A\omega \sin (\omega t)\\
a = -A\omega^2 \cos (\omega t)$$
Moto del pendolo
$$T = 2\pi\sqrt{\frac{l}{g}}$$
Moto elastico
$$T = 2\pi\sqrt{\frac{m}{k}}$$
Dinamica
Forze
$$F_{el} = kx\\
F_a = \mu P\\
P = mg$$
Leggi della dinamica
$$\mathrm{2°:} F = ma\\
\mathrm{3°:} F_{1,2} = -F_{2,1}$$
Relatività Galileiana
\begin{eqnarray}
\left\{
\begin{array}{l}
x' = x - vt \\
y' = y
\end{array}
\right.
\end{eqnarray}
Energia
Forme
$$K = \frac{1}{2} mv^2\\
U_g = mgh\\
L = F\cdot s\\
U_{el} = \frac{1}{2} kx^2$$
Teoremi
$$\mathrm{Forze vive}\\
\Delta K = \Delta L\\
\mathrm{Conservazione}\\
E = K + U \land \Delta E = 0\\
\Delta K = \Delta U$$
Urti Elastici
\begin{eqnarray}
\left\{
\begin{array}{l}
\frac{1}{2}m_i1v_i1^2 + \frac{1}{2}m_i2v_i2^2 = \frac{1}{2}m_f1v_f1^2 + \frac{1}{2}m_f2v_f2^2\\
m_i1v_i1 + m_i2v_i2 = m_f1v_f1 + m_f2v_f2\\
\end{array}
\right.
\end{eqnarray}
Dinamica rotazionale
$$\mathrm{Conversioni}\\
(n)giri/min=\frac{n}{60}giri/s=\frac{2n\pi}{60}rad/s\\
\mathrm{Grandezze}\\
\omega = \frac{2\pi}{T} = \frac{\Delta \theta}{\Delta t}\\
v_t = \omega r\\
\alpha = \frac{\Delta \omega}{\Delta t}\\
a_t = \alpha r\\
a_c = r\omega^2 = \frac{v^2}{r}\\
\mathrm{M. accellerato}\\
\omega = \omega_0 + \alpha t\\
\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2\\
\mathrm{Momenti}\\
I = \sum mr^2\\
L = I\omega = mv\land r\\
M = F\land b = I\alpha = \frac{\Delta L}{\Delta t}\\
\mathrm{Energia}\\
K_{rotolamento} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2
$$
Gravitazione
$$\mathrm{Grandezze}\\
F=G\frac{m_1 m_2}{r^2}\\
g=G\frac{M_p}{R_{p}^2}\\
v_{fuga}=\sqrt{\frac{2GM_p}{R_p}}\\
\mathrm{Energia}\\
U=-mgh=-G\frac{Mm}{r}\\
\mathrm{Keplero}\\
T=(\frac{2\pi}{\sqrt{GM_G}})r^{\frac{3}{2}}\\
\mathrm{Campo}\\
h=\frac{F}{m}=G\frac{M}{r^2}\\
$$
Idrodinamica
Legge di continuità
$$\rho_1A_1v_1=\rho_2A_2v_2\\
\Delta V = A_1v_1\Delta t\\
\Delta m = \rho_1A_1v_1\Delta t\\
\Delta m_1 = \Delta m_2\\
Q=\frac{\Delta V}{\Delta t}$$
Bernoulli
$$p_1+\frac{1}{2}\rho v_1^2 + \rho gy_1 = p_2+\frac{1}{2}\rho v_2^2 + \rho gy_2\\
1) p=\frac{F}{A} \Rightarrow \frac{F\Delta x}{A\Delta x} = \frac{W}{V}\\
2) K=\frac{1}{2}mv^2 \Rightarrow \frac{K}{V}=\frac{1}{2}\frac{m}{V}v^2=\frac{1}{2}\rho v^2\\
3) U=mgh \Rightarrow \frac{U}{V} = \frac{mgh}{V} = \rho gh$$
Onde sonore
$$v=\frac{\lambda}{T}=\lambda f=\sqrt{gh}=\sqrt{\frac{F}{\mu}}\\
\mu=\frac{m}{L}\\
y(x,t)=A cos(\frac{2\pi}{\lambda}x+-\frac{2\pi}{T}t)\\
I=\frac{E}{At}=\frac{P}{A}\\
I_p = \frac{P}{4\pi r^2}\\
\beta = (10dB)log(\frac{I}{I_0})\\
f'=\frac{v'}{\lambda}=(1+-\frac{u}{v})f\\
f'=\frac{v}{\lambda'}=\frac{f}{(1-+\frac{u}{v})}\\
f'=\frac{(1+-\frac{u}{v})}{(1-+\frac{u}{v})}f\\
\lambda_fondamentale = 2L\\
f_fondamentale = \frac{v}{2L}\\
f_n=nf\\
\lambda_n =\frac{\lambda}{n}=\frac{2L}{n}\\
f_bat = |f_1-f_2|$$
